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30,674 | networkx.convert_matrix | from_pandas_edgelist | Returns a graph from Pandas DataFrame containing an edge list.
The Pandas DataFrame should contain at least two columns of node names and
zero or more columns of edge attributes. Each row will be processed as one
edge instance.
Note: This function iterates over DataFrame.values, which is not
guaranteed to retain the data type across columns in the row. This is only
a problem if your row is entirely numeric and a mix of ints and floats. In
that case, all values will be returned as floats. See the
DataFrame.iterrows documentation for an example.
Parameters
----------
df : Pandas DataFrame
An edge list representation of a graph
source : str or int
A valid column name (string or integer) for the source nodes (for the
directed case).
target : str or int
A valid column name (string or integer) for the target nodes (for the
directed case).
edge_attr : str or int, iterable, True, or None
A valid column name (str or int) or iterable of column names that are
used to retrieve items and add them to the graph as edge attributes.
If `True`, all of the remaining columns will be added.
If `None`, no edge attributes are added to the graph.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
edge_key : str or None, optional (default=None)
A valid column name for the edge keys (for a MultiGraph). The values in
this column are used for the edge keys when adding edges if create_using
is a multigraph.
If you have node attributes stored in a separate dataframe `df_nodes`,
you can load those attributes to the graph `G` using the following code:
```
df_nodes = pd.DataFrame({"node_id": [1, 2, 3], "attribute1": ["A", "B", "C"]})
G.add_nodes_from((n, dict(d)) for n, d in df_nodes.iterrows())
```
See Also
--------
to_pandas_edgelist
Examples
--------
Simple integer weights on edges:
>>> import pandas as pd
>>> pd.options.display.max_columns = 20
>>> import numpy as np
>>> rng = np.random.RandomState(seed=5)
>>> ints = rng.randint(1, 11, size=(3, 2))
>>> a = ["A", "B", "C"]
>>> b = ["D", "A", "E"]
>>> df = pd.DataFrame(ints, columns=["weight", "cost"])
>>> df[0] = a
>>> df["b"] = b
>>> df[["weight", "cost", 0, "b"]]
weight cost 0 b
0 4 7 A D
1 7 1 B A
2 10 9 C E
>>> G = nx.from_pandas_edgelist(df, 0, "b", ["weight", "cost"])
>>> G["E"]["C"]["weight"]
10
>>> G["E"]["C"]["cost"]
9
>>> edges = pd.DataFrame(
... {
... "source": [0, 1, 2],
... "target": [2, 2, 3],
... "weight": [3, 4, 5],
... "color": ["red", "blue", "blue"],
... }
... )
>>> G = nx.from_pandas_edgelist(edges, edge_attr=True)
>>> G[0][2]["color"]
'red'
Build multigraph with custom keys:
>>> edges = pd.DataFrame(
... {
... "source": [0, 1, 2, 0],
... "target": [2, 2, 3, 2],
... "my_edge_key": ["A", "B", "C", "D"],
... "weight": [3, 4, 5, 6],
... "color": ["red", "blue", "blue", "blue"],
... }
... )
>>> G = nx.from_pandas_edgelist(
... edges,
... edge_key="my_edge_key",
... edge_attr=["weight", "color"],
... create_using=nx.MultiGraph(),
... )
>>> G[0][2]
AtlasView({'A': {'weight': 3, 'color': 'red'}, 'D': {'weight': 6, 'color': 'blue'}})
| def to_numpy_array(
G,
nodelist=None,
dtype=None,
order=None,
multigraph_weight=sum,
weight="weight",
nonedge=0.0,
):
"""Returns the graph adjacency matrix as a NumPy array.
Parameters
----------
G : graph
The NetworkX graph used to construct the NumPy array.
nodelist : list, optional
The rows and columns are ordered according to the nodes in `nodelist`.
If `nodelist` is ``None``, then the ordering is produced by ``G.nodes()``.
dtype : NumPy data type, optional
A NumPy data type used to initialize the array. If None, then the NumPy
default is used. The dtype can be structured if `weight=None`, in which
case the dtype field names are used to look up edge attributes. The
result is a structured array where each named field in the dtype
corresponds to the adjacency for that edge attribute. See examples for
details.
order : {'C', 'F'}, optional
Whether to store multidimensional data in C- or Fortran-contiguous
(row- or column-wise) order in memory. If None, then the NumPy default
is used.
multigraph_weight : callable, optional
An function that determines how weights in multigraphs are handled.
The function should accept a sequence of weights and return a single
value. The default is to sum the weights of the multiple edges.
weight : string or None optional (default = 'weight')
The edge attribute that holds the numerical value used for
the edge weight. If an edge does not have that attribute, then the
value 1 is used instead. `weight` must be ``None`` if a structured
dtype is used.
nonedge : array_like (default = 0.0)
The value used to represent non-edges in the adjacency matrix.
The array values corresponding to nonedges are typically set to zero.
However, this could be undesirable if there are array values
corresponding to actual edges that also have the value zero. If so,
one might prefer nonedges to have some other value, such as ``nan``.
Returns
-------
A : NumPy ndarray
Graph adjacency matrix
Raises
------
NetworkXError
If `dtype` is a structured dtype and `G` is a multigraph
ValueError
If `dtype` is a structured dtype and `weight` is not `None`
See Also
--------
from_numpy_array
Notes
-----
For directed graphs, entry ``i, j`` corresponds to an edge from ``i`` to ``j``.
Entries in the adjacency matrix are given by the `weight` edge attribute.
When an edge does not have a weight attribute, the value of the entry is
set to the number 1. For multiple (parallel) edges, the values of the
entries are determined by the `multigraph_weight` parameter. The default is
to sum the weight attributes for each of the parallel edges.
When `nodelist` does not contain every node in `G`, the adjacency matrix is
built from the subgraph of `G` that is induced by the nodes in `nodelist`.
The convention used for self-loop edges in graphs is to assign the
diagonal array entry value to the weight attribute of the edge
(or the number 1 if the edge has no weight attribute). If the
alternate convention of doubling the edge weight is desired the
resulting NumPy array can be modified as follows:
>>> import numpy as np
>>> G = nx.Graph([(1, 1)])
>>> A = nx.to_numpy_array(G)
>>> A
array([[1.]])
>>> A[np.diag_indices_from(A)] *= 2
>>> A
array([[2.]])
Examples
--------
>>> G = nx.MultiDiGraph()
>>> G.add_edge(0, 1, weight=2)
0
>>> G.add_edge(1, 0)
0
>>> G.add_edge(2, 2, weight=3)
0
>>> G.add_edge(2, 2)
1
>>> nx.to_numpy_array(G, nodelist=[0, 1, 2])
array([[0., 2., 0.],
[1., 0., 0.],
[0., 0., 4.]])
When `nodelist` argument is used, nodes of `G` which do not appear in the `nodelist`
and their edges are not included in the adjacency matrix. Here is an example:
>>> G = nx.Graph()
>>> G.add_edge(3, 1)
>>> G.add_edge(2, 0)
>>> G.add_edge(2, 1)
>>> G.add_edge(3, 0)
>>> nx.to_numpy_array(G, nodelist=[1, 2, 3])
array([[0., 1., 1.],
[1., 0., 0.],
[1., 0., 0.]])
This function can also be used to create adjacency matrices for multiple
edge attributes with structured dtypes:
>>> G = nx.Graph()
>>> G.add_edge(0, 1, weight=10)
>>> G.add_edge(1, 2, cost=5)
>>> G.add_edge(2, 3, weight=3, cost=-4.0)
>>> dtype = np.dtype([("weight", int), ("cost", float)])
>>> A = nx.to_numpy_array(G, dtype=dtype, weight=None)
>>> A["weight"]
array([[ 0, 10, 0, 0],
[10, 0, 1, 0],
[ 0, 1, 0, 3],
[ 0, 0, 3, 0]])
>>> A["cost"]
array([[ 0., 1., 0., 0.],
[ 1., 0., 5., 0.],
[ 0., 5., 0., -4.],
[ 0., 0., -4., 0.]])
As stated above, the argument "nonedge" is useful especially when there are
actually edges with weight 0 in the graph. Setting a nonedge value different than 0,
makes it much clearer to differentiate such 0-weighted edges and actual nonedge values.
>>> G = nx.Graph()
>>> G.add_edge(3, 1, weight=2)
>>> G.add_edge(2, 0, weight=0)
>>> G.add_edge(2, 1, weight=0)
>>> G.add_edge(3, 0, weight=1)
>>> nx.to_numpy_array(G, nonedge=-1.0)
array([[-1., 2., -1., 1.],
[ 2., -1., 0., -1.],
[-1., 0., -1., 0.],
[ 1., -1., 0., -1.]])
"""
import numpy as np
if nodelist is None:
nodelist = list(G)
nlen = len(nodelist)
# Input validation
nodeset = set(nodelist)
if nodeset - set(G):
raise nx.NetworkXError(f"Nodes {nodeset - set(G)} in nodelist is not in G")
if len(nodeset) < nlen:
raise nx.NetworkXError("nodelist contains duplicates.")
A = np.full((nlen, nlen), fill_value=nonedge, dtype=dtype, order=order)
# Corner cases: empty nodelist or graph without any edges
if nlen == 0 or G.number_of_edges() == 0:
return A
# If dtype is structured and weight is None, use dtype field names as
# edge attributes
edge_attrs = None # Only single edge attribute by default
if A.dtype.names:
if weight is None:
edge_attrs = dtype.names
else:
raise ValueError(
"Specifying `weight` not supported for structured dtypes\n."
"To create adjacency matrices from structured dtypes, use `weight=None`."
)
# Map nodes to row/col in matrix
idx = dict(zip(nodelist, range(nlen)))
if len(nodelist) < len(G):
G = G.subgraph(nodelist).copy()
# Collect all edge weights and reduce with `multigraph_weights`
if G.is_multigraph():
if edge_attrs:
raise nx.NetworkXError(
"Structured arrays are not supported for MultiGraphs"
)
d = defaultdict(list)
for u, v, wt in G.edges(data=weight, default=1.0):
d[(idx[u], idx[v])].append(wt)
i, j = np.array(list(d.keys())).T # indices
wts = [multigraph_weight(ws) for ws in d.values()] # reduced weights
else:
i, j, wts = [], [], []
# Special branch: multi-attr adjacency from structured dtypes
if edge_attrs:
# Extract edges with all data
for u, v, data in G.edges(data=True):
i.append(idx[u])
j.append(idx[v])
wts.append(data)
# Map each attribute to the appropriate named field in the
# structured dtype
for attr in edge_attrs:
attr_data = [wt.get(attr, 1.0) for wt in wts]
A[attr][i, j] = attr_data
if not G.is_directed():
A[attr][j, i] = attr_data
return A
for u, v, wt in G.edges(data=weight, default=1.0):
i.append(idx[u])
j.append(idx[v])
wts.append(wt)
# Set array values with advanced indexing
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|
30,676 | networkx.convert_matrix | from_scipy_sparse_array | Creates a new graph from an adjacency matrix given as a SciPy sparse
array.
Parameters
----------
A: scipy.sparse array
An adjacency matrix representation of a graph
parallel_edges : Boolean
If this is True, `create_using` is a multigraph, and `A` is an
integer matrix, then entry *(i, j)* in the matrix is interpreted as the
number of parallel edges joining vertices *i* and *j* in the graph.
If it is False, then the entries in the matrix are interpreted as
the weight of a single edge joining the vertices.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
edge_attribute: string
Name of edge attribute to store matrix numeric value. The data will
have the same type as the matrix entry (int, float, (real,imag)).
Notes
-----
For directed graphs, explicitly mention create_using=nx.DiGraph,
and entry i,j of A corresponds to an edge from i to j.
If `create_using` is :class:`networkx.MultiGraph` or
:class:`networkx.MultiDiGraph`, `parallel_edges` is True, and the
entries of `A` are of type :class:`int`, then this function returns a
multigraph (constructed from `create_using`) with parallel edges.
In this case, `edge_attribute` will be ignored.
If `create_using` indicates an undirected multigraph, then only the edges
indicated by the upper triangle of the matrix `A` will be added to the
graph.
Examples
--------
>>> import scipy as sp
>>> A = sp.sparse.eye(2, 2, 1)
>>> G = nx.from_scipy_sparse_array(A)
If `create_using` indicates a multigraph and the matrix has only integer
entries and `parallel_edges` is False, then the entries will be treated
as weights for edges joining the nodes (without creating parallel edges):
>>> A = sp.sparse.csr_array([[1, 1], [1, 2]])
>>> G = nx.from_scipy_sparse_array(A, create_using=nx.MultiGraph)
>>> G[1][1]
AtlasView({0: {'weight': 2}})
If `create_using` indicates a multigraph and the matrix has only integer
entries and `parallel_edges` is True, then the entries will be treated
as the number of parallel edges joining those two vertices:
>>> A = sp.sparse.csr_array([[1, 1], [1, 2]])
>>> G = nx.from_scipy_sparse_array(A, parallel_edges=True, create_using=nx.MultiGraph)
>>> G[1][1]
AtlasView({0: {'weight': 1}, 1: {'weight': 1}})
| def to_numpy_array(
G,
nodelist=None,
dtype=None,
order=None,
multigraph_weight=sum,
weight="weight",
nonedge=0.0,
):
"""Returns the graph adjacency matrix as a NumPy array.
Parameters
----------
G : graph
The NetworkX graph used to construct the NumPy array.
nodelist : list, optional
The rows and columns are ordered according to the nodes in `nodelist`.
If `nodelist` is ``None``, then the ordering is produced by ``G.nodes()``.
dtype : NumPy data type, optional
A NumPy data type used to initialize the array. If None, then the NumPy
default is used. The dtype can be structured if `weight=None`, in which
case the dtype field names are used to look up edge attributes. The
result is a structured array where each named field in the dtype
corresponds to the adjacency for that edge attribute. See examples for
details.
order : {'C', 'F'}, optional
Whether to store multidimensional data in C- or Fortran-contiguous
(row- or column-wise) order in memory. If None, then the NumPy default
is used.
multigraph_weight : callable, optional
An function that determines how weights in multigraphs are handled.
The function should accept a sequence of weights and return a single
value. The default is to sum the weights of the multiple edges.
weight : string or None optional (default = 'weight')
The edge attribute that holds the numerical value used for
the edge weight. If an edge does not have that attribute, then the
value 1 is used instead. `weight` must be ``None`` if a structured
dtype is used.
nonedge : array_like (default = 0.0)
The value used to represent non-edges in the adjacency matrix.
The array values corresponding to nonedges are typically set to zero.
However, this could be undesirable if there are array values
corresponding to actual edges that also have the value zero. If so,
one might prefer nonedges to have some other value, such as ``nan``.
Returns
-------
A : NumPy ndarray
Graph adjacency matrix
Raises
------
NetworkXError
If `dtype` is a structured dtype and `G` is a multigraph
ValueError
If `dtype` is a structured dtype and `weight` is not `None`
See Also
--------
from_numpy_array
Notes
-----
For directed graphs, entry ``i, j`` corresponds to an edge from ``i`` to ``j``.
Entries in the adjacency matrix are given by the `weight` edge attribute.
When an edge does not have a weight attribute, the value of the entry is
set to the number 1. For multiple (parallel) edges, the values of the
entries are determined by the `multigraph_weight` parameter. The default is
to sum the weight attributes for each of the parallel edges.
When `nodelist` does not contain every node in `G`, the adjacency matrix is
built from the subgraph of `G` that is induced by the nodes in `nodelist`.
The convention used for self-loop edges in graphs is to assign the
diagonal array entry value to the weight attribute of the edge
(or the number 1 if the edge has no weight attribute). If the
alternate convention of doubling the edge weight is desired the
resulting NumPy array can be modified as follows:
>>> import numpy as np
>>> G = nx.Graph([(1, 1)])
>>> A = nx.to_numpy_array(G)
>>> A
array([[1.]])
>>> A[np.diag_indices_from(A)] *= 2
>>> A
array([[2.]])
Examples
--------
>>> G = nx.MultiDiGraph()
>>> G.add_edge(0, 1, weight=2)
0
>>> G.add_edge(1, 0)
0
>>> G.add_edge(2, 2, weight=3)
0
>>> G.add_edge(2, 2)
1
>>> nx.to_numpy_array(G, nodelist=[0, 1, 2])
array([[0., 2., 0.],
[1., 0., 0.],
[0., 0., 4.]])
When `nodelist` argument is used, nodes of `G` which do not appear in the `nodelist`
and their edges are not included in the adjacency matrix. Here is an example:
>>> G = nx.Graph()
>>> G.add_edge(3, 1)
>>> G.add_edge(2, 0)
>>> G.add_edge(2, 1)
>>> G.add_edge(3, 0)
>>> nx.to_numpy_array(G, nodelist=[1, 2, 3])
array([[0., 1., 1.],
[1., 0., 0.],
[1., 0., 0.]])
This function can also be used to create adjacency matrices for multiple
edge attributes with structured dtypes:
>>> G = nx.Graph()
>>> G.add_edge(0, 1, weight=10)
>>> G.add_edge(1, 2, cost=5)
>>> G.add_edge(2, 3, weight=3, cost=-4.0)
>>> dtype = np.dtype([("weight", int), ("cost", float)])
>>> A = nx.to_numpy_array(G, dtype=dtype, weight=None)
>>> A["weight"]
array([[ 0, 10, 0, 0],
[10, 0, 1, 0],
[ 0, 1, 0, 3],
[ 0, 0, 3, 0]])
>>> A["cost"]
array([[ 0., 1., 0., 0.],
[ 1., 0., 5., 0.],
[ 0., 5., 0., -4.],
[ 0., 0., -4., 0.]])
As stated above, the argument "nonedge" is useful especially when there are
actually edges with weight 0 in the graph. Setting a nonedge value different than 0,
makes it much clearer to differentiate such 0-weighted edges and actual nonedge values.
>>> G = nx.Graph()
>>> G.add_edge(3, 1, weight=2)
>>> G.add_edge(2, 0, weight=0)
>>> G.add_edge(2, 1, weight=0)
>>> G.add_edge(3, 0, weight=1)
>>> nx.to_numpy_array(G, nonedge=-1.0)
array([[-1., 2., -1., 1.],
[ 2., -1., 0., -1.],
[-1., 0., -1., 0.],
[ 1., -1., 0., -1.]])
"""
import numpy as np
if nodelist is None:
nodelist = list(G)
nlen = len(nodelist)
# Input validation
nodeset = set(nodelist)
if nodeset - set(G):
raise nx.NetworkXError(f"Nodes {nodeset - set(G)} in nodelist is not in G")
if len(nodeset) < nlen:
raise nx.NetworkXError("nodelist contains duplicates.")
A = np.full((nlen, nlen), fill_value=nonedge, dtype=dtype, order=order)
# Corner cases: empty nodelist or graph without any edges
if nlen == 0 or G.number_of_edges() == 0:
return A
# If dtype is structured and weight is None, use dtype field names as
# edge attributes
edge_attrs = None # Only single edge attribute by default
if A.dtype.names:
if weight is None:
edge_attrs = dtype.names
else:
raise ValueError(
"Specifying `weight` not supported for structured dtypes\n."
"To create adjacency matrices from structured dtypes, use `weight=None`."
)
# Map nodes to row/col in matrix
idx = dict(zip(nodelist, range(nlen)))
if len(nodelist) < len(G):
G = G.subgraph(nodelist).copy()
# Collect all edge weights and reduce with `multigraph_weights`
if G.is_multigraph():
if edge_attrs:
raise nx.NetworkXError(
"Structured arrays are not supported for MultiGraphs"
)
d = defaultdict(list)
for u, v, wt in G.edges(data=weight, default=1.0):
d[(idx[u], idx[v])].append(wt)
i, j = np.array(list(d.keys())).T # indices
wts = [multigraph_weight(ws) for ws in d.values()] # reduced weights
else:
i, j, wts = [], [], []
# Special branch: multi-attr adjacency from structured dtypes
if edge_attrs:
# Extract edges with all data
for u, v, data in G.edges(data=True):
i.append(idx[u])
j.append(idx[v])
wts.append(data)
# Map each attribute to the appropriate named field in the
# structured dtype
for attr in edge_attrs:
attr_data = [wt.get(attr, 1.0) for wt in wts]
A[attr][i, j] = attr_data
if not G.is_directed():
A[attr][j, i] = attr_data
return A
for u, v, wt in G.edges(data=weight, default=1.0):
i.append(idx[u])
j.append(idx[v])
wts.append(wt)
# Set array values with advanced indexing
A[i, j] | (A, parallel_edges=False, create_using=None, edge_attribute='weight', *, backend=None, **backend_kwargs) | [
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|
30,678 | networkx.generators.small | frucht_graph |
Returns the Frucht Graph.
The Frucht Graph is the smallest cubical graph whose
automorphism group consists only of the identity element [1]_.
It has 12 nodes and 18 edges and no nontrivial symmetries.
It is planar and Hamiltonian [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Frucht Graph with 12 nodes and 18 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Frucht_graph
.. [2] https://mathworld.wolfram.com/FruchtGraph.html
| def sedgewick_maze_graph(create_using=None):
"""
Return a small maze with a cycle.
This is the maze used in Sedgewick, 3rd Edition, Part 5, Graph
Algorithms, Chapter 18, e.g. Figure 18.2 and following [1]_.
Nodes are numbered 0,..,7
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Small maze with a cycle
References
----------
.. [1] Figure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick
"""
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
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|
30,679 | networkx.drawing.layout | spring_layout | Position nodes using Fruchterman-Reingold force-directed algorithm.
The algorithm simulates a force-directed representation of the network
treating edges as springs holding nodes close, while treating nodes
as repelling objects, sometimes called an anti-gravity force.
Simulation continues until the positions are close to an equilibrium.
There are some hard-coded values: minimal distance between
nodes (0.01) and "temperature" of 0.1 to ensure nodes don't fly away.
During the simulation, `k` helps determine the distance between nodes,
though `scale` and `center` determine the size and place after
rescaling occurs at the end of the simulation.
Fixing some nodes doesn't allow them to move in the simulation.
It also turns off the rescaling feature at the simulation's end.
In addition, setting `scale` to `None` turns off rescaling.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
k : float (default=None)
Optimal distance between nodes. If None the distance is set to
1/sqrt(n) where n is the number of nodes. Increase this value
to move nodes farther apart.
pos : dict or None optional (default=None)
Initial positions for nodes as a dictionary with node as keys
and values as a coordinate list or tuple. If None, then use
random initial positions.
fixed : list or None optional (default=None)
Nodes to keep fixed at initial position.
Nodes not in ``G.nodes`` are ignored.
ValueError raised if `fixed` specified and `pos` not.
iterations : int optional (default=50)
Maximum number of iterations taken
threshold: float optional (default = 1e-4)
Threshold for relative error in node position changes.
The iteration stops if the error is below this threshold.
weight : string or None optional (default='weight')
The edge attribute that holds the numerical value used for
the edge weight. Larger means a stronger attractive force.
If None, then all edge weights are 1.
scale : number or None (default: 1)
Scale factor for positions. Not used unless `fixed is None`.
If scale is None, no rescaling is performed.
center : array-like or None
Coordinate pair around which to center the layout.
Not used unless `fixed is None`.
dim : int
Dimension of layout.
seed : int, RandomState instance or None optional (default=None)
Set the random state for deterministic node layouts.
If int, `seed` is the seed used by the random number generator,
if numpy.random.RandomState instance, `seed` is the random
number generator,
if None, the random number generator is the RandomState instance used
by numpy.random.
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.spring_layout(G)
# The same using longer but equivalent function name
>>> pos = nx.fruchterman_reingold_layout(G)
| def spectral_layout(G, weight="weight", scale=1, center=None, dim=2):
"""Position nodes using the eigenvectors of the graph Laplacian.
Using the unnormalized Laplacian, the layout shows possible clusters of
nodes which are an approximation of the ratio cut. If dim is the number of
dimensions then the positions are the entries of the dim eigenvectors
corresponding to the ascending eigenvalues starting from the second one.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
weight : string or None optional (default='weight')
The edge attribute that holds the numerical value used for
the edge weight. If None, then all edge weights are 1.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.spectral_layout(G)
Notes
-----
Directed graphs will be considered as undirected graphs when
positioning the nodes.
For larger graphs (>500 nodes) this will use the SciPy sparse
eigenvalue solver (ARPACK).
"""
# handle some special cases that break the eigensolvers
import numpy as np
G, center = _process_params(G, center, dim)
if len(G) <= 2:
if len(G) == 0:
pos = np.array([])
elif len(G) == 1:
pos = np.array([center])
else:
pos = np.array([np.zeros(dim), np.array(center) * 2.0])
return dict(zip(G, pos))
try:
# Sparse matrix
if len(G) < 500: # dense solver is faster for small graphs
raise ValueError
A = nx.to_scipy_sparse_array(G, weight=weight, dtype="d")
# Symmetrize directed graphs
if G.is_directed():
A = A + np.transpose(A)
pos = _sparse_spectral(A, dim)
except (ImportError, ValueError):
# Dense matrix
A = nx.to_numpy_array(G, weight=weight)
# Symmetrize directed graphs
if G.is_directed():
A += A.T
pos = _spectral(A, dim)
pos = rescale_layout(pos, scale=scale) + center
pos = dict(zip(G, pos))
return pos
| (G, k=None, pos=None, fixed=None, iterations=50, threshold=0.0001, weight='weight', scale=1, center=None, dim=2, seed=None) | [
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|
30,681 | networkx.generators.classic | full_rary_tree | Creates a full r-ary tree of `n` nodes.
Sometimes called a k-ary, n-ary, or m-ary tree.
"... all non-leaf nodes have exactly r children and all levels
are full except for some rightmost position of the bottom level
(if a leaf at the bottom level is missing, then so are all of the
leaves to its right." [1]_
.. plot::
>>> nx.draw(nx.full_rary_tree(2, 10))
Parameters
----------
r : int
branching factor of the tree
n : int
Number of nodes in the tree
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
An r-ary tree with n nodes
References
----------
.. [1] An introduction to data structures and algorithms,
James Andrew Storer, Birkhauser Boston 2001, (page 225).
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (r, n, create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,683 | networkx.generators.community | gaussian_random_partition_graph | Generate a Gaussian random partition graph.
A Gaussian random partition graph is created by creating k partitions
each with a size drawn from a normal distribution with mean s and variance
s/v. Nodes are connected within clusters with probability p_in and
between clusters with probability p_out[1]
Parameters
----------
n : int
Number of nodes in the graph
s : float
Mean cluster size
v : float
Shape parameter. The variance of cluster size distribution is s/v.
p_in : float
Probability of intra cluster connection.
p_out : float
Probability of inter cluster connection.
directed : boolean, optional default=False
Whether to create a directed graph or not
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
G : NetworkX Graph or DiGraph
gaussian random partition graph
Raises
------
NetworkXError
If s is > n
If p_in or p_out is not in [0,1]
Notes
-----
Note the number of partitions is dependent on s,v and n, and that the
last partition may be considerably smaller, as it is sized to simply
fill out the nodes [1]
See Also
--------
random_partition_graph
Examples
--------
>>> G = nx.gaussian_random_partition_graph(100, 10, 10, 0.25, 0.1)
>>> len(G)
100
References
----------
.. [1] Ulrik Brandes, Marco Gaertler, Dorothea Wagner,
Experiments on Graph Clustering Algorithms,
In the proceedings of the 11th Europ. Symp. Algorithms, 2003.
| def _generate_communities(degree_seq, community_sizes, mu, max_iters, seed):
"""Returns a list of sets, each of which represents a community.
``degree_seq`` is the degree sequence that must be met by the
graph.
``community_sizes`` is the community size distribution that must be
met by the generated list of sets.
``mu`` is a float in the interval [0, 1] indicating the fraction of
intra-community edges incident to each node.
``max_iters`` is the number of times to try to add a node to a
community. This must be greater than the length of
``degree_seq``, otherwise this function will always fail. If
the number of iterations exceeds this value,
:exc:`~networkx.exception.ExceededMaxIterations` is raised.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
The communities returned by this are sets of integers in the set {0,
..., *n* - 1}, where *n* is the length of ``degree_seq``.
"""
# This assumes the nodes in the graph will be natural numbers.
result = [set() for _ in community_sizes]
n = len(degree_seq)
free = list(range(n))
for i in range(max_iters):
v = free.pop()
c = seed.choice(range(len(community_sizes)))
# s = int(degree_seq[v] * (1 - mu) + 0.5)
s = round(degree_seq[v] * (1 - mu))
# If the community is large enough, add the node to the chosen
# community. Otherwise, return it to the list of unaffiliated
# nodes.
if s < community_sizes[c]:
result[c].add(v)
else:
free.append(v)
# If the community is too big, remove a node from it.
if len(result[c]) > community_sizes[c]:
free.append(result[c].pop())
if not free:
return result
msg = "Could not assign communities; try increasing min_community"
raise nx.ExceededMaxIterations(msg)
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|
30,686 | networkx.readwrite.adjlist | generate_adjlist | Generate a single line of the graph G in adjacency list format.
Parameters
----------
G : NetworkX graph
delimiter : string, optional
Separator for node labels
Returns
-------
lines : string
Lines of data in adjlist format.
Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> for line in nx.generate_adjlist(G):
... print(line)
0 1 2 3
1 2 3
2 3
3 4
4 5
5 6
6
See Also
--------
write_adjlist, read_adjlist
Notes
-----
The default `delimiter=" "` will result in unexpected results if node names contain
whitespace characters. To avoid this problem, specify an alternate delimiter when spaces are
valid in node names.
NB: This option is not available for data that isn't user-generated.
| def generate_adjlist(G, delimiter=" "):
"""Generate a single line of the graph G in adjacency list format.
Parameters
----------
G : NetworkX graph
delimiter : string, optional
Separator for node labels
Returns
-------
lines : string
Lines of data in adjlist format.
Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> for line in nx.generate_adjlist(G):
... print(line)
0 1 2 3
1 2 3
2 3
3 4
4 5
5 6
6
See Also
--------
write_adjlist, read_adjlist
Notes
-----
The default `delimiter=" "` will result in unexpected results if node names contain
whitespace characters. To avoid this problem, specify an alternate delimiter when spaces are
valid in node names.
NB: This option is not available for data that isn't user-generated.
"""
directed = G.is_directed()
seen = set()
for s, nbrs in G.adjacency():
line = str(s) + delimiter
for t, data in nbrs.items():
if not directed and t in seen:
continue
if G.is_multigraph():
for d in data.values():
line += str(t) + delimiter
else:
line += str(t) + delimiter
if not directed:
seen.add(s)
yield line[: -len(delimiter)]
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|
30,687 | networkx.readwrite.edgelist | generate_edgelist | Generate a single line of the graph G in edge list format.
Parameters
----------
G : NetworkX graph
delimiter : string, optional
Separator for node labels
data : bool or list of keys
If False generate no edge data. If True use a dictionary
representation of edge data. If a list of keys use a list of data
values corresponding to the keys.
Returns
-------
lines : string
Lines of data in adjlist format.
Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> G[1][2]["weight"] = 3
>>> G[3][4]["capacity"] = 12
>>> for line in nx.generate_edgelist(G, data=False):
... print(line)
0 1
0 2
0 3
1 2
1 3
2 3
3 4
4 5
5 6
>>> for line in nx.generate_edgelist(G):
... print(line)
0 1 {}
0 2 {}
0 3 {}
1 2 {'weight': 3}
1 3 {}
2 3 {}
3 4 {'capacity': 12}
4 5 {}
5 6 {}
>>> for line in nx.generate_edgelist(G, data=["weight"]):
... print(line)
0 1
0 2
0 3
1 2 3
1 3
2 3
3 4
4 5
5 6
See Also
--------
write_adjlist, read_adjlist
| def generate_edgelist(G, delimiter=" ", data=True):
"""Generate a single line of the graph G in edge list format.
Parameters
----------
G : NetworkX graph
delimiter : string, optional
Separator for node labels
data : bool or list of keys
If False generate no edge data. If True use a dictionary
representation of edge data. If a list of keys use a list of data
values corresponding to the keys.
Returns
-------
lines : string
Lines of data in adjlist format.
Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> G[1][2]["weight"] = 3
>>> G[3][4]["capacity"] = 12
>>> for line in nx.generate_edgelist(G, data=False):
... print(line)
0 1
0 2
0 3
1 2
1 3
2 3
3 4
4 5
5 6
>>> for line in nx.generate_edgelist(G):
... print(line)
0 1 {}
0 2 {}
0 3 {}
1 2 {'weight': 3}
1 3 {}
2 3 {}
3 4 {'capacity': 12}
4 5 {}
5 6 {}
>>> for line in nx.generate_edgelist(G, data=["weight"]):
... print(line)
0 1
0 2
0 3
1 2 3
1 3
2 3
3 4
4 5
5 6
See Also
--------
write_adjlist, read_adjlist
"""
if data is True:
for u, v, d in G.edges(data=True):
e = u, v, dict(d)
yield delimiter.join(map(str, e))
elif data is False:
for u, v in G.edges(data=False):
e = u, v
yield delimiter.join(map(str, e))
else:
for u, v, d in G.edges(data=True):
e = [u, v]
try:
e.extend(d[k] for k in data)
except KeyError:
pass # missing data for this edge, should warn?
yield delimiter.join(map(str, e))
| (G, delimiter=' ', data=True) | [
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|
30,688 | networkx.readwrite.gexf | generate_gexf | Generate lines of GEXF format representation of G.
"GEXF (Graph Exchange XML Format) is a language for describing
complex networks structures, their associated data and dynamics" [1]_.
Parameters
----------
G : graph
A NetworkX graph
encoding : string (optional, default: 'utf-8')
Encoding for text data.
prettyprint : bool (optional, default: True)
If True use line breaks and indenting in output XML.
version : string (default: 1.2draft)
Version of GEFX File Format (see http://gexf.net/schema.html)
Supported values: "1.1draft", "1.2draft"
Examples
--------
>>> G = nx.path_graph(4)
>>> linefeed = chr(10) # linefeed=
>>> s = linefeed.join(nx.generate_gexf(G))
>>> for line in nx.generate_gexf(G): # doctest: +SKIP
... print(line)
Notes
-----
This implementation does not support mixed graphs (directed and undirected
edges together).
The node id attribute is set to be the string of the node label.
If you want to specify an id use set it as node data, e.g.
node['a']['id']=1 to set the id of node 'a' to 1.
References
----------
.. [1] GEXF File Format, https://gephi.org/gexf/format/
| def generate_gexf(G, encoding="utf-8", prettyprint=True, version="1.2draft"):
"""Generate lines of GEXF format representation of G.
"GEXF (Graph Exchange XML Format) is a language for describing
complex networks structures, their associated data and dynamics" [1]_.
Parameters
----------
G : graph
A NetworkX graph
encoding : string (optional, default: 'utf-8')
Encoding for text data.
prettyprint : bool (optional, default: True)
If True use line breaks and indenting in output XML.
version : string (default: 1.2draft)
Version of GEFX File Format (see http://gexf.net/schema.html)
Supported values: "1.1draft", "1.2draft"
Examples
--------
>>> G = nx.path_graph(4)
>>> linefeed = chr(10) # linefeed=\n
>>> s = linefeed.join(nx.generate_gexf(G))
>>> for line in nx.generate_gexf(G): # doctest: +SKIP
... print(line)
Notes
-----
This implementation does not support mixed graphs (directed and undirected
edges together).
The node id attribute is set to be the string of the node label.
If you want to specify an id use set it as node data, e.g.
node['a']['id']=1 to set the id of node 'a' to 1.
References
----------
.. [1] GEXF File Format, https://gephi.org/gexf/format/
"""
writer = GEXFWriter(encoding=encoding, prettyprint=prettyprint, version=version)
writer.add_graph(G)
yield from str(writer).splitlines()
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|
30,689 | networkx.readwrite.gml | generate_gml | Generate a single entry of the graph `G` in GML format.
Parameters
----------
G : NetworkX graph
The graph to be converted to GML.
stringizer : callable, optional
A `stringizer` which converts non-int/non-float/non-dict values into
strings. If it cannot convert a value into a string, it should raise a
`ValueError` to indicate that. Default value: None.
Returns
-------
lines: generator of strings
Lines of GML data. Newlines are not appended.
Raises
------
NetworkXError
If `stringizer` cannot convert a value into a string, or the value to
convert is not a string while `stringizer` is None.
See Also
--------
literal_stringizer
Notes
-----
Graph attributes named 'directed', 'multigraph', 'node' or
'edge', node attributes named 'id' or 'label', edge attributes
named 'source' or 'target' (or 'key' if `G` is a multigraph)
are ignored because these attribute names are used to encode the graph
structure.
GML files are stored using a 7-bit ASCII encoding with any extended
ASCII characters (iso8859-1) appearing as HTML character entities.
Without specifying a `stringizer`/`destringizer`, the code is capable of
writing `int`/`float`/`str`/`dict`/`list` data as required by the GML
specification. For writing other data types, and for reading data other
than `str` you need to explicitly supply a `stringizer`/`destringizer`.
For additional documentation on the GML file format, please see the
`GML url <https://web.archive.org/web/20190207140002/http://www.fim.uni-passau.de/index.php?id=17297&L=1>`_.
See the module docstring :mod:`networkx.readwrite.gml` for more details.
Examples
--------
>>> G = nx.Graph()
>>> G.add_node("1")
>>> print("\n".join(nx.generate_gml(G)))
graph [
node [
id 0
label "1"
]
]
>>> G = nx.MultiGraph([("a", "b"), ("a", "b")])
>>> print("\n".join(nx.generate_gml(G)))
graph [
multigraph 1
node [
id 0
label "a"
]
node [
id 1
label "b"
]
edge [
source 0
target 1
key 0
]
edge [
source 0
target 1
key 1
]
]
| def generate_gml(G, stringizer=None):
r"""Generate a single entry of the graph `G` in GML format.
Parameters
----------
G : NetworkX graph
The graph to be converted to GML.
stringizer : callable, optional
A `stringizer` which converts non-int/non-float/non-dict values into
strings. If it cannot convert a value into a string, it should raise a
`ValueError` to indicate that. Default value: None.
Returns
-------
lines: generator of strings
Lines of GML data. Newlines are not appended.
Raises
------
NetworkXError
If `stringizer` cannot convert a value into a string, or the value to
convert is not a string while `stringizer` is None.
See Also
--------
literal_stringizer
Notes
-----
Graph attributes named 'directed', 'multigraph', 'node' or
'edge', node attributes named 'id' or 'label', edge attributes
named 'source' or 'target' (or 'key' if `G` is a multigraph)
are ignored because these attribute names are used to encode the graph
structure.
GML files are stored using a 7-bit ASCII encoding with any extended
ASCII characters (iso8859-1) appearing as HTML character entities.
Without specifying a `stringizer`/`destringizer`, the code is capable of
writing `int`/`float`/`str`/`dict`/`list` data as required by the GML
specification. For writing other data types, and for reading data other
than `str` you need to explicitly supply a `stringizer`/`destringizer`.
For additional documentation on the GML file format, please see the
`GML url <https://web.archive.org/web/20190207140002/http://www.fim.uni-passau.de/index.php?id=17297&L=1>`_.
See the module docstring :mod:`networkx.readwrite.gml` for more details.
Examples
--------
>>> G = nx.Graph()
>>> G.add_node("1")
>>> print("\n".join(nx.generate_gml(G)))
graph [
node [
id 0
label "1"
]
]
>>> G = nx.MultiGraph([("a", "b"), ("a", "b")])
>>> print("\n".join(nx.generate_gml(G)))
graph [
multigraph 1
node [
id 0
label "a"
]
node [
id 1
label "b"
]
edge [
source 0
target 1
key 0
]
edge [
source 0
target 1
key 1
]
]
"""
valid_keys = re.compile("^[A-Za-z][0-9A-Za-z_]*$")
def stringize(key, value, ignored_keys, indent, in_list=False):
if not isinstance(key, str):
raise NetworkXError(f"{key!r} is not a string")
if not valid_keys.match(key):
raise NetworkXError(f"{key!r} is not a valid key")
if not isinstance(key, str):
key = str(key)
if key not in ignored_keys:
if isinstance(value, int | bool):
if key == "label":
yield indent + key + ' "' + str(value) + '"'
elif value is True:
# python bool is an instance of int
yield indent + key + " 1"
elif value is False:
yield indent + key + " 0"
# GML only supports signed 32-bit integers
elif value < -(2**31) or value >= 2**31:
yield indent + key + ' "' + str(value) + '"'
else:
yield indent + key + " " + str(value)
elif isinstance(value, float):
text = repr(value).upper()
# GML matches INF to keys, so prepend + to INF. Use repr(float(*))
# instead of string literal to future proof against changes to repr.
if text == repr(float("inf")).upper():
text = "+" + text
else:
# GML requires that a real literal contain a decimal point, but
# repr may not output a decimal point when the mantissa is
# integral and hence needs fixing.
epos = text.rfind("E")
if epos != -1 and text.find(".", 0, epos) == -1:
text = text[:epos] + "." + text[epos:]
if key == "label":
yield indent + key + ' "' + text + '"'
else:
yield indent + key + " " + text
elif isinstance(value, dict):
yield indent + key + " ["
next_indent = indent + " "
for key, value in value.items():
yield from stringize(key, value, (), next_indent)
yield indent + "]"
elif isinstance(value, tuple) and key == "label":
yield indent + key + f" \"({','.join(repr(v) for v in value)})\""
elif isinstance(value, list | tuple) and key != "label" and not in_list:
if len(value) == 0:
yield indent + key + " " + f'"{value!r}"'
if len(value) == 1:
yield indent + key + " " + f'"{LIST_START_VALUE}"'
for val in value:
yield from stringize(key, val, (), indent, True)
else:
if stringizer:
try:
value = stringizer(value)
except ValueError as err:
raise NetworkXError(
f"{value!r} cannot be converted into a string"
) from err
if not isinstance(value, str):
raise NetworkXError(f"{value!r} is not a string")
yield indent + key + ' "' + escape(value) + '"'
multigraph = G.is_multigraph()
yield "graph ["
# Output graph attributes
if G.is_directed():
yield " directed 1"
if multigraph:
yield " multigraph 1"
ignored_keys = {"directed", "multigraph", "node", "edge"}
for attr, value in G.graph.items():
yield from stringize(attr, value, ignored_keys, " ")
# Output node data
node_id = dict(zip(G, range(len(G))))
ignored_keys = {"id", "label"}
for node, attrs in G.nodes.items():
yield " node ["
yield " id " + str(node_id[node])
yield from stringize("label", node, (), " ")
for attr, value in attrs.items():
yield from stringize(attr, value, ignored_keys, " ")
yield " ]"
# Output edge data
ignored_keys = {"source", "target"}
kwargs = {"data": True}
if multigraph:
ignored_keys.add("key")
kwargs["keys"] = True
for e in G.edges(**kwargs):
yield " edge ["
yield " source " + str(node_id[e[0]])
yield " target " + str(node_id[e[1]])
if multigraph:
yield from stringize("key", e[2], (), " ")
for attr, value in e[-1].items():
yield from stringize(attr, value, ignored_keys, " ")
yield " ]"
yield "]"
| (G, stringizer=None) | [
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0.11503682285547256,
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0.053678158670663834,
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0.08806972950696945,
0.00901925377547741,
0.02244412712752819,
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|
30,690 | networkx.readwrite.graphml | generate_graphml | Generate GraphML lines for G
Parameters
----------
G : graph
A networkx graph
encoding : string (optional)
Encoding for text data.
prettyprint : bool (optional)
If True use line breaks and indenting in output XML.
named_key_ids : bool (optional)
If True use attr.name as value for key elements' id attribute.
edge_id_from_attribute : dict key (optional)
If provided, the graphml edge id is set by looking up the corresponding
edge data attribute keyed by this parameter. If `None` or the key does not exist in edge data,
the edge id is set by the edge key if `G` is a MultiGraph, else the edge id is left unset.
Examples
--------
>>> G = nx.path_graph(4)
>>> linefeed = chr(10) # linefeed =
>>> s = linefeed.join(nx.generate_graphml(G))
>>> for line in nx.generate_graphml(G): # doctest: +SKIP
... print(line)
Notes
-----
This implementation does not support mixed graphs (directed and unidirected
edges together) hyperedges, nested graphs, or ports.
| def generate_graphml(
G,
encoding="utf-8",
prettyprint=True,
named_key_ids=False,
edge_id_from_attribute=None,
):
"""Generate GraphML lines for G
Parameters
----------
G : graph
A networkx graph
encoding : string (optional)
Encoding for text data.
prettyprint : bool (optional)
If True use line breaks and indenting in output XML.
named_key_ids : bool (optional)
If True use attr.name as value for key elements' id attribute.
edge_id_from_attribute : dict key (optional)
If provided, the graphml edge id is set by looking up the corresponding
edge data attribute keyed by this parameter. If `None` or the key does not exist in edge data,
the edge id is set by the edge key if `G` is a MultiGraph, else the edge id is left unset.
Examples
--------
>>> G = nx.path_graph(4)
>>> linefeed = chr(10) # linefeed = \n
>>> s = linefeed.join(nx.generate_graphml(G))
>>> for line in nx.generate_graphml(G): # doctest: +SKIP
... print(line)
Notes
-----
This implementation does not support mixed graphs (directed and unidirected
edges together) hyperedges, nested graphs, or ports.
"""
writer = GraphMLWriter(
encoding=encoding,
prettyprint=prettyprint,
named_key_ids=named_key_ids,
edge_id_from_attribute=edge_id_from_attribute,
)
writer.add_graph_element(G)
yield from str(writer).splitlines()
| (G, encoding='utf-8', prettyprint=True, named_key_ids=False, edge_id_from_attribute=None) | [
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|
30,691 | networkx.readwrite.multiline_adjlist | generate_multiline_adjlist | Generate a single line of the graph G in multiline adjacency list format.
Parameters
----------
G : NetworkX graph
delimiter : string, optional
Separator for node labels
Returns
-------
lines : string
Lines of data in multiline adjlist format.
Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> for line in nx.generate_multiline_adjlist(G):
... print(line)
0 3
1 {}
2 {}
3 {}
1 2
2 {}
3 {}
2 1
3 {}
3 1
4 {}
4 1
5 {}
5 1
6 {}
6 0
See Also
--------
write_multiline_adjlist, read_multiline_adjlist
| def generate_multiline_adjlist(G, delimiter=" "):
"""Generate a single line of the graph G in multiline adjacency list format.
Parameters
----------
G : NetworkX graph
delimiter : string, optional
Separator for node labels
Returns
-------
lines : string
Lines of data in multiline adjlist format.
Examples
--------
>>> G = nx.lollipop_graph(4, 3)
>>> for line in nx.generate_multiline_adjlist(G):
... print(line)
0 3
1 {}
2 {}
3 {}
1 2
2 {}
3 {}
2 1
3 {}
3 1
4 {}
4 1
5 {}
5 1
6 {}
6 0
See Also
--------
write_multiline_adjlist, read_multiline_adjlist
"""
if G.is_directed():
if G.is_multigraph():
for s, nbrs in G.adjacency():
nbr_edges = [
(u, data)
for u, datadict in nbrs.items()
for key, data in datadict.items()
]
deg = len(nbr_edges)
yield str(s) + delimiter + str(deg)
for u, d in nbr_edges:
if d is None:
yield str(u)
else:
yield str(u) + delimiter + str(d)
else: # directed single edges
for s, nbrs in G.adjacency():
deg = len(nbrs)
yield str(s) + delimiter + str(deg)
for u, d in nbrs.items():
if d is None:
yield str(u)
else:
yield str(u) + delimiter + str(d)
else: # undirected
if G.is_multigraph():
seen = set() # helper dict used to avoid duplicate edges
for s, nbrs in G.adjacency():
nbr_edges = [
(u, data)
for u, datadict in nbrs.items()
if u not in seen
for key, data in datadict.items()
]
deg = len(nbr_edges)
yield str(s) + delimiter + str(deg)
for u, d in nbr_edges:
if d is None:
yield str(u)
else:
yield str(u) + delimiter + str(d)
seen.add(s)
else: # undirected single edges
seen = set() # helper dict used to avoid duplicate edges
for s, nbrs in G.adjacency():
nbr_edges = [(u, d) for u, d in nbrs.items() if u not in seen]
deg = len(nbr_edges)
yield str(s) + delimiter + str(deg)
for u, d in nbr_edges:
if d is None:
yield str(u)
else:
yield str(u) + delimiter + str(d)
seen.add(s)
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|
30,692 | networkx.readwrite.text | generate_network_text | Generate lines in the "network text" format
This works via a depth-first traversal of the graph and writing a line for
each unique node encountered. Non-tree edges are written to the right of
each node, and connection to a non-tree edge is indicated with an ellipsis.
This representation works best when the input graph is a forest, but any
graph can be represented.
This notation is original to networkx, although it is simple enough that it
may be known in existing literature. See #5602 for details. The procedure
is summarized as follows:
1. Given a set of source nodes (which can be specified, or automatically
discovered via finding the (strongly) connected components and choosing one
node with minimum degree from each), we traverse the graph in depth first
order.
2. Each reachable node will be printed exactly once on it's own line.
3. Edges are indicated in one of four ways:
a. a parent "L-style" connection on the upper left. This corresponds to
a traversal in the directed DFS tree.
b. a backref "<-style" connection shown directly on the right. For
directed graphs, these are drawn for any incoming edges to a node that
is not a parent edge. For undirected graphs, these are drawn for only
the non-parent edges that have already been represented (The edges that
have not been represented will be handled in the recursive case).
c. a child "L-style" connection on the lower right. Drawing of the
children are handled recursively.
d. if ``vertical_chains`` is true, and a parent node only has one child
a "vertical-style" edge is drawn between them.
4. The children of each node (wrt the directed DFS tree) are drawn
underneath and to the right of it. In the case that a child node has already
been drawn the connection is replaced with an ellipsis ("...") to indicate
that there is one or more connections represented elsewhere.
5. If a maximum depth is specified, an edge to nodes past this maximum
depth will be represented by an ellipsis.
6. If a a node has a truthy "collapse" value, then we do not traverse past
that node.
Parameters
----------
graph : nx.DiGraph | nx.Graph
Graph to represent
with_labels : bool | str
If True will use the "label" attribute of a node to display if it
exists otherwise it will use the node value itself. If given as a
string, then that attribute name will be used instead of "label".
Defaults to True.
sources : List
Specifies which nodes to start traversal from. Note: nodes that are not
reachable from one of these sources may not be shown. If unspecified,
the minimal set of nodes needed to reach all others will be used.
max_depth : int | None
The maximum depth to traverse before stopping. Defaults to None.
ascii_only : Boolean
If True only ASCII characters are used to construct the visualization
vertical_chains : Boolean
If True, chains of nodes will be drawn vertically when possible.
Yields
------
str : a line of generated text
Examples
--------
>>> graph = nx.path_graph(10)
>>> graph.add_node("A")
>>> graph.add_node("B")
>>> graph.add_node("C")
>>> graph.add_node("D")
>>> graph.add_edge(9, "A")
>>> graph.add_edge(9, "B")
>>> graph.add_edge(9, "C")
>>> graph.add_edge("C", "D")
>>> graph.add_edge("C", "E")
>>> graph.add_edge("C", "F")
>>> nx.write_network_text(graph)
╙── 0
└── 1
└── 2
└── 3
└── 4
└── 5
└── 6
└── 7
└── 8
└── 9
├── A
├── B
└── C
├── D
├── E
└── F
>>> nx.write_network_text(graph, vertical_chains=True)
╙── 0
│
1
│
2
│
3
│
4
│
5
│
6
│
7
│
8
│
9
├── A
├── B
└── C
├── D
├── E
└── F
| def generate_network_text(
graph,
with_labels=True,
sources=None,
max_depth=None,
ascii_only=False,
vertical_chains=False,
):
"""Generate lines in the "network text" format
This works via a depth-first traversal of the graph and writing a line for
each unique node encountered. Non-tree edges are written to the right of
each node, and connection to a non-tree edge is indicated with an ellipsis.
This representation works best when the input graph is a forest, but any
graph can be represented.
This notation is original to networkx, although it is simple enough that it
may be known in existing literature. See #5602 for details. The procedure
is summarized as follows:
1. Given a set of source nodes (which can be specified, or automatically
discovered via finding the (strongly) connected components and choosing one
node with minimum degree from each), we traverse the graph in depth first
order.
2. Each reachable node will be printed exactly once on it's own line.
3. Edges are indicated in one of four ways:
a. a parent "L-style" connection on the upper left. This corresponds to
a traversal in the directed DFS tree.
b. a backref "<-style" connection shown directly on the right. For
directed graphs, these are drawn for any incoming edges to a node that
is not a parent edge. For undirected graphs, these are drawn for only
the non-parent edges that have already been represented (The edges that
have not been represented will be handled in the recursive case).
c. a child "L-style" connection on the lower right. Drawing of the
children are handled recursively.
d. if ``vertical_chains`` is true, and a parent node only has one child
a "vertical-style" edge is drawn between them.
4. The children of each node (wrt the directed DFS tree) are drawn
underneath and to the right of it. In the case that a child node has already
been drawn the connection is replaced with an ellipsis ("...") to indicate
that there is one or more connections represented elsewhere.
5. If a maximum depth is specified, an edge to nodes past this maximum
depth will be represented by an ellipsis.
6. If a a node has a truthy "collapse" value, then we do not traverse past
that node.
Parameters
----------
graph : nx.DiGraph | nx.Graph
Graph to represent
with_labels : bool | str
If True will use the "label" attribute of a node to display if it
exists otherwise it will use the node value itself. If given as a
string, then that attribute name will be used instead of "label".
Defaults to True.
sources : List
Specifies which nodes to start traversal from. Note: nodes that are not
reachable from one of these sources may not be shown. If unspecified,
the minimal set of nodes needed to reach all others will be used.
max_depth : int | None
The maximum depth to traverse before stopping. Defaults to None.
ascii_only : Boolean
If True only ASCII characters are used to construct the visualization
vertical_chains : Boolean
If True, chains of nodes will be drawn vertically when possible.
Yields
------
str : a line of generated text
Examples
--------
>>> graph = nx.path_graph(10)
>>> graph.add_node("A")
>>> graph.add_node("B")
>>> graph.add_node("C")
>>> graph.add_node("D")
>>> graph.add_edge(9, "A")
>>> graph.add_edge(9, "B")
>>> graph.add_edge(9, "C")
>>> graph.add_edge("C", "D")
>>> graph.add_edge("C", "E")
>>> graph.add_edge("C", "F")
>>> nx.write_network_text(graph)
╙── 0
└── 1
└── 2
└── 3
└── 4
└── 5
└── 6
└── 7
└── 8
└── 9
├── A
├── B
└── C
├── D
├── E
└── F
>>> nx.write_network_text(graph, vertical_chains=True)
╙── 0
│
1
│
2
│
3
│
4
│
5
│
6
│
7
│
8
│
9
├── A
├── B
└── C
├── D
├── E
└── F
"""
from typing import Any, NamedTuple
class StackFrame(NamedTuple):
parent: Any
node: Any
indents: list
this_islast: bool
this_vertical: bool
collapse_attr = "collapse"
is_directed = graph.is_directed()
if is_directed:
glyphs = AsciiDirectedGlyphs if ascii_only else UtfDirectedGlyphs
succ = graph.succ
pred = graph.pred
else:
glyphs = AsciiUndirectedGlyphs if ascii_only else UtfUndirectedGlyphs
succ = graph.adj
pred = graph.adj
if isinstance(with_labels, str):
label_attr = with_labels
elif with_labels:
label_attr = "label"
else:
label_attr = None
if max_depth == 0:
yield glyphs.empty + " ..."
elif len(graph.nodes) == 0:
yield glyphs.empty
else:
# If the nodes to traverse are unspecified, find the minimal set of
# nodes that will reach the entire graph
if sources is None:
sources = _find_sources(graph)
# Populate the stack with each:
# 1. parent node in the DFS tree (or None for root nodes),
# 2. the current node in the DFS tree
# 2. a list of indentations indicating depth
# 3. a flag indicating if the node is the final one to be written.
# Reverse the stack so sources are popped in the correct order.
last_idx = len(sources) - 1
stack = [
StackFrame(None, node, [], (idx == last_idx), False)
for idx, node in enumerate(sources)
][::-1]
num_skipped_children = defaultdict(lambda: 0)
seen_nodes = set()
while stack:
parent, node, indents, this_islast, this_vertical = stack.pop()
if node is not Ellipsis:
skip = node in seen_nodes
if skip:
# Mark that we skipped a parent's child
num_skipped_children[parent] += 1
if this_islast:
# If we reached the last child of a parent, and we skipped
# any of that parents children, then we should emit an
# ellipsis at the end after this.
if num_skipped_children[parent] and parent is not None:
# Append the ellipsis to be emitted last
next_islast = True
try_frame = StackFrame(
node, Ellipsis, indents, next_islast, False
)
stack.append(try_frame)
# Redo this frame, but not as a last object
next_islast = False
try_frame = StackFrame(
parent, node, indents, next_islast, this_vertical
)
stack.append(try_frame)
continue
if skip:
continue
seen_nodes.add(node)
if not indents:
# Top level items (i.e. trees in the forest) get different
# glyphs to indicate they are not actually connected
if this_islast:
this_vertical = False
this_prefix = indents + [glyphs.newtree_last]
next_prefix = indents + [glyphs.endof_forest]
else:
this_prefix = indents + [glyp | (graph, with_labels=True, sources=None, max_depth=None, ascii_only=False, vertical_chains=False) | [
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|
30,693 | networkx.readwrite.pajek | generate_pajek | Generate lines in Pajek graph format.
Parameters
----------
G : graph
A Networkx graph
References
----------
See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm
for format information.
| def generate_pajek(G):
"""Generate lines in Pajek graph format.
Parameters
----------
G : graph
A Networkx graph
References
----------
See http://vlado.fmf.uni-lj.si/pub/networks/pajek/doc/draweps.htm
for format information.
"""
if G.name == "":
name = "NetworkX"
else:
name = G.name
# Apparently many Pajek format readers can't process this line
# So we'll leave it out for now.
# yield '*network %s'%name
# write nodes with attributes
yield f"*vertices {G.order()}"
nodes = list(G)
# make dictionary mapping nodes to integers
nodenumber = dict(zip(nodes, range(1, len(nodes) + 1)))
for n in nodes:
# copy node attributes and pop mandatory attributes
# to avoid duplication.
na = G.nodes.get(n, {}).copy()
x = na.pop("x", 0.0)
y = na.pop("y", 0.0)
try:
id = int(na.pop("id", nodenumber[n]))
except ValueError as err:
err.args += (
(
"Pajek format requires 'id' to be an int()."
" Refer to the 'Relabeling nodes' section."
),
)
raise
nodenumber[n] = id
shape = na.pop("shape", "ellipse")
s = " ".join(map(make_qstr, (id, n, x, y, shape)))
# only optional attributes are left in na.
for k, v in na.items():
if isinstance(v, str) and v.strip() != "":
s += f" {make_qstr(k)} {make_qstr(v)}"
else:
warnings.warn(
f"Node attribute {k} is not processed. {('Empty attribute' if isinstance(v, str) else 'Non-string attribute')}."
)
yield s
# write edges with attributes
if G.is_directed():
yield "*arcs"
else:
yield "*edges"
for u, v, edgedata in G.edges(data=True):
d = edgedata.copy()
value = d.pop("weight", 1.0) # use 1 as default edge value
s = " ".join(map(make_qstr, (nodenumber[u], nodenumber[v], value)))
for k, v in d.items():
if isinstance(v, str) and v.strip() != "":
s += f" {make_qstr(k)} {make_qstr(v)}"
else:
warnings.warn(
f"Edge attribute {k} is not processed. {('Empty attribute' if isinstance(v, str) else 'Non-string attribute')}."
)
yield s
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|
30,694 | networkx.algorithms.similarity | generate_random_paths | Randomly generate `sample_size` paths of length `path_length`.
Parameters
----------
G : NetworkX graph
A NetworkX graph
sample_size : integer
The number of paths to generate. This is ``R`` in [1]_.
path_length : integer (default = 5)
The maximum size of the path to randomly generate.
This is ``T`` in [1]_. According to the paper, ``T >= 5`` is
recommended.
index_map : dictionary, optional
If provided, this will be populated with the inverted
index of nodes mapped to the set of generated random path
indices within ``paths``.
weight : string or None, optional (default="weight")
The name of an edge attribute that holds the numerical value
used as a weight. If None then each edge has weight 1.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
paths : generator of lists
Generator of `sample_size` paths each with length `path_length`.
Examples
--------
Note that the return value is the list of paths:
>>> G = nx.star_graph(3)
>>> random_path = nx.generate_random_paths(G, 2)
By passing a dictionary into `index_map`, it will build an
inverted index mapping of nodes to the paths in which that node is present:
>>> G = nx.star_graph(3)
>>> index_map = {}
>>> random_path = nx.generate_random_paths(G, 3, index_map=index_map)
>>> paths_containing_node_0 = [
... random_path[path_idx] for path_idx in index_map.get(0, [])
... ]
References
----------
.. [1] Zhang, J., Tang, J., Ma, C., Tong, H., Jing, Y., & Li, J.
Panther: Fast top-k similarity search on large networks.
In Proceedings of the ACM SIGKDD International Conference
on Knowledge Discovery and Data Mining (Vol. 2015-August, pp. 1445–1454).
Association for Computing Machinery. https://doi.org/10.1145/2783258.2783267.
| def optimize_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
strictly_decreasing=True,
roots=None,
timeout=None,
):
"""GED (graph edit distance) calculation: advanced interface.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Graph edit distance is defined as minimum cost of edit path.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost. Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
See Also
--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
"""
# TODO: support DiGraph
import numpy as np
import scipy as sp
@dataclass
class CostMatrix:
C: ...
lsa_row_ind: ...
lsa_col_ind: ...
ls: ...
def make_CostMatrix(C, m, n):
# assert(C.shape == (m + n, m + n))
lsa_row_ind, lsa_col_ind = sp.optimize.linear_sum_assignment(C)
# Fixup dummy assignments:
# each substitution i<->j should have dummy assignment m+j<->n+i
# NOTE: fast reduce of Cv relies on it
# assert len(lsa_row_ind) == len(lsa_col_ind)
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
subst_ind = [k for k, i, j in indexes if i < m and j < n]
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
dummy_ind = [k for k, i, j in indexes if i >= m and j >= n]
# assert len(subst_ind) == len(dummy_ind)
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n
return CostMatrix(
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
)
def extract_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k in i or k - m in j for k in range(m + n)]
col_ind = [k in j or k - n in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k not in i and k - m not in j for k in range(m + n)]
col_ind = [k not in j and k - n not in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_ind(ind, i):
# assert set(ind) == set(range(len(ind)))
rind = ind[[k not in i for k in ind]]
for k in set(i):
rind[rind >= k] -= 1
return rind
def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=None):
"""
Parameters:
u, v: matched vertices, u=None or v=None for
deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_uv: partial vertex edit path
list of tuples (u, v) of previously matched vertex
mappings u<->v, u=None or v=None for
deletion/insertion
Returns:
list of (i, j): indices of edge mappings g<->h
localCe: local CostMatrix of edge mappings
(basically submatrix of Ce at cross of rows i, cols j)
"""
M = len(pending_g)
N = len(pending_h)
# assert Ce.C.shape == (M + N, M + N)
# only attempt to match edges after one node match has been made
# this will stop self-edges on the first node being automatically deleted
# even when a substitution is the better option
if matched_uv is None or len(matched_uv) == 0:
g_ind = []
h_ind = []
else:
| (G, sample_size, path_length=5, index_map=None, weight='weight', seed=None, *, backend=None, **backend_kwargs) | [
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|
30,698 | networkx.generators.geometric | geographical_threshold_graph | Returns a geographical threshold graph.
The geographical threshold graph model places $n$ nodes uniformly at
random in a rectangular domain. Each node $u$ is assigned a weight
$w_u$. Two nodes $u$ and $v$ are joined by an edge if
.. math::
(w_u + w_v)p_{dist}(r) \ge \theta
where `r` is the distance between `u` and `v`, `p_dist` is any function of
`r`, and :math:`\theta` as the threshold parameter. `p_dist` is used to
give weight to the distance between nodes when deciding whether or not
they should be connected. The larger `p_dist` is, the more prone nodes
separated by `r` are to be connected, and vice versa.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
theta: float
Threshold value
dim : int, optional
Dimension of graph
pos : dict
Node positions as a dictionary of tuples keyed by node.
weight : dict
Node weights as a dictionary of numbers keyed by node.
metric : function
A metric on vectors of numbers (represented as lists or
tuples). This must be a function that accepts two lists (or
tuples) as input and yields a number as output. The function
must also satisfy the four requirements of a `metric`_.
Specifically, if $d$ is the function and $x$, $y$,
and $z$ are vectors in the graph, then $d$ must satisfy
1. $d(x, y) \ge 0$,
2. $d(x, y) = 0$ if and only if $x = y$,
3. $d(x, y) = d(y, x)$,
4. $d(x, z) \le d(x, y) + d(y, z)$.
If this argument is not specified, the Euclidean distance metric is
used.
.. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29
p_dist : function, optional
Any function used to give weight to the distance between nodes when
deciding whether or not they should be connected. `p_dist` was
originally conceived as a probability density function giving the
probability of connecting two nodes that are of metric distance `r`
apart. The implementation here allows for more arbitrary definitions
of `p_dist` that do not need to correspond to valid probability
density functions. The :mod:`scipy.stats` package has many
probability density functions implemented and tools for custom
probability density definitions, and passing the ``.pdf`` method of
scipy.stats distributions can be used here. If ``p_dist=None``
(the default), the exponential function :math:`r^{-2}` is used.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
pos_name : string, default="pos"
The name of the node attribute which represents the position
in 2D coordinates of the node in the returned graph.
weight_name : string, default="weight"
The name of the node attribute which represents the weight
of the node in the returned graph.
Returns
-------
Graph
A random geographic threshold graph, undirected and without
self-loops.
Each node has a node attribute ``pos`` that stores the
position of that node in Euclidean space as provided by the
``pos`` keyword argument or, if ``pos`` was not provided, as
generated by this function. Similarly, each node has a node
attribute ``weight`` that stores the weight of that node as
provided or as generated.
Examples
--------
Specify an alternate distance metric using the ``metric`` keyword
argument. For example, to use the `taxicab metric`_ instead of the
default `Euclidean metric`_::
>>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y))
>>> G = nx.geographical_threshold_graph(10, 0.1, metric=dist)
.. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry
.. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance
Notes
-----
If weights are not specified they are assigned to nodes by drawing randomly
from the exponential distribution with rate parameter $\lambda=1$.
To specify weights from a different distribution, use the `weight` keyword
argument::
>>> import random
>>> n = 20
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.geographical_threshold_graph(20, 50, weight=w)
If node positions are not specified they are randomly assigned from the
uniform distribution.
References
----------
.. [1] Masuda, N., Miwa, H., Konno, N.:
Geographical threshold graphs with small-world and scale-free
properties.
Physical Review E 71, 036108 (2005)
.. [2] Milan Bradonjić, Aric Hagberg and Allon G. Percus,
Giant component and connectivity in geographical threshold graphs,
in Algorithms and Models for the Web-Graph (WAW 2007),
Antony Bonato and Fan Chung (Eds), pp. 209--216, 2007
| def thresholded_random_geometric_graph(
n,
radius,
theta,
dim=2,
pos=None,
weight=None,
p=2,
seed=None,
*,
pos_name="pos",
weight_name="weight",
):
r"""Returns a thresholded random geometric graph in the unit cube.
The thresholded random geometric graph [1] model places `n` nodes
uniformly at random in the unit cube of dimensions `dim`. Each node
`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
joined by an edge if they are within the maximum connection distance,
`radius` computed by the `p`-Minkowski distance and the summation of
weights :math:`w_u` + :math:`w_v` is greater than or equal
to the threshold parameter `theta`.
Edges within `radius` of each other are determined using a KDTree when
SciPy is available. This reduces the time complexity from :math:`O(n^2)`
to :math:`O(n)`.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
radius: float
Distance threshold value
theta: float
Threshold value
dim : int, optional
Dimension of graph
pos : dict, optional
A dictionary keyed by node with node positions as values.
weight : dict, optional
Node weights as a dictionary of numbers keyed by node.
p : float, optional (default 2)
Which Minkowski distance metric to use. `p` has to meet the condition
``1 <= p <= infinity``.
If this argument is not specified, the :math:`L^2` metric
(the Euclidean distance metric), p = 2 is used.
This should not be confused with the `p` of an Erdős-Rényi random
graph, which represents probability.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
pos_name : string, default="pos"
The name of the node attribute which represents the position
in 2D coordinates of the node in the returned graph.
weight_name : string, default="weight"
The name of the node attribute which represents the weight
of the node in the returned graph.
Returns
-------
Graph
A thresholded random geographic graph, undirected and without
self-loops.
Each node has a node attribute ``'pos'`` that stores the
position of that node in Euclidean space as provided by the
``pos`` keyword argument or, if ``pos`` was not provided, as
generated by this function. Similarly, each node has a nodethre
attribute ``'weight'`` that stores the weight of that node as
provided or as generated.
Examples
--------
Default Graph:
G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
Custom Graph:
Create a thresholded random geometric graph on 50 uniformly distributed
nodes where nodes are joined by an edge if their sum weights drawn from
a exponential distribution with rate = 5 are >= theta = 0.1 and their
Euclidean distance is at most 0.2.
Notes
-----
This uses a *k*-d tree to build the graph.
The `pos` keyword argument can be used to specify node positions so you
can create an arbitrary distribution and domain for positions.
For example, to use a 2D Gaussian distribution of node positions with mean
(0, 0) and standard deviation 2
If weights are not specified they are assigned to nodes by drawing randomly
from the exponential distribution with rate parameter :math:`\lambda=1`.
To specify weights from a different distribution, use the `weight` keyword
argument::
::
>>> import random
>>> import math
>>> n = 50
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
References
----------
.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
"""
G = nx.empty_graph(n)
G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})"
# If no weights are provided, choose them from an exponential
# distribution.
if weight is None:
weight = {v: seed.expovariate(1) for v in G}
# If no positions are provided, choose uniformly random vectors in
# Euclidean space of the specified dimension.
if pos is None:
pos = {v: [seed.random() for i in range(dim)] for v in G}
# If no distance metric is provided, use Euclidean distance.
nx.set_node_attributes(G, weight, weight_name)
nx.set_node_attributes(G, pos, pos_name)
edges = (
(u, v)
for u, v in _geometric_edges(G, radius, p, pos_name)
if weight[u] + weight[v] >= theta
)
G.add_edges_from(edges)
return G
| (n, theta, dim=2, pos=None, weight=None, metric=None, p_dist=None, seed=None, *, pos_name='pos', weight_name='weight', backend=None, **backend_kwargs) | [
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|
30,700 | networkx.generators.geometric | geometric_edges | Returns edge list of node pairs within `radius` of each other.
Parameters
----------
G : networkx graph
The graph from which to generate the edge list. The nodes in `G` should
have an attribute ``pos`` corresponding to the node position, which is
used to compute the distance to other nodes.
radius : scalar
The distance threshold. Edges are included in the edge list if the
distance between the two nodes is less than `radius`.
pos_name : string, default="pos"
The name of the node attribute which represents the position of each
node in 2D coordinates. Every node in the Graph must have this attribute.
p : scalar, default=2
The `Minkowski distance metric
<https://en.wikipedia.org/wiki/Minkowski_distance>`_ used to compute
distances. The default value is 2, i.e. Euclidean distance.
Returns
-------
edges : list
List of edges whose distances are less than `radius`
Notes
-----
Radius uses Minkowski distance metric `p`.
If scipy is available, `scipy.spatial.cKDTree` is used to speed computation.
Examples
--------
Create a graph with nodes that have a "pos" attribute representing 2D
coordinates.
>>> G = nx.Graph()
>>> G.add_nodes_from(
... [
... (0, {"pos": (0, 0)}),
... (1, {"pos": (3, 0)}),
... (2, {"pos": (8, 0)}),
... ]
... )
>>> nx.geometric_edges(G, radius=1)
[]
>>> nx.geometric_edges(G, radius=4)
[(0, 1)]
>>> nx.geometric_edges(G, radius=6)
[(0, 1), (1, 2)]
>>> nx.geometric_edges(G, radius=9)
[(0, 1), (0, 2), (1, 2)]
| def thresholded_random_geometric_graph(
n,
radius,
theta,
dim=2,
pos=None,
weight=None,
p=2,
seed=None,
*,
pos_name="pos",
weight_name="weight",
):
r"""Returns a thresholded random geometric graph in the unit cube.
The thresholded random geometric graph [1] model places `n` nodes
uniformly at random in the unit cube of dimensions `dim`. Each node
`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
joined by an edge if they are within the maximum connection distance,
`radius` computed by the `p`-Minkowski distance and the summation of
weights :math:`w_u` + :math:`w_v` is greater than or equal
to the threshold parameter `theta`.
Edges within `radius` of each other are determined using a KDTree when
SciPy is available. This reduces the time complexity from :math:`O(n^2)`
to :math:`O(n)`.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
radius: float
Distance threshold value
theta: float
Threshold value
dim : int, optional
Dimension of graph
pos : dict, optional
A dictionary keyed by node with node positions as values.
weight : dict, optional
Node weights as a dictionary of numbers keyed by node.
p : float, optional (default 2)
Which Minkowski distance metric to use. `p` has to meet the condition
``1 <= p <= infinity``.
If this argument is not specified, the :math:`L^2` metric
(the Euclidean distance metric), p = 2 is used.
This should not be confused with the `p` of an Erdős-Rényi random
graph, which represents probability.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
pos_name : string, default="pos"
The name of the node attribute which represents the position
in 2D coordinates of the node in the returned graph.
weight_name : string, default="weight"
The name of the node attribute which represents the weight
of the node in the returned graph.
Returns
-------
Graph
A thresholded random geographic graph, undirected and without
self-loops.
Each node has a node attribute ``'pos'`` that stores the
position of that node in Euclidean space as provided by the
``pos`` keyword argument or, if ``pos`` was not provided, as
generated by this function. Similarly, each node has a nodethre
attribute ``'weight'`` that stores the weight of that node as
provided or as generated.
Examples
--------
Default Graph:
G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
Custom Graph:
Create a thresholded random geometric graph on 50 uniformly distributed
nodes where nodes are joined by an edge if their sum weights drawn from
a exponential distribution with rate = 5 are >= theta = 0.1 and their
Euclidean distance is at most 0.2.
Notes
-----
This uses a *k*-d tree to build the graph.
The `pos` keyword argument can be used to specify node positions so you
can create an arbitrary distribution and domain for positions.
For example, to use a 2D Gaussian distribution of node positions with mean
(0, 0) and standard deviation 2
If weights are not specified they are assigned to nodes by drawing randomly
from the exponential distribution with rate parameter :math:`\lambda=1`.
To specify weights from a different distribution, use the `weight` keyword
argument::
::
>>> import random
>>> import math
>>> n = 50
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
References
----------
.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
"""
G = nx.empty_graph(n)
G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})"
# If no weights are provided, choose them from an exponential
# distribution.
if weight is None:
weight = {v: seed.expovariate(1) for v in G}
# If no positions are provided, choose uniformly random vectors in
# Euclidean space of the specified dimension.
if pos is None:
pos = {v: [seed.random() for i in range(dim)] for v in G}
# If no distance metric is provided, use Euclidean distance.
nx.set_node_attributes(G, weight, weight_name)
nx.set_node_attributes(G, pos, pos_name)
edges = (
(u, v)
for u, v in _geometric_edges(G, radius, p, pos_name)
if weight[u] + weight[v] >= theta
)
G.add_edges_from(edges)
return G
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|
30,701 | networkx.generators.geometric | geometric_soft_configuration_graph | Returns a random graph from the geometric soft configuration model.
The $\mathbb{S}^1$ model [1]_ is the geometric soft configuration model
which is able to explain many fundamental features of real networks such as
small-world property, heteregenous degree distributions, high level of
clustering, and self-similarity.
In the geometric soft configuration model, a node $i$ is assigned two hidden
variables: a hidden degree $\kappa_i$, quantifying its popularity, influence,
or importance, and an angular position $\theta_i$ in a circle abstracting the
similarity space, where angular distances between nodes are a proxy for their
similarity. Focusing on the angular position, this model is often called
the $\mathbb{S}^1$ model (a one-dimensional sphere). The circle's radius is
adjusted to $R = N/2\pi$, where $N$ is the number of nodes, so that the density
is set to 1 without loss of generality.
The connection probability between any pair of nodes increases with
the product of their hidden degrees (i.e., their combined popularities),
and decreases with the angular distance between the two nodes.
Specifically, nodes $i$ and $j$ are connected with the probability
$p_{ij} = \frac{1}{1 + \frac{d_{ij}^\beta}{\left(\mu \kappa_i \kappa_j\right)^{\max(1, \beta)}}}$
where $d_{ij} = R\Delta\theta_{ij}$ is the arc length of the circle between
nodes $i$ and $j$ separated by an angular distance $\Delta\theta_{ij}$.
Parameters $\mu$ and $\beta$ (also called inverse temperature) control the
average degree and the clustering coefficient, respectively.
It can be shown [2]_ that the model undergoes a structural phase transition
at $\beta=1$ so that for $\beta<1$ networks are unclustered in the thermodynamic
limit (when $N\to \infty$) whereas for $\beta>1$ the ensemble generates
networks with finite clustering coefficient.
The $\mathbb{S}^1$ model can be expressed as a purely geometric model
$\mathbb{H}^2$ in the hyperbolic plane [3]_ by mapping the hidden degree of
each node into a radial coordinate as
$r_i = \hat{R} - \frac{2 \max(1, \beta)}{\beta \zeta} \ln \left(\frac{\kappa_i}{\kappa_0}\right)$
where $\hat{R}$ is the radius of the hyperbolic disk and $\zeta$ is the curvature,
$\hat{R} = \frac{2}{\zeta} \ln \left(\frac{N}{\pi}\right)
- \frac{2\max(1, \beta)}{\beta \zeta} \ln (\mu \kappa_0^2)$
The connection probability then reads
$p_{ij} = \frac{1}{1 + \exp\left({\frac{\beta\zeta}{2} (x_{ij} - \hat{R})}\right)}$
where
$x_{ij} = r_i + r_j + \frac{2}{\zeta} \ln \frac{\Delta\theta_{ij}}{2}$
is a good approximation of the hyperbolic distance between two nodes separated
by an angular distance $\Delta\theta_{ij}$ with radial coordinates $r_i$ and $r_j$.
For $\beta > 1$, the curvature $\zeta = 1$, for $\beta < 1$, $\zeta = \beta^{-1}$.
Parameters
----------
Either `n`, `gamma`, `mean_degree` are provided or `kappas`. The values of
`n`, `gamma`, `mean_degree` (if provided) are used to construct a random
kappa-dict keyed by node with values sampled from a power-law distribution.
beta : positive number
Inverse temperature, controlling the clustering coefficient.
n : int (default: None)
Size of the network (number of nodes).
If not provided, `kappas` must be provided and holds the nodes.
gamma : float (default: None)
Exponent of the power-law distribution for hidden degrees `kappas`.
If not provided, `kappas` must be provided directly.
mean_degree : float (default: None)
The mean degree in the network.
If not provided, `kappas` must be provided directly.
kappas : dict (default: None)
A dict keyed by node to its hidden degree value.
If not provided, random values are computed based on a power-law
distribution using `n`, `gamma` and `mean_degree`.
seed : int, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
Graph
A random geometric soft configuration graph (undirected with no self-loops).
Each node has three node-attributes:
- ``kappa`` that represents the hidden degree.
- ``theta`` the position in the similarity space ($\mathbb{S}^1$) which is
also the angular position in the hyperbolic plane.
- ``radius`` the radial position in the hyperbolic plane
(based on the hidden degree).
Examples
--------
Generate a network with specified parameters:
>>> G = nx.geometric_soft_configuration_graph(beta=1.5, n=100, gamma=2.7, mean_degree=5)
Create a geometric soft configuration graph with 100 nodes. The $\beta$ parameter
is set to 1.5 and the exponent of the powerlaw distribution of the hidden
degrees is 2.7 with mean value of 5.
Generate a network with predefined hidden degrees:
>>> kappas = {i: 10 for i in range(100)}
>>> G = nx.geometric_soft_configuration_graph(beta=2.5, kappas=kappas)
Create a geometric soft configuration graph with 100 nodes. The $\beta$ parameter
is set to 2.5 and all nodes with hidden degree $\kappa=10$.
References
----------
.. [1] Serrano, M. Á., Krioukov, D., & Boguñá, M. (2008). Self-similarity
of complex networks and hidden metric spaces. Physical review letters, 100(7), 078701.
.. [2] van der Kolk, J., Serrano, M. Á., & Boguñá, M. (2022). An anomalous
topological phase transition in spatial random graphs. Communications Physics, 5(1), 245.
.. [3] Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., & Boguná, M. (2010).
Hyperbolic geometry of complex networks. Physical Review E, 82(3), 036106.
| def thresholded_random_geometric_graph(
n,
radius,
theta,
dim=2,
pos=None,
weight=None,
p=2,
seed=None,
*,
pos_name="pos",
weight_name="weight",
):
r"""Returns a thresholded random geometric graph in the unit cube.
The thresholded random geometric graph [1] model places `n` nodes
uniformly at random in the unit cube of dimensions `dim`. Each node
`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
joined by an edge if they are within the maximum connection distance,
`radius` computed by the `p`-Minkowski distance and the summation of
weights :math:`w_u` + :math:`w_v` is greater than or equal
to the threshold parameter `theta`.
Edges within `radius` of each other are determined using a KDTree when
SciPy is available. This reduces the time complexity from :math:`O(n^2)`
to :math:`O(n)`.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
radius: float
Distance threshold value
theta: float
Threshold value
dim : int, optional
Dimension of graph
pos : dict, optional
A dictionary keyed by node with node positions as values.
weight : dict, optional
Node weights as a dictionary of numbers keyed by node.
p : float, optional (default 2)
Which Minkowski distance metric to use. `p` has to meet the condition
``1 <= p <= infinity``.
If this argument is not specified, the :math:`L^2` metric
(the Euclidean distance metric), p = 2 is used.
This should not be confused with the `p` of an Erdős-Rényi random
graph, which represents probability.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
pos_name : string, default="pos"
The name of the node attribute which represents the position
in 2D coordinates of the node in the returned graph.
weight_name : string, default="weight"
The name of the node attribute which represents the weight
of the node in the returned graph.
Returns
-------
Graph
A thresholded random geographic graph, undirected and without
self-loops.
Each node has a node attribute ``'pos'`` that stores the
position of that node in Euclidean space as provided by the
``pos`` keyword argument or, if ``pos`` was not provided, as
generated by this function. Similarly, each node has a nodethre
attribute ``'weight'`` that stores the weight of that node as
provided or as generated.
Examples
--------
Default Graph:
G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
Custom Graph:
Create a thresholded random geometric graph on 50 uniformly distributed
nodes where nodes are joined by an edge if their sum weights drawn from
a exponential distribution with rate = 5 are >= theta = 0.1 and their
Euclidean distance is at most 0.2.
Notes
-----
This uses a *k*-d tree to build the graph.
The `pos` keyword argument can be used to specify node positions so you
can create an arbitrary distribution and domain for positions.
For example, to use a 2D Gaussian distribution of node positions with mean
(0, 0) and standard deviation 2
If weights are not specified they are assigned to nodes by drawing randomly
from the exponential distribution with rate parameter :math:`\lambda=1`.
To specify weights from a different distribution, use the `weight` keyword
argument::
::
>>> import random
>>> import math
>>> n = 50
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
References
----------
.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
"""
G = nx.empty_graph(n)
G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})"
# If no weights are provided, choose them from an exponential
# distribution.
if weight is None:
weight = {v: seed.expovariate(1) for v in G}
# If no positions are provided, choose uniformly random vectors in
# Euclidean space of the specified dimension.
if pos is None:
pos = {v: [seed.random() for i in range(dim)] for v in G}
# If no distance metric is provided, use Euclidean distance.
nx.set_node_attributes(G, weight, weight_name)
nx.set_node_attributes(G, pos, pos_name)
edges = (
(u, v)
for u, v in _geometric_edges(G, radius, p, pos_name)
if weight[u] + weight[v] >= theta
)
G.add_edges_from(edges)
return G
| (*, beta, n=None, gamma=None, mean_degree=None, kappas=None, seed=None, backend=None, **backend_kwargs) | [
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|
30,702 | networkx.classes.function | get_edge_attributes | Get edge attributes from graph
Parameters
----------
G : NetworkX Graph
name : string
Attribute name
default: object (default=None)
Default value of the edge attribute if there is no value set for that
edge in graph. If `None` then edges without this attribute are not
included in the returned dict.
Returns
-------
Dictionary of attributes keyed by edge. For (di)graphs, the keys are
2-tuples of the form: (u, v). For multi(di)graphs, the keys are 3-tuples of
the form: (u, v, key).
Examples
--------
>>> G = nx.Graph()
>>> nx.add_path(G, [1, 2, 3], color="red")
>>> color = nx.get_edge_attributes(G, "color")
>>> color[(1, 2)]
'red'
>>> G.add_edge(3, 4)
>>> color = nx.get_edge_attributes(G, "color", default="yellow")
>>> color[(3, 4)]
'yellow'
| def get_edge_attributes(G, name, default=None):
"""Get edge attributes from graph
Parameters
----------
G : NetworkX Graph
name : string
Attribute name
default: object (default=None)
Default value of the edge attribute if there is no value set for that
edge in graph. If `None` then edges without this attribute are not
included in the returned dict.
Returns
-------
Dictionary of attributes keyed by edge. For (di)graphs, the keys are
2-tuples of the form: (u, v). For multi(di)graphs, the keys are 3-tuples of
the form: (u, v, key).
Examples
--------
>>> G = nx.Graph()
>>> nx.add_path(G, [1, 2, 3], color="red")
>>> color = nx.get_edge_attributes(G, "color")
>>> color[(1, 2)]
'red'
>>> G.add_edge(3, 4)
>>> color = nx.get_edge_attributes(G, "color", default="yellow")
>>> color[(3, 4)]
'yellow'
"""
if G.is_multigraph():
edges = G.edges(keys=True, data=True)
else:
edges = G.edges(data=True)
if default is not None:
return {x[:-1]: x[-1].get(name, default) for x in edges}
return {x[:-1]: x[-1][name] for x in edges if name in x[-1]}
| (G, name, default=None) | [
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|
30,703 | networkx.classes.function | get_node_attributes | Get node attributes from graph
Parameters
----------
G : NetworkX Graph
name : string
Attribute name
default: object (default=None)
Default value of the node attribute if there is no value set for that
node in graph. If `None` then nodes without this attribute are not
included in the returned dict.
Returns
-------
Dictionary of attributes keyed by node.
Examples
--------
>>> G = nx.Graph()
>>> G.add_nodes_from([1, 2, 3], color="red")
>>> color = nx.get_node_attributes(G, "color")
>>> color[1]
'red'
>>> G.add_node(4)
>>> color = nx.get_node_attributes(G, "color", default="yellow")
>>> color[4]
'yellow'
| def get_node_attributes(G, name, default=None):
"""Get node attributes from graph
Parameters
----------
G : NetworkX Graph
name : string
Attribute name
default: object (default=None)
Default value of the node attribute if there is no value set for that
node in graph. If `None` then nodes without this attribute are not
included in the returned dict.
Returns
-------
Dictionary of attributes keyed by node.
Examples
--------
>>> G = nx.Graph()
>>> G.add_nodes_from([1, 2, 3], color="red")
>>> color = nx.get_node_attributes(G, "color")
>>> color[1]
'red'
>>> G.add_node(4)
>>> color = nx.get_node_attributes(G, "color", default="yellow")
>>> color[4]
'yellow'
"""
if default is not None:
return {n: d.get(name, default) for n, d in G.nodes.items()}
return {n: d[name] for n, d in G.nodes.items() if name in d}
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|
30,705 | networkx.algorithms.cycles | girth | Returns the girth of the graph.
The girth of a graph is the length of its shortest cycle, or infinity if
the graph is acyclic. The algorithm follows the description given on the
Wikipedia page [1]_, and runs in time O(mn) on a graph with m edges and n
nodes.
Parameters
----------
G : NetworkX Graph
Returns
-------
int or math.inf
Examples
--------
All examples below (except P_5) can easily be checked using Wikipedia,
which has a page for each of these famous graphs.
>>> nx.girth(nx.chvatal_graph())
4
>>> nx.girth(nx.tutte_graph())
4
>>> nx.girth(nx.petersen_graph())
5
>>> nx.girth(nx.heawood_graph())
6
>>> nx.girth(nx.pappus_graph())
6
>>> nx.girth(nx.path_graph(5))
inf
References
----------
.. [1] `Wikipedia: Girth <https://en.wikipedia.org/wiki/Girth_(graph_theory)>`_
| def recursive_simple_cycles(G):
"""Find simple cycles (elementary circuits) of a directed graph.
A `simple cycle`, or `elementary circuit`, is a closed path where
no node appears twice. Two elementary circuits are distinct if they
are not cyclic permutations of each other.
This version uses a recursive algorithm to build a list of cycles.
You should probably use the iterator version called simple_cycles().
Warning: This recursive version uses lots of RAM!
It appears in NetworkX for pedagogical value.
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
A list of cycles, where each cycle is represented by a list of nodes
along the cycle.
Example:
>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
>>> G = nx.DiGraph(edges)
>>> nx.recursive_simple_cycles(G)
[[0], [2], [0, 1, 2], [0, 2], [1, 2]]
Notes
-----
The implementation follows pp. 79-80 in [1]_.
The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
elementary circuits.
References
----------
.. [1] Finding all the elementary circuits of a directed graph.
D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
https://doi.org/10.1137/0204007
See Also
--------
simple_cycles, cycle_basis
"""
# Jon Olav Vik, 2010-08-09
def _unblock(thisnode):
"""Recursively unblock and remove nodes from B[thisnode]."""
if blocked[thisnode]:
blocked[thisnode] = False
while B[thisnode]:
_unblock(B[thisnode].pop())
def circuit(thisnode, startnode, component):
closed = False # set to True if elementary path is closed
path.append(thisnode)
blocked[thisnode] = True
for nextnode in component[thisnode]: # direct successors of thisnode
if nextnode == startnode:
result.append(path[:])
closed = True
elif not blocked[nextnode]:
if circuit(nextnode, startnode, component):
closed = True
if closed:
_unblock(thisnode)
else:
for nextnode in component[thisnode]:
if thisnode not in B[nextnode]: # TODO: use set for speedup?
B[nextnode].append(thisnode)
path.pop() # remove thisnode from path
return closed
path = [] # stack of nodes in current path
blocked = defaultdict(bool) # vertex: blocked from search?
B = defaultdict(list) # graph portions that yield no elementary circuit
result = [] # list to accumulate the circuits found
# Johnson's algorithm exclude self cycle edges like (v, v)
# To be backward compatible, we record those cycles in advance
# and then remove from subG
for v in G:
if G.has_edge(v, v):
result.append([v])
G.remove_edge(v, v)
# Johnson's algorithm requires some ordering of the nodes.
# They might not be sortable so we assign an arbitrary ordering.
ordering = dict(zip(G, range(len(G))))
for s in ordering:
# Build the subgraph induced by s and following nodes in the ordering
subgraph = G.subgraph(node for node in G if ordering[node] >= ordering[s])
# Find the strongly connected component in the subgraph
# that contains the least node according to the ordering
strongcomp = nx.strongly_connected_components(subgraph)
mincomp = min(strongcomp, key=lambda ns: min(ordering[n] for n in ns))
component = G.subgraph(mincomp)
if len(component) > 1:
# smallest node in the component according to the ordering
startnode = min(component, key=ordering.__getitem__)
for node in component:
blocked[node] = False
B[node][:] = []
dummy = circuit(startnode, startnode, component)
return result
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|
30,707 | networkx.algorithms.distance_regular | global_parameters | Returns global parameters for a given intersection array.
Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d
such that for any 2 vertices x,y in G at a distance i=d(x,y), there
are exactly c_i neighbors of y at a distance of i-1 from x and b_i
neighbors of y at a distance of i+1 from x.
Thus, a distance regular graph has the global parameters,
[[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for the
intersection array [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
where a_i+b_i+c_i=k , k= degree of every vertex.
Parameters
----------
b : list
c : list
Returns
-------
iterable
An iterable over three tuples.
Examples
--------
>>> G = nx.dodecahedral_graph()
>>> b, c = nx.intersection_array(G)
>>> list(nx.global_parameters(b, c))
[(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)]
References
----------
.. [1] Weisstein, Eric W. "Global Parameters."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/GlobalParameters.html
See Also
--------
intersection_array
| def global_parameters(b, c):
"""Returns global parameters for a given intersection array.
Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d
such that for any 2 vertices x,y in G at a distance i=d(x,y), there
are exactly c_i neighbors of y at a distance of i-1 from x and b_i
neighbors of y at a distance of i+1 from x.
Thus, a distance regular graph has the global parameters,
[[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for the
intersection array [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
where a_i+b_i+c_i=k , k= degree of every vertex.
Parameters
----------
b : list
c : list
Returns
-------
iterable
An iterable over three tuples.
Examples
--------
>>> G = nx.dodecahedral_graph()
>>> b, c = nx.intersection_array(G)
>>> list(nx.global_parameters(b, c))
[(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)]
References
----------
.. [1] Weisstein, Eric W. "Global Parameters."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/GlobalParameters.html
See Also
--------
intersection_array
"""
return ((y, b[0] - x - y, x) for x, y in zip(b + [0], [0] + c))
| (b, c) | [
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|
30,712 | networkx.generators.random_graphs | gnm_random_graph | Returns a $G_{n,m}$ random graph.
In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set
of all graphs with $n$ nodes and $m$ edges.
This algorithm should be faster than :func:`dense_gnm_random_graph` for
sparse graphs.
Parameters
----------
n : int
The number of nodes.
m : int
The number of edges.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
directed : bool, optional (default=False)
If True return a directed graph
See also
--------
dense_gnm_random_graph
| def dual_barabasi_albert_graph(n, m1, m2, p, seed=None, initial_graph=None):
"""Returns a random graph using dual Barabási–Albert preferential attachment
A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$
edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that
are preferentially attached to existing nodes with high degree.
Parameters
----------
n : int
Number of nodes
m1 : int
Number of edges to link each new node to existing nodes with probability $p$
m2 : int
Number of edges to link each new node to existing nodes with probability $1-p$
p : float
The probability of attaching $m_1$ edges (as opposed to $m_2$ edges)
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
initial_graph : Graph or None (default)
Initial network for Barabási–Albert algorithm.
A copy of `initial_graph` is used.
It should be connected for most use cases.
If None, starts from an star graph on max(m1, m2) + 1 nodes.
Returns
-------
G : Graph
Raises
------
NetworkXError
If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n``, or
`p` does not satisfy ``0 <= p <= 1``, or
the initial graph number of nodes m0 does not satisfy m1, m2 <= m0 <= n.
References
----------
.. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538.
"""
if m1 < 1 or m1 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}"
)
if m2 < 1 or m2 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}"
)
if p < 0 or p > 1:
raise nx.NetworkXError(
f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}"
)
# For simplicity, if p == 0 or 1, just return BA
if p == 1:
return barabasi_albert_graph(n, m1, seed)
elif p == 0:
return barabasi_albert_graph(n, m2, seed)
if initial_graph is None:
# Default initial graph : empty graph on max(m1, m2) nodes
G = star_graph(max(m1, m2))
else:
if len(initial_graph) < max(m1, m2) or len(initial_graph) > n:
raise nx.NetworkXError(
f"Barabási–Albert initial graph must have between "
f"max(m1, m2) = {max(m1, m2)} and n = {n} nodes"
)
G = initial_graph.copy()
# Target nodes for new edges
targets = list(G)
# List of existing nodes, with nodes repeated once for each adjacent edge
repeated_nodes = [n for n, d in G.degree() for _ in range(d)]
# Start adding the remaining nodes.
source = len(G)
while source < n:
# Pick which m to use (m1 or m2)
if seed.random() < p:
m = m1
else:
m = m2
# Now choose m unique nodes from the existing nodes
# Pick uniformly from repeated_nodes (preferential attachment)
targets = _random_subset(repeated_nodes, m, seed)
# Add edges to m nodes from the source.
G.add_edges_from(zip([source] * m, targets))
# Add one node to the list for each new edge just created.
repeated_nodes.extend(targets)
# And the new node "source" has m edges to add to the list.
repeated_nodes.extend([source] * m)
source += 1
return G
| (n, m, seed=None, directed=False, *, backend=None, **backend_kwargs) | [
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|
30,715 | networkx.algorithms.shortest_paths.weighted | goldberg_radzik | Compute shortest path lengths and predecessors on shortest paths
in weighted graphs.
The algorithm has a running time of $O(mn)$ where $n$ is the number of
nodes and $m$ is the number of edges. It is slower than Dijkstra but
can handle negative edge weights.
Parameters
----------
G : NetworkX graph
The algorithm works for all types of graphs, including directed
graphs and multigraphs.
source: node label
Starting node for path
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
Returns
-------
pred, dist : dictionaries
Returns two dictionaries keyed by node to predecessor in the
path and to the distance from the source respectively.
Raises
------
NodeNotFound
If `source` is not in `G`.
NetworkXUnbounded
If the (di)graph contains a negative (di)cycle, the
algorithm raises an exception to indicate the presence of the
negative (di)cycle. Note: any negative weight edge in an
undirected graph is a negative cycle.
As of NetworkX v3.2, a zero weight cycle is no longer
incorrectly reported as a negative weight cycle.
Examples
--------
>>> G = nx.path_graph(5, create_using=nx.DiGraph())
>>> pred, dist = nx.goldberg_radzik(G, 0)
>>> sorted(pred.items())
[(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
>>> sorted(dist.items())
[(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]
>>> G = nx.cycle_graph(5, create_using=nx.DiGraph())
>>> G[1][2]["weight"] = -7
>>> nx.goldberg_radzik(G, 0)
Traceback (most recent call last):
...
networkx.exception.NetworkXUnbounded: Negative cycle detected.
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The dictionaries returned only have keys for nodes reachable from
the source.
In the case where the (di)graph is not connected, if a component
not containing the source contains a negative (di)cycle, it
will not be detected.
| def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths
Parameters
----------
G : NetworkX graph
sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.
weight: function
Function with (u, v, data) input that returns that edge's weight
or None to indicate a hidden edge
pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.
paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.
target : node label, optional
Ending node for path. Search is halted when target is found.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.
Raises
------
NodeNotFound
If any of `sources` is not in `G`.
Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.
"""
G_succ = G._adj # For speed-up (and works for both directed and undirected graphs)
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
seen[source] = 0
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist
| (G, source, weight='weight', *, backend=None, **backend_kwargs) | [
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|
30,722 | networkx.algorithms.similarity | graph_edit_distance | Returns GED (graph edit distance) between graphs G1 and G2.
Graph edit distance is a graph similarity measure analogous to
Levenshtein distance for strings. It is defined as minimum cost
of edit path (sequence of node and edge edit operations)
transforming graph G1 to graph isomorphic to G2.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
upper_bound : numeric
Maximum edit distance to consider. Return None if no edit
distance under or equal to upper_bound exists.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Examples
--------
>>> G1 = nx.cycle_graph(6)
>>> G2 = nx.wheel_graph(7)
>>> nx.graph_edit_distance(G1, G2)
7.0
>>> G1 = nx.star_graph(5)
>>> G2 = nx.star_graph(5)
>>> nx.graph_edit_distance(G1, G2, roots=(0, 0))
0.0
>>> nx.graph_edit_distance(G1, G2, roots=(1, 0))
8.0
See Also
--------
optimal_edit_paths, optimize_graph_edit_distance,
is_isomorphic: test for graph edit distance of 0
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
| def optimize_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
strictly_decreasing=True,
roots=None,
timeout=None,
):
"""GED (graph edit distance) calculation: advanced interface.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Graph edit distance is defined as minimum cost of edit path.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost. Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
See Also
--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
"""
# TODO: support DiGraph
import numpy as np
import scipy as sp
@dataclass
class CostMatrix:
C: ...
lsa_row_ind: ...
lsa_col_ind: ...
ls: ...
def make_CostMatrix(C, m, n):
# assert(C.shape == (m + n, m + n))
lsa_row_ind, lsa_col_ind = sp.optimize.linear_sum_assignment(C)
# Fixup dummy assignments:
# each substitution i<->j should have dummy assignment m+j<->n+i
# NOTE: fast reduce of Cv relies on it
# assert len(lsa_row_ind) == len(lsa_col_ind)
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
subst_ind = [k for k, i, j in indexes if i < m and j < n]
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
dummy_ind = [k for k, i, j in indexes if i >= m and j >= n]
# assert len(subst_ind) == len(dummy_ind)
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n
return CostMatrix(
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
)
def extract_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k in i or k - m in j for k in range(m + n)]
col_ind = [k in j or k - n in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k not in i and k - m not in j for k in range(m + n)]
col_ind = [k not in j and k - n not in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_ind(ind, i):
# assert set(ind) == set(range(len(ind)))
rind = ind[[k not in i for k in ind]]
for k in set(i):
rind[rind >= k] -= 1
return rind
def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=None):
"""
Parameters:
u, v: matched vertices, u=None or v=None for
deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_uv: partial vertex edit path
list of tuples (u, v) of previously matched vertex
mappings u<->v, u=None or v=None for
deletion/insertion
Returns:
list of (i, j): indices of edge mappings g<->h
localCe: local CostMatrix of edge mappings
(basically submatrix of Ce at cross of rows i, cols j)
"""
M = len(pending_g)
N = len(pending_h)
# assert Ce.C.shape == (M + N, M + N)
# only attempt to match edges after one node match has been made
# this will stop self-edges on the first node being automatically deleted
# even when a substitution is the better option
if matched_uv is None or len(matched_uv) == 0:
g_ind = []
h_ind = []
else:
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|
30,744 | networkx.generators.degree_seq | havel_hakimi_graph | Returns a simple graph with given degree sequence constructed
using the Havel-Hakimi algorithm.
Parameters
----------
deg_sequence: list of integers
Each integer corresponds to the degree of a node (need not be sorted).
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Directed graphs are not allowed.
Raises
------
NetworkXException
For a non-graphical degree sequence (i.e. one
not realizable by some simple graph).
Notes
-----
The Havel-Hakimi algorithm constructs a simple graph by
successively connecting the node of highest degree to other nodes
of highest degree, resorting remaining nodes by degree, and
repeating the process. The resulting graph has a high
degree-associativity. Nodes are labeled 1,.., len(deg_sequence),
corresponding to their position in deg_sequence.
The basic algorithm is from Hakimi [1]_ and was generalized by
Kleitman and Wang [2]_.
References
----------
.. [1] Hakimi S., On Realizability of a Set of Integers as
Degrees of the Vertices of a Linear Graph. I,
Journal of SIAM, 10(3), pp. 496-506 (1962)
.. [2] Kleitman D.J. and Wang D.L.
Algorithms for Constructing Graphs and Digraphs with Given Valences
and Factors Discrete Mathematics, 6(1), pp. 79-88 (1973)
| def generate(self):
# remaining_degree is mapping from int->remaining degree
self.remaining_degree = dict(enumerate(self.degree))
# add all nodes to make sure we get isolated nodes
self.graph = nx.Graph()
self.graph.add_nodes_from(self.remaining_degree)
# remove zero degree nodes
for n, d in list(self.remaining_degree.items()):
if d == 0:
del self.remaining_degree[n]
if len(self.remaining_degree) > 0:
# build graph in three phases according to how many unmatched edges
self.phase1()
self.phase2()
self.phase3()
return self.graph
| (deg_sequence, create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,745 | networkx.generators.small | heawood_graph |
Returns the Heawood Graph, a (3,6) cage.
The Heawood Graph is an undirected graph with 14 nodes and 21 edges,
named after Percy John Heawood [1]_.
It is cubic symmetric, nonplanar, Hamiltonian, and can be represented
in LCF notation as ``[5,-5]^7`` [2]_.
It is the unique (3,6)-cage: the regular cubic graph of girth 6 with
minimal number of vertices [3]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Heawood Graph with 14 nodes and 21 edges
References
----------
.. [1] https://en.wikipedia.org/wiki/Heawood_graph
.. [2] https://mathworld.wolfram.com/HeawoodGraph.html
.. [3] https://www.win.tue.nl/~aeb/graphs/Heawood.html
| def sedgewick_maze_graph(create_using=None):
"""
Return a small maze with a cycle.
This is the maze used in Sedgewick, 3rd Edition, Part 5, Graph
Algorithms, Chapter 18, e.g. Figure 18.2 and following [1]_.
Nodes are numbered 0,..,7
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Small maze with a cycle
References
----------
.. [1] Figure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick
"""
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
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|
30,752 | networkx.generators.small | hoffman_singleton_graph |
Returns the Hoffman-Singleton Graph.
The Hoffman–Singleton graph is a symmetrical undirected graph
with 50 nodes and 175 edges.
All indices lie in ``Z % 5``: that is, the integers mod 5 [1]_.
It is the only regular graph of vertex degree 7, diameter 2, and girth 5.
It is the unique (7,5)-cage graph and Moore graph, and contains many
copies of the Petersen graph [2]_.
Returns
-------
G : networkx Graph
Hoffman–Singleton Graph with 50 nodes and 175 edges
Notes
-----
Constructed from pentagon and pentagram as follows: Take five pentagons $P_h$
and five pentagrams $Q_i$ . Join vertex $j$ of $P_h$ to vertex $h·i+j$ of $Q_i$ [3]_.
References
----------
.. [1] https://blogs.ams.org/visualinsight/2016/02/01/hoffman-singleton-graph/
.. [2] https://mathworld.wolfram.com/Hoffman-SingletonGraph.html
.. [3] https://en.wikipedia.org/wiki/Hoffman%E2%80%93Singleton_graph
| def sedgewick_maze_graph(create_using=None):
"""
Return a small maze with a cycle.
This is the maze used in Sedgewick, 3rd Edition, Part 5, Graph
Algorithms, Chapter 18, e.g. Figure 18.2 and following [1]_.
Nodes are numbered 0,..,7
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Small maze with a cycle
References
----------
.. [1] Figure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick
"""
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
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|
30,753 | networkx.generators.small | house_graph |
Returns the House graph (square with triangle on top)
The house graph is a simple undirected graph with
5 nodes and 6 edges [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
House graph in the form of a square with a triangle on top
References
----------
.. [1] https://mathworld.wolfram.com/HouseGraph.html
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,754 | networkx.generators.small | house_x_graph |
Returns the House graph with a cross inside the house square.
The House X-graph is the House graph plus the two edges connecting diagonally
opposite vertices of the square base. It is also one of the two graphs
obtained by removing two edges from the pentatope graph [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
House graph with diagonal vertices connected
References
----------
.. [1] https://mathworld.wolfram.com/HouseGraph.html
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,757 | networkx.generators.small | icosahedral_graph |
Returns the Platonic Icosahedral graph.
The icosahedral graph has 12 nodes and 30 edges. It is a Platonic graph
whose nodes have the connectivity of the icosahedron. It is undirected,
regular and Hamiltonian [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Icosahedral graph with 12 nodes and 30 edges.
References
----------
.. [1] https://mathworld.wolfram.com/IcosahedralGraph.html
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,763 | networkx.classes.function | induced_subgraph | Returns a SubGraph view of `G` showing only nodes in nbunch.
The induced subgraph of a graph on a set of nodes N is the
graph with nodes N and edges from G which have both ends in N.
Parameters
----------
G : NetworkX Graph
nbunch : node, container of nodes or None (for all nodes)
Returns
-------
subgraph : SubGraph View
A read-only view of the subgraph in `G` induced by the nodes.
Changes to the graph `G` will be reflected in the view.
Notes
-----
To create a mutable subgraph with its own copies of nodes
edges and attributes use `subgraph.copy()` or `Graph(subgraph)`
For an inplace reduction of a graph to a subgraph you can remove nodes:
`G.remove_nodes_from(n in G if n not in set(nbunch))`
If you are going to compute subgraphs of your subgraphs you could
end up with a chain of views that can be very slow once the chain
has about 15 views in it. If they are all induced subgraphs, you
can short-cut the chain by making them all subgraphs of the original
graph. The graph class method `G.subgraph` does this when `G` is
a subgraph. In contrast, this function allows you to choose to build
chains or not, as you wish. The returned subgraph is a view on `G`.
Examples
--------
>>> G = nx.path_graph(4) # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> H = nx.induced_subgraph(G, [0, 1, 3])
>>> list(H.edges)
[(0, 1)]
>>> list(H.nodes)
[0, 1, 3]
| def induced_subgraph(G, nbunch):
"""Returns a SubGraph view of `G` showing only nodes in nbunch.
The induced subgraph of a graph on a set of nodes N is the
graph with nodes N and edges from G which have both ends in N.
Parameters
----------
G : NetworkX Graph
nbunch : node, container of nodes or None (for all nodes)
Returns
-------
subgraph : SubGraph View
A read-only view of the subgraph in `G` induced by the nodes.
Changes to the graph `G` will be reflected in the view.
Notes
-----
To create a mutable subgraph with its own copies of nodes
edges and attributes use `subgraph.copy()` or `Graph(subgraph)`
For an inplace reduction of a graph to a subgraph you can remove nodes:
`G.remove_nodes_from(n in G if n not in set(nbunch))`
If you are going to compute subgraphs of your subgraphs you could
end up with a chain of views that can be very slow once the chain
has about 15 views in it. If they are all induced subgraphs, you
can short-cut the chain by making them all subgraphs of the original
graph. The graph class method `G.subgraph` does this when `G` is
a subgraph. In contrast, this function allows you to choose to build
chains or not, as you wish. The returned subgraph is a view on `G`.
Examples
--------
>>> G = nx.path_graph(4) # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> H = nx.induced_subgraph(G, [0, 1, 3])
>>> list(H.edges)
[(0, 1)]
>>> list(H.nodes)
[0, 1, 3]
"""
induced_nodes = nx.filters.show_nodes(G.nbunch_iter(nbunch))
return nx.subgraph_view(G, filter_node=induced_nodes)
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|
30,771 | networkx.algorithms.dag | is_aperiodic | Returns True if `G` is aperiodic.
A directed graph is aperiodic if there is no integer k > 1 that
divides the length of every cycle in the graph.
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
bool
True if the graph is aperiodic False otherwise
Raises
------
NetworkXError
If `G` is not directed
Examples
--------
A graph consisting of one cycle, the length of which is 2. Therefore ``k = 2``
divides the length of every cycle in the graph and thus the graph
is *not aperiodic*::
>>> DG = nx.DiGraph([(1, 2), (2, 1)])
>>> nx.is_aperiodic(DG)
False
A graph consisting of two cycles: one of length 2 and the other of length 3.
The cycle lengths are coprime, so there is no single value of k where ``k > 1``
that divides each cycle length and therefore the graph is *aperiodic*::
>>> DG = nx.DiGraph([(1, 2), (2, 3), (3, 1), (1, 4), (4, 1)])
>>> nx.is_aperiodic(DG)
True
A graph consisting of two cycles: one of length 2 and the other of length 4.
The lengths of the cycles share a common factor ``k = 2``, and therefore
the graph is *not aperiodic*::
>>> DG = nx.DiGraph([(1, 2), (2, 1), (3, 4), (4, 5), (5, 6), (6, 3)])
>>> nx.is_aperiodic(DG)
False
An acyclic graph, therefore the graph is *not aperiodic*::
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> nx.is_aperiodic(DG)
False
Notes
-----
This uses the method outlined in [1]_, which runs in $O(m)$ time
given $m$ edges in `G`. Note that a graph is not aperiodic if it is
acyclic as every integer trivial divides length 0 cycles.
References
----------
.. [1] Jarvis, J. P.; Shier, D. R. (1996),
"Graph-theoretic analysis of finite Markov chains,"
in Shier, D. R.; Wallenius, K. T., Applied Mathematical Modeling:
A Multidisciplinary Approach, CRC Press.
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
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|
30,782 | networkx.classes.function | is_directed | Return True if graph is directed. | def is_directed(G):
"""Return True if graph is directed."""
return G.is_directed()
| (G) | [
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|
30,783 | networkx.algorithms.dag | is_directed_acyclic_graph | Returns True if the graph `G` is a directed acyclic graph (DAG) or
False if not.
Parameters
----------
G : NetworkX graph
Returns
-------
bool
True if `G` is a DAG, False otherwise
Examples
--------
Undirected graph::
>>> G = nx.Graph([(1, 2), (2, 3)])
>>> nx.is_directed_acyclic_graph(G)
False
Directed graph with cycle::
>>> G = nx.DiGraph([(1, 2), (2, 3), (3, 1)])
>>> nx.is_directed_acyclic_graph(G)
False
Directed acyclic graph::
>>> G = nx.DiGraph([(1, 2), (2, 3)])
>>> nx.is_directed_acyclic_graph(G)
True
See also
--------
topological_sort
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
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|
30,787 | networkx.classes.function | is_empty | Returns True if `G` has no edges.
Parameters
----------
G : graph
A NetworkX graph.
Returns
-------
bool
True if `G` has no edges, and False otherwise.
Notes
-----
An empty graph can have nodes but not edges. The empty graph with zero
nodes is known as the null graph. This is an $O(n)$ operation where n
is the number of nodes in the graph.
| def is_empty(G):
"""Returns True if `G` has no edges.
Parameters
----------
G : graph
A NetworkX graph.
Returns
-------
bool
True if `G` has no edges, and False otherwise.
Notes
-----
An empty graph can have nodes but not edges. The empty graph with zero
nodes is known as the null graph. This is an $O(n)$ operation where n
is the number of nodes in the graph.
"""
return not any(G._adj.values())
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|
30,790 | networkx.classes.function | is_frozen | Returns True if graph is frozen.
Parameters
----------
G : graph
A NetworkX graph
See Also
--------
freeze
| def is_frozen(G):
"""Returns True if graph is frozen.
Parameters
----------
G : graph
A NetworkX graph
See Also
--------
freeze
"""
try:
return G.frozen
except AttributeError:
return False
| (G) | [
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|
30,794 | networkx.algorithms.connectivity.edge_augmentation | is_k_edge_connected | Tests to see if a graph is k-edge-connected.
Is it impossible to disconnect the graph by removing fewer than k edges?
If so, then G is k-edge-connected.
Parameters
----------
G : NetworkX graph
An undirected graph.
k : integer
edge connectivity to test for
Returns
-------
boolean
True if G is k-edge-connected.
See Also
--------
:func:`is_locally_k_edge_connected`
Examples
--------
>>> G = nx.barbell_graph(10, 0)
>>> nx.is_k_edge_connected(G, k=1)
True
>>> nx.is_k_edge_connected(G, k=2)
False
| def unconstrained_bridge_augmentation(G):
"""Finds an optimal 2-edge-augmentation of G using the fewest edges.
This is an implementation of the algorithm detailed in [1]_.
The basic idea is to construct a meta-graph of bridge-ccs, connect leaf
nodes of the trees to connect the entire graph, and finally connect the
leafs of the tree in dfs-preorder to bridge connect the entire graph.
Parameters
----------
G : NetworkX graph
An undirected graph.
Yields
------
edge : tuple
Edges in the bridge augmentation of G
Notes
-----
Input: a graph G.
First find the bridge components of G and collapse each bridge-cc into a
node of a metagraph graph C, which is guaranteed to be a forest of trees.
C contains p "leafs" --- nodes with exactly one incident edge.
C contains q "isolated nodes" --- nodes with no incident edges.
Theorem: If p + q > 1, then at least :math:`ceil(p / 2) + q` edges are
needed to bridge connect C. This algorithm achieves this min number.
The method first adds enough edges to make G into a tree and then pairs
leafs in a simple fashion.
Let n be the number of trees in C. Let v(i) be an isolated vertex in the
i-th tree if one exists, otherwise it is a pair of distinct leafs nodes
in the i-th tree. Alternating edges from these sets (i.e. adding edges
A1 = [(v(i)[0], v(i + 1)[1]), v(i + 1)[0], v(i + 2)[1])...]) connects C
into a tree T. This tree has p' = p + 2q - 2(n -1) leafs and no isolated
vertices. A1 has n - 1 edges. The next step finds ceil(p' / 2) edges to
biconnect any tree with p' leafs.
Convert T into an arborescence T' by picking an arbitrary root node with
degree >= 2 and directing all edges away from the root. Note the
implementation implicitly constructs T'.
The leafs of T are the nodes with no existing edges in T'.
Order the leafs of T' by DFS preorder. Then break this list in half
and add the zipped pairs to A2.
The set A = A1 + A2 is the minimum augmentation in the metagraph.
To convert this to edges in the original graph
References
----------
.. [1] Eswaran, Kapali P., and R. Endre Tarjan. (1975) Augmentation problems.
http://epubs.siam.org/doi/abs/10.1137/0205044
See Also
--------
:func:`bridge_augmentation`
:func:`k_edge_augmentation`
Examples
--------
>>> G = nx.path_graph((1, 2, 3, 4, 5, 6, 7))
>>> sorted(unconstrained_bridge_augmentation(G))
[(1, 7)]
>>> G = nx.path_graph((1, 2, 3, 2, 4, 5, 6, 7))
>>> sorted(unconstrained_bridge_augmentation(G))
[(1, 3), (3, 7)]
>>> G = nx.Graph([(0, 1), (0, 2), (1, 2)])
>>> G.add_node(4)
>>> sorted(unconstrained_bridge_augmentation(G))
[(1, 4), (4, 0)]
"""
# -----
# Mapping of terms from (Eswaran and Tarjan):
# G = G_0 - the input graph
# C = G_0' - the bridge condensation of G. (This is a forest of trees)
# A1 = A_1 - the edges to connect the forest into a tree
# leaf = pendant - a node with degree of 1
# alpha(v) = maps the node v in G to its meta-node in C
# beta(x) = maps the meta-node x in C to any node in the bridge
# component of G corresponding to x.
# find the 2-edge-connected components of G
bridge_ccs = list(nx.connectivity.bridge_components(G))
# condense G into an forest C
C = collapse(G, bridge_ccs)
# Choose pairs of distinct leaf nodes in each tree. If this is not
# possible then make a pair using the single isolated node in the tree.
vset1 = [
tuple(cc) * 2 # case1: an isolated node
if len(cc) == 1
else sorted(cc, key=C.degree)[0:2] # case2: pair of leaf nodes
for cc in nx.connected_components(C)
]
if len(vset1) > 1:
# Use this set to construct edges that connect C into a tree.
nodes1 = [vs[0] for vs in vset1]
nodes2 = [vs[1] for vs in vset1]
A1 = list(zip(nodes1[1:], nodes2))
else:
A1 = []
# Connect each tree in the forest to construct an arborescence
T = C.copy()
T.add_edges_from(A1)
# If there are only two leaf nodes, we simply connect them.
leafs = [n for n, d in T.degree() if d == 1]
if len(leafs) == 1:
A2 = []
if len(leafs) == 2:
A2 = [tuple(leafs)]
else:
# Choose an arbitrary non-leaf root
try:
root = next(n for n, d in T.degree() if d > 1)
except StopIteration: # no nodes found with degree > 1
return
# order the leaves of C by (induced directed) preorder
v2 = [n for n in nx.dfs_preorder_nodes(T, root) if T.degree(n) == 1]
# connecting first half of the leafs in pre-order to the second
# half will bridge connect the tree with the fewest edges.
half = math.ceil(len(v2) / 2)
A2 = list(zip(v2[:half], v2[-half:]))
# collect the edges used to augment the original forest
aug_tree_edges = A1 + A2
# Construct the mapping (beta) from meta-nodes to regular nodes
inverse = defaultdict(list)
for k, v in C.graph["mapping"].items():
inverse[v].append(k)
# sort so we choose minimum degree nodes first
inverse = {
mu: sorted(mapped, key=lambda u: (G.degree(u), u))
for mu, mapped in inverse.items()
}
# For each meta-edge, map back to an arbitrary pair in the original graph
G2 = G.copy()
for mu, mv in aug_tree_edges:
# Find the first available edge that doesn't exist and return it
for u, v in it.product(inverse[mu], inverse[mv]):
if not G2.has_edge(u, v):
G2.add_edge(u, v)
yield u, v
break
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|
30,797 | networkx.algorithms.matching | is_matching | Return True if ``matching`` is a valid matching of ``G``
A *matching* in a graph is a set of edges in which no two distinct
edges share a common endpoint. Each node is incident to at most one
edge in the matching. The edges are said to be independent.
Parameters
----------
G : NetworkX graph
matching : dict or set
A dictionary or set representing a matching. If a dictionary, it
must have ``matching[u] == v`` and ``matching[v] == u`` for each
edge ``(u, v)`` in the matching. If a set, it must have elements
of the form ``(u, v)``, where ``(u, v)`` is an edge in the
matching.
Returns
-------
bool
Whether the given set or dictionary represents a valid matching
in the graph.
Raises
------
NetworkXError
If the proposed matching has an edge to a node not in G.
Or if the matching is not a collection of 2-tuple edges.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5)])
>>> nx.is_maximal_matching(G, {1: 3, 2: 4}) # using dict to represent matching
True
>>> nx.is_matching(G, {(1, 3), (2, 4)}) # using set to represent matching
True
| @not_implemented_for("multigraph")
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def max_weight_matching(G, maxcardinality=False, weight="weight"):
"""Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
Parameters
----------
G : NetworkX graph
Undirected graph
maxcardinality: bool, optional (default=False)
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]
Notes
-----
If G has edges with weight attributes the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primal-dual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
Bipartite graphs can also be matched using the functions present in
:mod:`networkx.algorithms.bipartite.matching`.
References
----------
.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primal-dual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
class Blossom:
"""Representation of a non-trivial blossom or sub-blossom."""
__slots__ = ["childs", "edges", "mybestedges"]
# b.childs is an ordered list of b's sub-blossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a top-level S-blossom,
# b.mybestedges is a list of least-slack edges to neighboring
# S-blossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
stack = [*self.childs]
while stack:
t = stack.pop()
if isinstance(t, Blossom):
stack.extend(t.childs)
else:
yield t
# Get a list of vertices.
gnodes = list(G)
if not gnodes:
return set() # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for i, j, d in G.edges(data=True):
wt = d.get(weight, 1)
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (str(type(wt)).split("'")[1] in ("int", "long"))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = {}
# If b is a top-level blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an S-blossom,
# 2 if b is a T-blossom.
# The label of a vertex is found by looking at the label of its top-level
# containing blossom.
# If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
# from an S-vertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = {}
# If b is a labeled top-level blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a T-blossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = {}
# If v is a vertex, inblossom[v] is the top-level blossom to which v
# belongs.
# If v is a top-level vertex, inblossom[v] == v since v is itself
# a (trivial) top-level blossom.
# Initially all vertices are top-level trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a sub-blossom,
# blossomparent[b] is its immediate parent (sub-)blossom.
# If b is a top-level blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub-)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a T-blossom),
# bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
# or None if there is no such edge.
# If b is a (possibly trivial) top-level S-blossom,
# bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = {}
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a non-trivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = {}
# If (v, w) in allowedge or (w, v) in allowedg, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = {}
# Queue of newly discovered S-vertices.
queue = []
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
def slack(v, w):
return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1)
# Assign label t to the top-level blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an S-vertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a T-vertex/blossom; assign label S to its mate.
# (If b is a non-trivial blos | (G, matching, *, backend=None, **backend_kwargs) | [
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|
30,798 | networkx.algorithms.matching | is_maximal_matching | Return True if ``matching`` is a maximal matching of ``G``
A *maximal matching* in a graph is a matching in which adding any
edge would cause the set to no longer be a valid matching.
Parameters
----------
G : NetworkX graph
matching : dict or set
A dictionary or set representing a matching. If a dictionary, it
must have ``matching[u] == v`` and ``matching[v] == u`` for each
edge ``(u, v)`` in the matching. If a set, it must have elements
of the form ``(u, v)``, where ``(u, v)`` is an edge in the
matching.
Returns
-------
bool
Whether the given set or dictionary represents a valid maximal
matching in the graph.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (3, 4), (3, 5)])
>>> nx.is_maximal_matching(G, {(1, 2), (3, 4)})
True
| @not_implemented_for("multigraph")
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def max_weight_matching(G, maxcardinality=False, weight="weight"):
"""Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
Parameters
----------
G : NetworkX graph
Undirected graph
maxcardinality: bool, optional (default=False)
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]
Notes
-----
If G has edges with weight attributes the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primal-dual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
Bipartite graphs can also be matched using the functions present in
:mod:`networkx.algorithms.bipartite.matching`.
References
----------
.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primal-dual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
class Blossom:
"""Representation of a non-trivial blossom or sub-blossom."""
__slots__ = ["childs", "edges", "mybestedges"]
# b.childs is an ordered list of b's sub-blossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a top-level S-blossom,
# b.mybestedges is a list of least-slack edges to neighboring
# S-blossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
stack = [*self.childs]
while stack:
t = stack.pop()
if isinstance(t, Blossom):
stack.extend(t.childs)
else:
yield t
# Get a list of vertices.
gnodes = list(G)
if not gnodes:
return set() # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for i, j, d in G.edges(data=True):
wt = d.get(weight, 1)
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (str(type(wt)).split("'")[1] in ("int", "long"))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = {}
# If b is a top-level blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an S-blossom,
# 2 if b is a T-blossom.
# The label of a vertex is found by looking at the label of its top-level
# containing blossom.
# If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
# from an S-vertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = {}
# If b is a labeled top-level blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a T-blossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = {}
# If v is a vertex, inblossom[v] is the top-level blossom to which v
# belongs.
# If v is a top-level vertex, inblossom[v] == v since v is itself
# a (trivial) top-level blossom.
# Initially all vertices are top-level trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a sub-blossom,
# blossomparent[b] is its immediate parent (sub-)blossom.
# If b is a top-level blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub-)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a T-blossom),
# bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
# or None if there is no such edge.
# If b is a (possibly trivial) top-level S-blossom,
# bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = {}
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a non-trivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = {}
# If (v, w) in allowedge or (w, v) in allowedg, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = {}
# Queue of newly discovered S-vertices.
queue = []
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
def slack(v, w):
return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1)
# Assign label t to the top-level blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an S-vertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a T-vertex/blossom; assign label S to its mate.
# (If b is a non-trivial blos | (G, matching, *, backend=None, **backend_kwargs) | [
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|
30,801 | networkx.classes.function | is_negatively_weighted | Returns True if `G` has negatively weighted edges.
Parameters
----------
G : graph
A NetworkX graph.
edge : tuple, optional
A 2-tuple specifying the only edge in `G` that will be tested. If
None, then every edge in `G` is tested.
weight: string, optional
The attribute name used to query for edge weights.
Returns
-------
bool
A boolean signifying if `G`, or the specified edge, is negatively
weighted.
Raises
------
NetworkXError
If the specified edge does not exist.
Examples
--------
>>> G = nx.Graph()
>>> G.add_edges_from([(1, 3), (2, 4), (2, 6)])
>>> G.add_edge(1, 2, weight=4)
>>> nx.is_negatively_weighted(G, (1, 2))
False
>>> G[2][4]["weight"] = -2
>>> nx.is_negatively_weighted(G)
True
>>> G = nx.DiGraph()
>>> edges = [("0", "3", 3), ("0", "1", -5), ("1", "0", -2)]
>>> G.add_weighted_edges_from(edges)
>>> nx.is_negatively_weighted(G)
True
| def set_edge_attributes(G, values, name=None):
"""Sets edge attributes from a given value or dictionary of values.
.. Warning:: The call order of arguments `values` and `name`
switched between v1.x & v2.x.
Parameters
----------
G : NetworkX Graph
values : scalar value, dict-like
What the edge attribute should be set to. If `values` is
not a dictionary, then it is treated as a single attribute value
that is then applied to every edge in `G`. This means that if
you provide a mutable object, like a list, updates to that object
will be reflected in the edge attribute for each edge. The attribute
name will be `name`.
If `values` is a dict or a dict of dict, it should be keyed
by edge tuple to either an attribute value or a dict of attribute
key/value pairs used to update the edge's attributes.
For multigraphs, the edge tuples must be of the form ``(u, v, key)``,
where `u` and `v` are nodes and `key` is the edge key.
For non-multigraphs, the keys must be tuples of the form ``(u, v)``.
name : string (optional, default=None)
Name of the edge attribute to set if values is a scalar.
Examples
--------
After computing some property of the edges of a graph, you may want
to assign a edge attribute to store the value of that property for
each edge::
>>> G = nx.path_graph(3)
>>> bb = nx.edge_betweenness_centrality(G, normalized=False)
>>> nx.set_edge_attributes(G, bb, "betweenness")
>>> G.edges[1, 2]["betweenness"]
2.0
If you provide a list as the second argument, updates to the list
will be reflected in the edge attribute for each edge::
>>> labels = []
>>> nx.set_edge_attributes(G, labels, "labels")
>>> labels.append("foo")
>>> G.edges[0, 1]["labels"]
['foo']
>>> G.edges[1, 2]["labels"]
['foo']
If you provide a dictionary of dictionaries as the second argument,
the entire dictionary will be used to update edge attributes::
>>> G = nx.path_graph(3)
>>> attrs = {(0, 1): {"attr1": 20, "attr2": "nothing"}, (1, 2): {"attr2": 3}}
>>> nx.set_edge_attributes(G, attrs)
>>> G[0][1]["attr1"]
20
>>> G[0][1]["attr2"]
'nothing'
>>> G[1][2]["attr2"]
3
The attributes of one Graph can be used to set those of another.
>>> H = nx.path_graph(3)
>>> nx.set_edge_attributes(H, G.edges)
Note that if the dict contains edges that are not in `G`, they are
silently ignored::
>>> G = nx.Graph([(0, 1)])
>>> nx.set_edge_attributes(G, {(1, 2): {"weight": 2.0}})
>>> (1, 2) in G.edges()
False
For multigraphs, the `values` dict is expected to be keyed by 3-tuples
including the edge key::
>>> MG = nx.MultiGraph()
>>> edges = [(0, 1), (0, 1)]
>>> MG.add_edges_from(edges) # Returns list of edge keys
[0, 1]
>>> attributes = {(0, 1, 0): {"cost": 21}, (0, 1, 1): {"cost": 7}}
>>> nx.set_edge_attributes(MG, attributes)
>>> MG[0][1][0]["cost"]
21
>>> MG[0][1][1]["cost"]
7
If MultiGraph attributes are desired for a Graph, you must convert the 3-tuple
multiedge to a 2-tuple edge and the last multiedge's attribute value will
overwrite the previous values. Continuing from the previous case we get::
>>> H = nx.path_graph([0, 1, 2])
>>> nx.set_edge_attributes(H, {(u, v): ed for u, v, ed in MG.edges.data()})
>>> nx.get_edge_attributes(H, "cost")
{(0, 1): 7}
"""
if name is not None:
# `values` does not contain attribute names
try:
# if `values` is a dict using `.items()` => {edge: value}
if G.is_multigraph():
for (u, v, key), value in values.items():
try:
G._adj[u][v][key][name] = value
except KeyError:
pass
else:
for (u, v), value in values.items():
try:
G._adj[u][v][name] = value
except KeyError:
pass
except AttributeError:
# treat `values` as a constant
for u, v, data in G.edges(data=True):
data[name] = values
else:
# `values` consists of doct-of-dict {edge: {attr: value}} shape
if G.is_multigraph():
for (u, v, key), d in values.items():
try:
G._adj[u][v][key].update(d)
except KeyError:
pass
else:
for (u, v), d in values.items():
try:
G._adj[u][v].update(d)
except KeyError:
pass
nx._clear_cache(G)
| (G, edge=None, weight='weight', *, backend=None, **backend_kwargs) | [
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|
30,802 | networkx.classes.function | is_path | Returns whether or not the specified path exists.
For it to return True, every node on the path must exist and
each consecutive pair must be connected via one or more edges.
Parameters
----------
G : graph
A NetworkX graph.
path : list
A list of nodes which defines the path to traverse
Returns
-------
bool
True if `path` is a valid path in `G`
| def is_path(G, path):
"""Returns whether or not the specified path exists.
For it to return True, every node on the path must exist and
each consecutive pair must be connected via one or more edges.
Parameters
----------
G : graph
A NetworkX graph.
path : list
A list of nodes which defines the path to traverse
Returns
-------
bool
True if `path` is a valid path in `G`
"""
try:
return all(nbr in G._adj[node] for node, nbr in nx.utils.pairwise(path))
except (KeyError, TypeError):
return False
| (G, path) | [
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|
30,803 | networkx.algorithms.matching | is_perfect_matching | Return True if ``matching`` is a perfect matching for ``G``
A *perfect matching* in a graph is a matching in which exactly one edge
is incident upon each vertex.
Parameters
----------
G : NetworkX graph
matching : dict or set
A dictionary or set representing a matching. If a dictionary, it
must have ``matching[u] == v`` and ``matching[v] == u`` for each
edge ``(u, v)`` in the matching. If a set, it must have elements
of the form ``(u, v)``, where ``(u, v)`` is an edge in the
matching.
Returns
-------
bool
Whether the given set or dictionary represents a valid perfect
matching in the graph.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5), (4, 6)])
>>> my_match = {1: 2, 3: 5, 4: 6}
>>> nx.is_perfect_matching(G, my_match)
True
| @not_implemented_for("multigraph")
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def max_weight_matching(G, maxcardinality=False, weight="weight"):
"""Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
Parameters
----------
G : NetworkX graph
Undirected graph
maxcardinality: bool, optional (default=False)
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]
Notes
-----
If G has edges with weight attributes the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primal-dual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
Bipartite graphs can also be matched using the functions present in
:mod:`networkx.algorithms.bipartite.matching`.
References
----------
.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primal-dual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
class Blossom:
"""Representation of a non-trivial blossom or sub-blossom."""
__slots__ = ["childs", "edges", "mybestedges"]
# b.childs is an ordered list of b's sub-blossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a top-level S-blossom,
# b.mybestedges is a list of least-slack edges to neighboring
# S-blossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
stack = [*self.childs]
while stack:
t = stack.pop()
if isinstance(t, Blossom):
stack.extend(t.childs)
else:
yield t
# Get a list of vertices.
gnodes = list(G)
if not gnodes:
return set() # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for i, j, d in G.edges(data=True):
wt = d.get(weight, 1)
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (str(type(wt)).split("'")[1] in ("int", "long"))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = {}
# If b is a top-level blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an S-blossom,
# 2 if b is a T-blossom.
# The label of a vertex is found by looking at the label of its top-level
# containing blossom.
# If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
# from an S-vertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = {}
# If b is a labeled top-level blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a T-blossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = {}
# If v is a vertex, inblossom[v] is the top-level blossom to which v
# belongs.
# If v is a top-level vertex, inblossom[v] == v since v is itself
# a (trivial) top-level blossom.
# Initially all vertices are top-level trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a sub-blossom,
# blossomparent[b] is its immediate parent (sub-)blossom.
# If b is a top-level blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub-)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a T-blossom),
# bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
# or None if there is no such edge.
# If b is a (possibly trivial) top-level S-blossom,
# bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = {}
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a non-trivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = {}
# If (v, w) in allowedge or (w, v) in allowedg, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = {}
# Queue of newly discovered S-vertices.
queue = []
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
def slack(v, w):
return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1)
# Assign label t to the top-level blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an S-vertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a T-vertex/blossom; assign label S to its mate.
# (If b is a non-trivial blos | (G, matching, *, backend=None, **backend_kwargs) | [
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|
30,804 | networkx.algorithms.planarity | is_planar | Returns True if and only if `G` is planar.
A graph is *planar* iff it can be drawn in a plane without
any edge intersections.
Parameters
----------
G : NetworkX graph
Returns
-------
bool
Whether the graph is planar.
Examples
--------
>>> G = nx.Graph([(0, 1), (0, 2)])
>>> nx.is_planar(G)
True
>>> nx.is_planar(nx.complete_graph(5))
False
See Also
--------
check_planarity :
Check if graph is planar *and* return a `PlanarEmbedding` instance if True.
| def sign_recursive(self, e):
"""Recursive version of :meth:`sign`."""
if self.ref[e] is not None:
self.side[e] = self.side[e] * self.sign_recursive(self.ref[e])
self.ref[e] = None
return self.side[e]
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|
30,810 | networkx.algorithms.simple_paths | is_simple_path | Returns True if and only if `nodes` form a simple path in `G`.
A *simple path* in a graph is a nonempty sequence of nodes in which
no node appears more than once in the sequence, and each adjacent
pair of nodes in the sequence is adjacent in the graph.
Parameters
----------
G : graph
A NetworkX graph.
nodes : list
A list of one or more nodes in the graph `G`.
Returns
-------
bool
Whether the given list of nodes represents a simple path in `G`.
Notes
-----
An empty list of nodes is not a path but a list of one node is a
path. Here's an explanation why.
This function operates on *node paths*. One could also consider
*edge paths*. There is a bijection between node paths and edge
paths.
The *length of a path* is the number of edges in the path, so a list
of nodes of length *n* corresponds to a path of length *n* - 1.
Thus the smallest edge path would be a list of zero edges, the empty
path. This corresponds to a list of one node.
To convert between a node path and an edge path, you can use code
like the following::
>>> from networkx.utils import pairwise
>>> nodes = [0, 1, 2, 3]
>>> edges = list(pairwise(nodes))
>>> edges
[(0, 1), (1, 2), (2, 3)]
>>> nodes = [edges[0][0]] + [v for u, v in edges]
>>> nodes
[0, 1, 2, 3]
Examples
--------
>>> G = nx.cycle_graph(4)
>>> nx.is_simple_path(G, [2, 3, 0])
True
>>> nx.is_simple_path(G, [0, 2])
False
| def _bidirectional_dijkstra(
G, source, target, weight="weight", ignore_nodes=None, ignore_edges=None
):
"""Dijkstra's algorithm for shortest paths using bidirectional search.
This function returns the shortest path between source and target
ignoring nodes and edges in the containers ignore_nodes and
ignore_edges.
This is a custom modification of the standard Dijkstra bidirectional
shortest path implementation at networkx.algorithms.weighted
Parameters
----------
G : NetworkX graph
source : node
Starting node.
target : node
Ending node.
weight: string, function, optional (default='weight')
Edge data key or weight function corresponding to the edge weight
ignore_nodes : container of nodes
nodes to ignore, optional
ignore_edges : container of edges
edges to ignore, optional
Returns
-------
length : number
Shortest path length.
Returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from the source.
The second stores the path from the source to that node.
Raises
------
NetworkXNoPath
If no path exists between source and target.
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
In practice bidirectional Dijkstra is much more than twice as fast as
ordinary Dijkstra.
Ordinary Dijkstra expands nodes in a sphere-like manner from the
source. The radius of this sphere will eventually be the length
of the shortest path. Bidirectional Dijkstra will expand nodes
from both the source and the target, making two spheres of half
this radius. Volume of the first sphere is pi*r*r while the
others are 2*pi*r/2*r/2, making up half the volume.
This algorithm is not guaranteed to work if edge weights
are negative or are floating point numbers
(overflows and roundoff errors can cause problems).
See Also
--------
shortest_path
shortest_path_length
"""
if ignore_nodes and (source in ignore_nodes or target in ignore_nodes):
raise nx.NetworkXNoPath(f"No path between {source} and {target}.")
if source == target:
if source not in G:
raise nx.NodeNotFound(f"Node {source} not in graph")
return (0, [source])
# handle either directed or undirected
if G.is_directed():
Gpred = G.predecessors
Gsucc = G.successors
else:
Gpred = G.neighbors
Gsucc = G.neighbors
# support optional nodes filter
if ignore_nodes:
def filter_iter(nodes):
def iterate(v):
for w in nodes(v):
if w not in ignore_nodes:
yield w
return iterate
Gpred = filter_iter(Gpred)
Gsucc = filter_iter(Gsucc)
# support optional edges filter
if ignore_edges:
if G.is_directed():
def filter_pred_iter(pred_iter):
def iterate(v):
for w in pred_iter(v):
if (w, v) not in ignore_edges:
yield w
return iterate
def filter_succ_iter(succ_iter):
def iterate(v):
for w in succ_iter(v):
if (v, w) not in ignore_edges:
yield w
return iterate
Gpred = filter_pred_iter(Gpred)
Gsucc = filter_succ_iter(Gsucc)
else:
def filter_iter(nodes):
def iterate(v):
for w in nodes(v):
if (v, w) not in ignore_edges and (w, v) not in ignore_edges:
yield w
return iterate
Gpred = filter_iter(Gpred)
Gsucc = filter_iter(Gsucc)
push = heappush
pop = heappop
# Init: Forward Backward
dists = [{}, {}] # dictionary of final distances
paths = [{source: [source]}, {target: [target]}] # dictionary of paths
fringe = [[], []] # heap of (distance, node) tuples for
# extracting next node to expand
seen = [{source: 0}, {target: 0}] # dictionary of distances to
# nodes seen
c = count()
# initialize fringe heap
push(fringe[0], (0, next(c), source))
push(fringe[1], (0, next(c), target))
# neighs for extracting correct neighbor information
neighs = [Gsucc, Gpred]
# variables to hold shortest discovered path
# finaldist = 1e30000
finalpath = []
dir = 1
while fringe[0] and fringe[1]:
# choose direction
# dir == 0 is forward direction and dir == 1 is back
dir = 1 - dir
# extract closest to expand
(dist, _, v) = pop(fringe[dir])
if v in dists[dir]:
# Shortest path to v has already been found
continue
# update distance
dists[dir][v] = dist # equal to seen[dir][v]
if v in dists[1 - dir]:
# if we have scanned v in both directions we are done
# we have now discovered the shortest path
return (finaldist, finalpath)
wt = _weight_function(G, weight)
for w in neighs[dir](v):
if dir == 0: # forward
minweight = wt(v, w, G.get_edge_data(v, w))
vwLength = dists[dir][v] + minweight
else: # back, must remember to change v,w->w,v
minweight = wt(w, v, G.get_edge_data(w, v))
vwLength = dists[dir][v] + minweight
if w in dists[dir]:
if vwLength < dists[dir][w]:
raise ValueError("Contradictory paths found: negative weights?")
elif w not in seen[dir] or vwLength < seen[dir][w]:
# relaxing
seen[dir][w] = vwLength
push(fringe[dir], (vwLength, next(c), w))
paths[dir][w] = paths[dir][v] + [w]
if w in seen[0] and w in seen[1]:
# see if this path is better than the already
# discovered shortest path
totaldist = seen[0][w] + seen[1][w]
if finalpath == [] or finaldist > totaldist:
finaldist = totaldist
revpath = paths[1][w][:]
revpath.reverse()
finalpath = paths[0][w] + revpath[1:]
raise nx.NetworkXNoPath(f"No path between {source} and {target}.")
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|
30,821 | networkx.classes.function | is_weighted | Returns True if `G` has weighted edges.
Parameters
----------
G : graph
A NetworkX graph.
edge : tuple, optional
A 2-tuple specifying the only edge in `G` that will be tested. If
None, then every edge in `G` is tested.
weight: string, optional
The attribute name used to query for edge weights.
Returns
-------
bool
A boolean signifying if `G`, or the specified edge, is weighted.
Raises
------
NetworkXError
If the specified edge does not exist.
Examples
--------
>>> G = nx.path_graph(4)
>>> nx.is_weighted(G)
False
>>> nx.is_weighted(G, (2, 3))
False
>>> G = nx.DiGraph()
>>> G.add_edge(1, 2, weight=1)
>>> nx.is_weighted(G)
True
| def is_weighted(G, edge=None, weight="weight"):
"""Returns True if `G` has weighted edges.
Parameters
----------
G : graph
A NetworkX graph.
edge : tuple, optional
A 2-tuple specifying the only edge in `G` that will be tested. If
None, then every edge in `G` is tested.
weight: string, optional
The attribute name used to query for edge weights.
Returns
-------
bool
A boolean signifying if `G`, or the specified edge, is weighted.
Raises
------
NetworkXError
If the specified edge does not exist.
Examples
--------
>>> G = nx.path_graph(4)
>>> nx.is_weighted(G)
False
>>> nx.is_weighted(G, (2, 3))
False
>>> G = nx.DiGraph()
>>> G.add_edge(1, 2, weight=1)
>>> nx.is_weighted(G)
True
"""
if edge is not None:
data = G.get_edge_data(*edge)
if data is None:
msg = f"Edge {edge!r} does not exist."
raise nx.NetworkXError(msg)
return weight in data
if is_empty(G):
# Special handling required since: all([]) == True
return False
return all(weight in data for u, v, data in G.edges(data=True))
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|
30,826 | networkx.algorithms.shortest_paths.weighted | johnson | Uses Johnson's Algorithm to compute shortest paths.
Johnson's Algorithm finds a shortest path between each pair of
nodes in a weighted graph even if negative weights are present.
Parameters
----------
G : NetworkX graph
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
Returns
-------
distance : dictionary
Dictionary, keyed by source and target, of shortest paths.
Examples
--------
>>> graph = nx.DiGraph()
>>> graph.add_weighted_edges_from(
... [("0", "3", 3), ("0", "1", -5), ("0", "2", 2), ("1", "2", 4), ("2", "3", 1)]
... )
>>> paths = nx.johnson(graph, weight="weight")
>>> paths["0"]["2"]
['0', '1', '2']
Notes
-----
Johnson's algorithm is suitable even for graphs with negative weights. It
works by using the Bellman–Ford algorithm to compute a transformation of
the input graph that removes all negative weights, allowing Dijkstra's
algorithm to be used on the transformed graph.
The time complexity of this algorithm is $O(n^2 \log n + n m)$,
where $n$ is the number of nodes and $m$ the number of edges in the
graph. For dense graphs, this may be faster than the Floyd–Warshall
algorithm.
See Also
--------
floyd_warshall_predecessor_and_distance
floyd_warshall_numpy
all_pairs_shortest_path
all_pairs_shortest_path_length
all_pairs_dijkstra_path
bellman_ford_predecessor_and_distance
all_pairs_bellman_ford_path
all_pairs_bellman_ford_path_length
| def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths
Parameters
----------
G : NetworkX graph
sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.
weight: function
Function with (u, v, data) input that returns that edge's weight
or None to indicate a hidden edge
pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.
paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.
target : node label, optional
Ending node for path. Search is halted when target is found.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.
Raises
------
NodeNotFound
If any of `sources` is not in `G`.
Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.
"""
G_succ = G._adj # For speed-up (and works for both directed and undirected graphs)
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
seen[source] = 0
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist
| (G, weight='weight', *, backend=None, **backend_kwargs) | [
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|
30,827 | networkx.algorithms.tree.operations | join | A deprecated name for `join_trees`
Returns a new rooted tree with a root node joined with the roots
of each of the given rooted trees.
.. deprecated:: 3.2
`join` is deprecated in NetworkX v3.2 and will be removed in v3.4.
It has been renamed join_trees with the same syntax/interface.
| def join(rooted_trees, label_attribute=None):
"""A deprecated name for `join_trees`
Returns a new rooted tree with a root node joined with the roots
of each of the given rooted trees.
.. deprecated:: 3.2
`join` is deprecated in NetworkX v3.2 and will be removed in v3.4.
It has been renamed join_trees with the same syntax/interface.
"""
import warnings
warnings.warn(
"The function `join` is deprecated and is renamed `join_trees`.\n"
"The ``join`` function itself will be removed in v3.4",
DeprecationWarning,
stacklevel=2,
)
return join_trees(rooted_trees, label_attribute=label_attribute)
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|
30,837 | networkx.algorithms.connectivity.edge_augmentation | k_edge_augmentation | Finds set of edges to k-edge-connect G.
Adding edges from the augmentation to G make it impossible to disconnect G
unless k or more edges are removed. This function uses the most efficient
function available (depending on the value of k and if the problem is
weighted or unweighted) to search for a minimum weight subset of available
edges that k-edge-connects G. In general, finding a k-edge-augmentation is
NP-hard, so solutions are not guaranteed to be minimal. Furthermore, a
k-edge-augmentation may not exist.
Parameters
----------
G : NetworkX graph
An undirected graph.
k : integer
Desired edge connectivity
avail : dict or a set of 2 or 3 tuples
The available edges that can be used in the augmentation.
If unspecified, then all edges in the complement of G are available.
Otherwise, each item is an available edge (with an optional weight).
In the unweighted case, each item is an edge ``(u, v)``.
In the weighted case, each item is a 3-tuple ``(u, v, d)`` or a dict
with items ``(u, v): d``. The third item, ``d``, can be a dictionary
or a real number. If ``d`` is a dictionary ``d[weight]``
correspondings to the weight.
weight : string
key to use to find weights if ``avail`` is a set of 3-tuples where the
third item in each tuple is a dictionary.
partial : boolean
If partial is True and no feasible k-edge-augmentation exists, then all
a partial k-edge-augmentation is generated. Adding the edges in a
partial augmentation to G, minimizes the number of k-edge-connected
components and maximizes the edge connectivity between those
components. For details, see :func:`partial_k_edge_augmentation`.
Yields
------
edge : tuple
Edges that, once added to G, would cause G to become k-edge-connected.
If partial is False, an error is raised if this is not possible.
Otherwise, generated edges form a partial augmentation, which
k-edge-connects any part of G where it is possible, and maximally
connects the remaining parts.
Raises
------
NetworkXUnfeasible
If partial is False and no k-edge-augmentation exists.
NetworkXNotImplemented
If the input graph is directed or a multigraph.
ValueError:
If k is less than 1
Notes
-----
When k=1 this returns an optimal solution.
When k=2 and ``avail`` is None, this returns an optimal solution.
Otherwise when k=2, this returns a 2-approximation of the optimal solution.
For k>3, this problem is NP-hard and this uses a randomized algorithm that
produces a feasible solution, but provides no guarantees on the
solution weight.
Examples
--------
>>> # Unweighted cases
>>> G = nx.path_graph((1, 2, 3, 4))
>>> G.add_node(5)
>>> sorted(nx.k_edge_augmentation(G, k=1))
[(1, 5)]
>>> sorted(nx.k_edge_augmentation(G, k=2))
[(1, 5), (5, 4)]
>>> sorted(nx.k_edge_augmentation(G, k=3))
[(1, 4), (1, 5), (2, 5), (3, 5), (4, 5)]
>>> complement = list(nx.k_edge_augmentation(G, k=5, partial=True))
>>> G.add_edges_from(complement)
>>> nx.edge_connectivity(G)
4
>>> # Weighted cases
>>> G = nx.path_graph((1, 2, 3, 4))
>>> G.add_node(5)
>>> # avail can be a tuple with a dict
>>> avail = [(1, 5, {"weight": 11}), (2, 5, {"weight": 10})]
>>> sorted(nx.k_edge_augmentation(G, k=1, avail=avail, weight="weight"))
[(2, 5)]
>>> # or avail can be a 3-tuple with a real number
>>> avail = [(1, 5, 11), (2, 5, 10), (4, 3, 1), (4, 5, 51)]
>>> sorted(nx.k_edge_augmentation(G, k=2, avail=avail))
[(1, 5), (2, 5), (4, 5)]
>>> # or avail can be a dict
>>> avail = {(1, 5): 11, (2, 5): 10, (4, 3): 1, (4, 5): 51}
>>> sorted(nx.k_edge_augmentation(G, k=2, avail=avail))
[(1, 5), (2, 5), (4, 5)]
>>> # If augmentation is infeasible, then a partial solution can be found
>>> avail = {(1, 5): 11}
>>> sorted(nx.k_edge_augmentation(G, k=2, avail=avail, partial=True))
[(1, 5)]
| def unconstrained_bridge_augmentation(G):
"""Finds an optimal 2-edge-augmentation of G using the fewest edges.
This is an implementation of the algorithm detailed in [1]_.
The basic idea is to construct a meta-graph of bridge-ccs, connect leaf
nodes of the trees to connect the entire graph, and finally connect the
leafs of the tree in dfs-preorder to bridge connect the entire graph.
Parameters
----------
G : NetworkX graph
An undirected graph.
Yields
------
edge : tuple
Edges in the bridge augmentation of G
Notes
-----
Input: a graph G.
First find the bridge components of G and collapse each bridge-cc into a
node of a metagraph graph C, which is guaranteed to be a forest of trees.
C contains p "leafs" --- nodes with exactly one incident edge.
C contains q "isolated nodes" --- nodes with no incident edges.
Theorem: If p + q > 1, then at least :math:`ceil(p / 2) + q` edges are
needed to bridge connect C. This algorithm achieves this min number.
The method first adds enough edges to make G into a tree and then pairs
leafs in a simple fashion.
Let n be the number of trees in C. Let v(i) be an isolated vertex in the
i-th tree if one exists, otherwise it is a pair of distinct leafs nodes
in the i-th tree. Alternating edges from these sets (i.e. adding edges
A1 = [(v(i)[0], v(i + 1)[1]), v(i + 1)[0], v(i + 2)[1])...]) connects C
into a tree T. This tree has p' = p + 2q - 2(n -1) leafs and no isolated
vertices. A1 has n - 1 edges. The next step finds ceil(p' / 2) edges to
biconnect any tree with p' leafs.
Convert T into an arborescence T' by picking an arbitrary root node with
degree >= 2 and directing all edges away from the root. Note the
implementation implicitly constructs T'.
The leafs of T are the nodes with no existing edges in T'.
Order the leafs of T' by DFS preorder. Then break this list in half
and add the zipped pairs to A2.
The set A = A1 + A2 is the minimum augmentation in the metagraph.
To convert this to edges in the original graph
References
----------
.. [1] Eswaran, Kapali P., and R. Endre Tarjan. (1975) Augmentation problems.
http://epubs.siam.org/doi/abs/10.1137/0205044
See Also
--------
:func:`bridge_augmentation`
:func:`k_edge_augmentation`
Examples
--------
>>> G = nx.path_graph((1, 2, 3, 4, 5, 6, 7))
>>> sorted(unconstrained_bridge_augmentation(G))
[(1, 7)]
>>> G = nx.path_graph((1, 2, 3, 2, 4, 5, 6, 7))
>>> sorted(unconstrained_bridge_augmentation(G))
[(1, 3), (3, 7)]
>>> G = nx.Graph([(0, 1), (0, 2), (1, 2)])
>>> G.add_node(4)
>>> sorted(unconstrained_bridge_augmentation(G))
[(1, 4), (4, 0)]
"""
# -----
# Mapping of terms from (Eswaran and Tarjan):
# G = G_0 - the input graph
# C = G_0' - the bridge condensation of G. (This is a forest of trees)
# A1 = A_1 - the edges to connect the forest into a tree
# leaf = pendant - a node with degree of 1
# alpha(v) = maps the node v in G to its meta-node in C
# beta(x) = maps the meta-node x in C to any node in the bridge
# component of G corresponding to x.
# find the 2-edge-connected components of G
bridge_ccs = list(nx.connectivity.bridge_components(G))
# condense G into an forest C
C = collapse(G, bridge_ccs)
# Choose pairs of distinct leaf nodes in each tree. If this is not
# possible then make a pair using the single isolated node in the tree.
vset1 = [
tuple(cc) * 2 # case1: an isolated node
if len(cc) == 1
else sorted(cc, key=C.degree)[0:2] # case2: pair of leaf nodes
for cc in nx.connected_components(C)
]
if len(vset1) > 1:
# Use this set to construct edges that connect C into a tree.
nodes1 = [vs[0] for vs in vset1]
nodes2 = [vs[1] for vs in vset1]
A1 = list(zip(nodes1[1:], nodes2))
else:
A1 = []
# Connect each tree in the forest to construct an arborescence
T = C.copy()
T.add_edges_from(A1)
# If there are only two leaf nodes, we simply connect them.
leafs = [n for n, d in T.degree() if d == 1]
if len(leafs) == 1:
A2 = []
if len(leafs) == 2:
A2 = [tuple(leafs)]
else:
# Choose an arbitrary non-leaf root
try:
root = next(n for n, d in T.degree() if d > 1)
except StopIteration: # no nodes found with degree > 1
return
# order the leaves of C by (induced directed) preorder
v2 = [n for n in nx.dfs_preorder_nodes(T, root) if T.degree(n) == 1]
# connecting first half of the leafs in pre-order to the second
# half will bridge connect the tree with the fewest edges.
half = math.ceil(len(v2) / 2)
A2 = list(zip(v2[:half], v2[-half:]))
# collect the edges used to augment the original forest
aug_tree_edges = A1 + A2
# Construct the mapping (beta) from meta-nodes to regular nodes
inverse = defaultdict(list)
for k, v in C.graph["mapping"].items():
inverse[v].append(k)
# sort so we choose minimum degree nodes first
inverse = {
mu: sorted(mapped, key=lambda u: (G.degree(u), u))
for mu, mapped in inverse.items()
}
# For each meta-edge, map back to an arbitrary pair in the original graph
G2 = G.copy()
for mu, mv in aug_tree_edges:
# Find the first available edge that doesn't exist and return it
for u, v in it.product(inverse[mu], inverse[mv]):
if not G2.has_edge(u, v):
G2.add_edge(u, v)
yield u, v
break
| (G, k, avail=None, weight=None, partial=False, *, backend=None, **backend_kwargs) | [
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|
30,844 | networkx.drawing.layout | kamada_kawai_layout | Position nodes using Kamada-Kawai path-length cost-function.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
dist : dict (default=None)
A two-level dictionary of optimal distances between nodes,
indexed by source and destination node.
If None, the distance is computed using shortest_path_length().
pos : dict or None optional (default=None)
Initial positions for nodes as a dictionary with node as keys
and values as a coordinate list or tuple. If None, then use
circular_layout() for dim >= 2 and a linear layout for dim == 1.
weight : string or None optional (default='weight')
The edge attribute that holds the numerical value used for
the edge weight. If None, then all edge weights are 1.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.kamada_kawai_layout(G)
| def kamada_kawai_layout(
G, dist=None, pos=None, weight="weight", scale=1, center=None, dim=2
):
"""Position nodes using Kamada-Kawai path-length cost-function.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
dist : dict (default=None)
A two-level dictionary of optimal distances between nodes,
indexed by source and destination node.
If None, the distance is computed using shortest_path_length().
pos : dict or None optional (default=None)
Initial positions for nodes as a dictionary with node as keys
and values as a coordinate list or tuple. If None, then use
circular_layout() for dim >= 2 and a linear layout for dim == 1.
weight : string or None optional (default='weight')
The edge attribute that holds the numerical value used for
the edge weight. If None, then all edge weights are 1.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
dim : int
Dimension of layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node
Examples
--------
>>> G = nx.path_graph(4)
>>> pos = nx.kamada_kawai_layout(G)
"""
import numpy as np
G, center = _process_params(G, center, dim)
nNodes = len(G)
if nNodes == 0:
return {}
if dist is None:
dist = dict(nx.shortest_path_length(G, weight=weight))
dist_mtx = 1e6 * np.ones((nNodes, nNodes))
for row, nr in enumerate(G):
if nr not in dist:
continue
rdist = dist[nr]
for col, nc in enumerate(G):
if nc not in rdist:
continue
dist_mtx[row][col] = rdist[nc]
if pos is None:
if dim >= 3:
pos = random_layout(G, dim=dim)
elif dim == 2:
pos = circular_layout(G, dim=dim)
else:
pos = dict(zip(G, np.linspace(0, 1, len(G))))
pos_arr = np.array([pos[n] for n in G])
pos = _kamada_kawai_solve(dist_mtx, pos_arr, dim)
pos = rescale_layout(pos, scale=scale) + center
return dict(zip(G, pos))
| (G, dist=None, pos=None, weight='weight', scale=1, center=None, dim=2) | [
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|
30,849 | networkx.algorithms.distance_measures | kemeny_constant | Returns the Kemeny constant of the given graph.
The *Kemeny constant* (or Kemeny's constant) of a graph `G`
can be computed by regarding the graph as a Markov chain.
The Kemeny constant is then the expected number of time steps
to transition from a starting state i to a random destination state
sampled from the Markov chain's stationary distribution.
The Kemeny constant is independent of the chosen initial state [1]_.
The Kemeny constant measures the time needed for spreading
across a graph. Low values indicate a closely connected graph
whereas high values indicate a spread-out graph.
If weight is not provided, then a weight of 1 is used for all edges.
Since `G` represents a Markov chain, the weights must be positive.
Parameters
----------
G : NetworkX graph
weight : string or None, optional (default=None)
The edge data key used to compute the Kemeny constant.
If None, then each edge has weight 1.
Returns
-------
float
The Kemeny constant of the graph `G`.
Raises
------
NetworkXNotImplemented
If the graph `G` is directed.
NetworkXError
If the graph `G` is not connected, or contains no nodes,
or has edges with negative weights.
Examples
--------
>>> G = nx.complete_graph(5)
>>> round(nx.kemeny_constant(G), 10)
3.2
Notes
-----
The implementation is based on equation (3.3) in [2]_.
Self-loops are allowed and indicate a Markov chain where
the state can remain the same. Multi-edges are contracted
in one edge with weight equal to the sum of the weights.
References
----------
.. [1] Wikipedia
"Kemeny's constant."
https://en.wikipedia.org/wiki/Kemeny%27s_constant
.. [2] Lovász L.
Random walks on graphs: A survey.
Paul Erdös is Eighty, vol. 2, Bolyai Society,
Mathematical Studies, Keszthely, Hungary (1993), pp. 1-46
| def effective_graph_resistance(G, weight=None, invert_weight=True):
"""Returns the Effective graph resistance of G.
Also known as the Kirchhoff index.
The effective graph resistance is defined as the sum
of the resistance distance of every node pair in G [1]_.
If weight is not provided, then a weight of 1 is used for all edges.
The effective graph resistance of a disconnected graph is infinite.
Parameters
----------
G : NetworkX graph
A graph
weight : string or None, optional (default=None)
The edge data key used to compute the effective graph resistance.
If None, then each edge has weight 1.
invert_weight : boolean (default=True)
Proper calculation of resistance distance requires building the
Laplacian matrix with the reciprocal of the weight. Not required
if the weight is already inverted. Weight cannot be zero.
Returns
-------
RG : float
The effective graph resistance of `G`.
Raises
------
NetworkXNotImplemented
If `G` is a directed graph.
NetworkXError
If `G` does not contain any nodes.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (1, 4), (3, 4), (3, 5), (4, 5)])
>>> round(nx.effective_graph_resistance(G), 10)
10.25
Notes
-----
The implementation is based on Theorem 2.2 in [2]_. Self-loops are ignored.
Multi-edges are contracted in one edge with weight equal to the harmonic sum of the weights.
References
----------
.. [1] Wolfram
"Kirchhoff Index."
https://mathworld.wolfram.com/KirchhoffIndex.html
.. [2] W. Ellens, F. M. Spieksma, P. Van Mieghem, A. Jamakovic, R. E. Kooij.
Effective graph resistance.
Lin. Alg. Appl. 435:2491-2506, 2011.
"""
import numpy as np
if len(G) == 0:
raise nx.NetworkXError("Graph G must contain at least one node.")
# Disconnected graphs have infinite Effective graph resistance
if not nx.is_connected(G):
return float("inf")
# Invert weights
G = G.copy()
if invert_weight and weight is not None:
if G.is_multigraph():
for u, v, k, d in G.edges(keys=True, data=True):
d[weight] = 1 / d[weight]
else:
for u, v, d in G.edges(data=True):
d[weight] = 1 / d[weight]
# Get Laplacian eigenvalues
mu = np.sort(nx.laplacian_spectrum(G, weight=weight))
# Compute Effective graph resistance based on spectrum of the Laplacian
# Self-loops are ignored
return float(np.sum(1 / mu[1:]) * G.number_of_nodes())
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|
30,851 | networkx.generators.classic | kneser_graph | Returns the Kneser Graph with parameters `n` and `k`.
The Kneser Graph has nodes that are k-tuples (subsets) of the integers
between 0 and ``n-1``. Nodes are adjacent if their corresponding sets are disjoint.
Parameters
----------
n: int
Number of integers from which to make node subsets.
Subsets are drawn from ``set(range(n))``.
k: int
Size of the subsets.
Returns
-------
G : NetworkX Graph
Examples
--------
>>> G = nx.kneser_graph(5, 2)
>>> G.number_of_nodes()
10
>>> G.number_of_edges()
15
>>> nx.is_isomorphic(G, nx.petersen_graph())
True
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (n, k, *, backend=None, **backend_kwargs) | [
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|
30,853 | networkx.generators.small | krackhardt_kite_graph |
Returns the Krackhardt Kite Social Network.
A 10 actor social network introduced by David Krackhardt
to illustrate different centrality measures [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Krackhardt Kite graph with 10 nodes and 18 edges
Notes
-----
The traditional labeling is:
Andre=1, Beverley=2, Carol=3, Diane=4,
Ed=5, Fernando=6, Garth=7, Heather=8, Ike=9, Jane=10.
References
----------
.. [1] Krackhardt, David. "Assessing the Political Landscape: Structure,
Cognition, and Power in Organizations". Administrative Science Quarterly.
35 (2): 342–369. doi:10.2307/2393394. JSTOR 2393394. June 1990.
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,854 | networkx.generators.classic | ladder_graph | Returns the Ladder graph of length n.
This is two paths of n nodes, with
each pair connected by a single edge.
Node labels are the integers 0 to 2*n - 1.
.. plot::
>>> nx.draw(nx.ladder_graph(5))
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (n, create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,867 | networkx.algorithms.dag | lexicographical_topological_sort | Generate the nodes in the unique lexicographical topological sort order.
Generates a unique ordering of nodes by first sorting topologically (for which there are often
multiple valid orderings) and then additionally by sorting lexicographically.
A topological sort arranges the nodes of a directed graph so that the
upstream node of each directed edge precedes the downstream node.
It is always possible to find a solution for directed graphs that have no cycles.
There may be more than one valid solution.
Lexicographical sorting is just sorting alphabetically. It is used here to break ties in the
topological sort and to determine a single, unique ordering. This can be useful in comparing
sort results.
The lexicographical order can be customized by providing a function to the `key=` parameter.
The definition of the key function is the same as used in python's built-in `sort()`.
The function takes a single argument and returns a key to use for sorting purposes.
Lexicographical sorting can fail if the node names are un-sortable. See the example below.
The solution is to provide a function to the `key=` argument that returns sortable keys.
Parameters
----------
G : NetworkX digraph
A directed acyclic graph (DAG)
key : function, optional
A function of one argument that converts a node name to a comparison key.
It defines and resolves ambiguities in the sort order. Defaults to the identity function.
Yields
------
nodes
Yields the nodes of G in lexicographical topological sort order.
Raises
------
NetworkXError
Topological sort is defined for directed graphs only. If the graph `G`
is undirected, a :exc:`NetworkXError` is raised.
NetworkXUnfeasible
If `G` is not a directed acyclic graph (DAG) no topological sort exists
and a :exc:`NetworkXUnfeasible` exception is raised. This can also be
raised if `G` is changed while the returned iterator is being processed
RuntimeError
If `G` is changed while the returned iterator is being processed.
TypeError
Results from un-sortable node names.
Consider using `key=` parameter to resolve ambiguities in the sort order.
Examples
--------
>>> DG = nx.DiGraph([(2, 1), (2, 5), (1, 3), (1, 4), (5, 4)])
>>> list(nx.lexicographical_topological_sort(DG))
[2, 1, 3, 5, 4]
>>> list(nx.lexicographical_topological_sort(DG, key=lambda x: -x))
[2, 5, 1, 4, 3]
The sort will fail for any graph with integer and string nodes. Comparison of integer to strings
is not defined in python. Is 3 greater or less than 'red'?
>>> DG = nx.DiGraph([(1, "red"), (3, "red"), (1, "green"), (2, "blue")])
>>> list(nx.lexicographical_topological_sort(DG))
Traceback (most recent call last):
...
TypeError: '<' not supported between instances of 'str' and 'int'
...
Incomparable nodes can be resolved using a `key` function. This example function
allows comparison of integers and strings by returning a tuple where the first
element is True for `str`, False otherwise. The second element is the node name.
This groups the strings and integers separately so they can be compared only among themselves.
>>> key = lambda node: (isinstance(node, str), node)
>>> list(nx.lexicographical_topological_sort(DG, key=key))
[1, 2, 3, 'blue', 'green', 'red']
Notes
-----
This algorithm is based on a description and proof in
"Introduction to Algorithms: A Creative Approach" [1]_ .
See also
--------
topological_sort
References
----------
.. [1] Manber, U. (1989).
*Introduction to Algorithms - A Creative Approach.* Addison-Wesley.
| def transitive_closure_dag(G, topo_order=None):
"""Returns the transitive closure of a directed acyclic graph.
This function is faster than the function `transitive_closure`, but fails
if the graph has a cycle.
The transitive closure of G = (V,E) is a graph G+ = (V,E+) such that
for all v, w in V there is an edge (v, w) in E+ if and only if there
is a non-null path from v to w in G.
Parameters
----------
G : NetworkX DiGraph
A directed acyclic graph (DAG)
topo_order: list or tuple, optional
A topological order for G (if None, the function will compute one)
Returns
-------
NetworkX DiGraph
The transitive closure of `G`
Raises
------
NetworkXNotImplemented
If `G` is not directed
NetworkXUnfeasible
If `G` has a cycle
Examples
--------
>>> DG = nx.DiGraph([(1, 2), (2, 3)])
>>> TC = nx.transitive_closure_dag(DG)
>>> TC.edges()
OutEdgeView([(1, 2), (1, 3), (2, 3)])
Notes
-----
This algorithm is probably simple enough to be well-known but I didn't find
a mention in the literature.
"""
if topo_order is None:
topo_order = list(topological_sort(G))
TC = G.copy()
# idea: traverse vertices following a reverse topological order, connecting
# each vertex to its descendants at distance 2 as we go
for v in reversed(topo_order):
TC.add_edges_from((v, u) for u in nx.descendants_at_distance(TC, v, 2))
return TC
| (G, key=None, *, backend=None, **backend_kwargs) | [
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|
30,879 | networkx.generators.classic | lollipop_graph | Returns the Lollipop Graph; ``K_m`` connected to ``P_n``.
This is the Barbell Graph without the right barbell.
.. plot::
>>> nx.draw(nx.lollipop_graph(3, 4))
Parameters
----------
m, n : int or iterable container of nodes
If an integer, nodes are from ``range(m)`` and ``range(m, m+n)``.
If a container of nodes, those nodes appear in the graph.
Warning: `m` and `n` are not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
The nodes for `m` appear in the complete graph $K_m$ and the nodes
for `n` appear in the path $P_n$
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
Networkx graph
A complete graph with `m` nodes connected to a path of length `n`.
Notes
-----
The 2 subgraphs are joined via an edge ``(m-1, m)``.
If ``n=0``, this is merely a complete graph.
(This graph is an extremal example in David Aldous and Jim
Fill's etext on Random Walks on Graphs.)
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (m, n, create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,888 | networkx.algorithms.matching | max_weight_matching | Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
Parameters
----------
G : NetworkX graph
Undirected graph
maxcardinality: bool, optional (default=False)
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]
Notes
-----
If G has edges with weight attributes the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primal-dual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
Bipartite graphs can also be matched using the functions present in
:mod:`networkx.algorithms.bipartite.matching`.
References
----------
.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
| @not_implemented_for("multigraph")
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def max_weight_matching(G, maxcardinality=False, weight="weight"):
"""Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
Parameters
----------
G : NetworkX graph
Undirected graph
maxcardinality: bool, optional (default=False)
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]
Notes
-----
If G has edges with weight attributes the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primal-dual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
Bipartite graphs can also be matched using the functions present in
:mod:`networkx.algorithms.bipartite.matching`.
References
----------
.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primal-dual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
class Blossom:
"""Representation of a non-trivial blossom or sub-blossom."""
__slots__ = ["childs", "edges", "mybestedges"]
# b.childs is an ordered list of b's sub-blossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a top-level S-blossom,
# b.mybestedges is a list of least-slack edges to neighboring
# S-blossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
stack = [*self.childs]
while stack:
t = stack.pop()
if isinstance(t, Blossom):
stack.extend(t.childs)
else:
yield t
# Get a list of vertices.
gnodes = list(G)
if not gnodes:
return set() # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for i, j, d in G.edges(data=True):
wt = d.get(weight, 1)
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (str(type(wt)).split("'")[1] in ("int", "long"))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = {}
# If b is a top-level blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an S-blossom,
# 2 if b is a T-blossom.
# The label of a vertex is found by looking at the label of its top-level
# containing blossom.
# If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
# from an S-vertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = {}
# If b is a labeled top-level blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a T-blossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = {}
# If v is a vertex, inblossom[v] is the top-level blossom to which v
# belongs.
# If v is a top-level vertex, inblossom[v] == v since v is itself
# a (trivial) top-level blossom.
# Initially all vertices are top-level trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a sub-blossom,
# blossomparent[b] is its immediate parent (sub-)blossom.
# If b is a top-level blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub-)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a T-blossom),
# bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
# or None if there is no such edge.
# If b is a (possibly trivial) top-level S-blossom,
# bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = {}
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a non-trivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = {}
# If (v, w) in allowedge or (w, v) in allowedg, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = {}
# Queue of newly discovered S-vertices.
queue = []
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
def slack(v, w):
return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1)
# Assign label t to the top-level blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an S-vertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a T-vertex/blossom; assign label S to its mate.
# (If b is a non-trivial blos | (G, maxcardinality=False, weight='weight', *, backend=None, **backend_kwargs) | [
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|
30,890 | networkx.algorithms.matching | maximal_matching | Find a maximal matching in the graph.
A matching is a subset of edges in which no node occurs more than once.
A maximal matching cannot add more edges and still be a matching.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph([(1, 2), (1, 3), (2, 3), (2, 4), (3, 5), (4, 5)])
>>> sorted(nx.maximal_matching(G))
[(1, 2), (3, 5)]
Notes
-----
The algorithm greedily selects a maximal matching M of the graph G
(i.e. no superset of M exists). It runs in $O(|E|)$ time.
| @not_implemented_for("multigraph")
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def max_weight_matching(G, maxcardinality=False, weight="weight"):
"""Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
Parameters
----------
G : NetworkX graph
Undirected graph
maxcardinality: bool, optional (default=False)
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]
Notes
-----
If G has edges with weight attributes the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primal-dual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
Bipartite graphs can also be matched using the functions present in
:mod:`networkx.algorithms.bipartite.matching`.
References
----------
.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primal-dual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
class Blossom:
"""Representation of a non-trivial blossom or sub-blossom."""
__slots__ = ["childs", "edges", "mybestedges"]
# b.childs is an ordered list of b's sub-blossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a top-level S-blossom,
# b.mybestedges is a list of least-slack edges to neighboring
# S-blossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
stack = [*self.childs]
while stack:
t = stack.pop()
if isinstance(t, Blossom):
stack.extend(t.childs)
else:
yield t
# Get a list of vertices.
gnodes = list(G)
if not gnodes:
return set() # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for i, j, d in G.edges(data=True):
wt = d.get(weight, 1)
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (str(type(wt)).split("'")[1] in ("int", "long"))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = {}
# If b is a top-level blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an S-blossom,
# 2 if b is a T-blossom.
# The label of a vertex is found by looking at the label of its top-level
# containing blossom.
# If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
# from an S-vertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = {}
# If b is a labeled top-level blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a T-blossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = {}
# If v is a vertex, inblossom[v] is the top-level blossom to which v
# belongs.
# If v is a top-level vertex, inblossom[v] == v since v is itself
# a (trivial) top-level blossom.
# Initially all vertices are top-level trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a sub-blossom,
# blossomparent[b] is its immediate parent (sub-)blossom.
# If b is a top-level blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub-)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a T-blossom),
# bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
# or None if there is no such edge.
# If b is a (possibly trivial) top-level S-blossom,
# bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = {}
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a non-trivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = {}
# If (v, w) in allowedge or (w, v) in allowedg, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = {}
# Queue of newly discovered S-vertices.
queue = []
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
def slack(v, w):
return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1)
# Assign label t to the top-level blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an S-vertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a T-vertex/blossom; assign label S to its mate.
# (If b is a non-trivial blos | (G, *, backend=None, **backend_kwargs) | [
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|
30,891 | networkx.algorithms.tree.branchings | maximum_branching |
Returns a maximum branching from G.
Parameters
----------
G : (multi)digraph-like
The graph to be searched.
attr : str
The edge attribute used to in determining optimality.
default : float
The value of the edge attribute used if an edge does not have
the attribute `attr`.
preserve_attrs : bool
If True, preserve the other attributes of the original graph (that are not
passed to `attr`)
partition : str
The key for the edge attribute containing the partition
data on the graph. Edges can be included, excluded or open using the
`EdgePartition` enum.
Returns
-------
B : (multi)digraph-like
A maximum branching.
| @nx._dispatchable(preserve_edge_attrs=True, returns_graph=True)
def maximum_branching(
G,
attr="weight",
default=1,
preserve_attrs=False,
partition=None,
):
#######################################
### Data Structure Helper Functions ###
#######################################
def edmonds_add_edge(G, edge_index, u, v, key, **d):
"""
Adds an edge to `G` while also updating the edge index.
This algorithm requires the use of an external dictionary to track
the edge keys since it is possible that the source or destination
node of an edge will be changed and the default key-handling
capabilities of the MultiDiGraph class do not account for this.
Parameters
----------
G : MultiDiGraph
The graph to insert an edge into.
edge_index : dict
A mapping from integers to the edges of the graph.
u : node
The source node of the new edge.
v : node
The destination node of the new edge.
key : int
The key to use from `edge_index`.
d : keyword arguments, optional
Other attributes to store on the new edge.
"""
if key in edge_index:
uu, vv, _ = edge_index[key]
if (u != uu) or (v != vv):
raise Exception(f"Key {key!r} is already in use.")
G.add_edge(u, v, key, **d)
edge_index[key] = (u, v, G.succ[u][v][key])
def edmonds_remove_node(G, edge_index, n):
"""
Remove a node from the graph, updating the edge index to match.
Parameters
----------
G : MultiDiGraph
The graph to remove an edge from.
edge_index : dict
A mapping from integers to the edges of the graph.
n : node
The node to remove from `G`.
"""
keys = set()
for keydict in G.pred[n].values():
keys.update(keydict)
for keydict in G.succ[n].values():
keys.update(keydict)
for key in keys:
del edge_index[key]
G.remove_node(n)
#######################
### Algorithm Setup ###
#######################
# Pick an attribute name that the original graph is unlikly to have
candidate_attr = "edmonds' secret candidate attribute"
new_node_base_name = "edmonds new node base name "
G_original = G
G = nx.MultiDiGraph()
G.__networkx_cache__ = None # Disable caching
# A dict to reliably track mutations to the edges using the key of the edge.
G_edge_index = {}
# Each edge is given an arbitrary numerical key
for key, (u, v, data) in enumerate(G_original.edges(data=True)):
d = {attr: data.get(attr, default)}
if data.get(partition) is not None:
d[partition] = data.get(partition)
if preserve_attrs:
for d_k, d_v in data.items():
if d_k != attr:
d[d_k] = d_v
edmonds_add_edge(G, G_edge_index, u, v, key, **d)
level = 0 # Stores the number of contracted nodes
# These are the buckets from the paper.
#
# In the paper, G^i are modified versions of the original graph.
# D^i and E^i are the nodes and edges of the maximal edges that are
# consistent with G^i. In this implementation, D^i and E^i are stored
# together as the graph B^i. We will have strictly more B^i then the
# paper will have.
#
# Note that the data in graphs and branchings are tuples with the graph as
# the first element and the edge index as the second.
B = nx.MultiDiGraph()
B_edge_index = {}
graphs = [] # G^i list
branchings = [] # B^i list
selected_nodes = set() # D^i bucket
uf = nx.utils.UnionFind()
# A list of lists of edge indices. Each list is a circuit for graph G^i.
# Note the edge list is not required to be a circuit in G^0.
circuits = []
# Stores the index of the minimum edge in the circuit found in G^i and B^i.
# The ordering of the edges seems to preserver the weight ordering from
# G^0. So even if the circuit does not form a circuit in G^0, it is still
# true that the minimum edges in circuit G^0 (despite their weights being
# different)
minedge_circuit = []
###########################
### Algorithm Structure ###
###########################
# Each step listed in the algorithm is an inner function. Thus, the overall
# loop structure is:
#
# while True:
# step_I1()
# if cycle detected:
# step_I2()
# elif every node of G is in D and E is a branching:
# break
##################################
### Algorithm Helper Functions ###
##################################
def edmonds_find_desired_edge(v):
"""
Find the edge directed towards v with maximal weight.
If an edge partition exists in this graph, return the included
edge if it exists and never return any excluded edge.
Note: There can only be one included edge for each vertex otherwise
the edge partition is empty.
Parameters
----------
v : node
The node to search for the maximal weight incoming edge.
"""
edge = None
max_weight = -INF
for u, _, key, data in G.in_edges(v, data=True, keys=True):
# Skip excluded edges
if data.get(partition) == nx.EdgePartition.EXCLUDED:
continue
new_weight = data[attr]
# Return the included edge
if data.get(partition) == nx.EdgePartition.INCLUDED:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
break
# Find the best open edge
if new_weight > max_weight:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
return edge, max_weight
def edmonds_step_I2(v, desired_edge, level):
"""
Perform step I2 from Edmonds' paper
First, check if the last step I1 created a cycle. If it did not, do nothing.
If it did, store the cycle for later reference and contract it.
Parameters
----------
v : node
The current node to consider
desired_edge : edge
The minimum desired edge to remove from the cycle.
level : int
The current level, i.e. the number of cycles that have already been removed.
"""
u = desired_edge[0]
Q_nodes = nx.shortest_path(B, v, u)
Q_edges = [
list(B[Q_nodes[i]][vv].keys())[0] for i, vv in enumerate(Q_nodes[1:])
]
Q_edges.append(desired_edge[2]) # Add the new edge key to complete the circuit
# Get the edge in the circuit with the minimum weight.
# Also, save the incoming weights for each node.
minweight = INF
minedge = None
Q_incoming_weight = {}
for edge_key in Q_edges:
u, v, data = B_edge_index[edge_key]
w = data[attr]
# We cannot remove an included edge, even if it is the
# minimum edge in the circuit
Q_incoming_weight[v] = w
if data.get(partition) == nx.EdgePartition.INCLUDED:
continue
if w < minweight:
minweight = w
minedge = edge_key
circuits.append(Q_edges)
minedge_circuit.append(minedge)
graphs.append((G.copy(), G_edge_index.copy()))
branchings.append((B.copy(), B_edge_index.copy()))
# Mutate the graph to contract the circuit
new_node = new_node_base_name + str(level)
G.add_node(new_node)
new_edges = []
for u, v, key, data in G.edges(data=True, keys=True):
if u in Q_incoming_weight:
if v in Q_incoming_weight:
# Circuit edge. For the moment do nothing,
# eventually it will be removed.
continue
else:
# | (G, attr='weight', default=1, preserve_attrs=False, partition=None, *, backend=None, **backend_kwargs) | [
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|
30,894 | networkx.algorithms.tree.branchings | maximum_spanning_arborescence |
Returns a maximum spanning arborescence from G.
Parameters
----------
G : (multi)digraph-like
The graph to be searched.
attr : str
The edge attribute used to in determining optimality.
default : float
The value of the edge attribute used if an edge does not have
the attribute `attr`.
preserve_attrs : bool
If True, preserve the other attributes of the original graph (that are not
passed to `attr`)
partition : str
The key for the edge attribute containing the partition
data on the graph. Edges can be included, excluded or open using the
`EdgePartition` enum.
Returns
-------
B : (multi)digraph-like
A maximum spanning arborescence.
Raises
------
NetworkXException
If the graph does not contain a maximum spanning arborescence.
| @nx._dispatchable(preserve_edge_attrs=True, returns_graph=True)
def maximum_branching(
G,
attr="weight",
default=1,
preserve_attrs=False,
partition=None,
):
#######################################
### Data Structure Helper Functions ###
#######################################
def edmonds_add_edge(G, edge_index, u, v, key, **d):
"""
Adds an edge to `G` while also updating the edge index.
This algorithm requires the use of an external dictionary to track
the edge keys since it is possible that the source or destination
node of an edge will be changed and the default key-handling
capabilities of the MultiDiGraph class do not account for this.
Parameters
----------
G : MultiDiGraph
The graph to insert an edge into.
edge_index : dict
A mapping from integers to the edges of the graph.
u : node
The source node of the new edge.
v : node
The destination node of the new edge.
key : int
The key to use from `edge_index`.
d : keyword arguments, optional
Other attributes to store on the new edge.
"""
if key in edge_index:
uu, vv, _ = edge_index[key]
if (u != uu) or (v != vv):
raise Exception(f"Key {key!r} is already in use.")
G.add_edge(u, v, key, **d)
edge_index[key] = (u, v, G.succ[u][v][key])
def edmonds_remove_node(G, edge_index, n):
"""
Remove a node from the graph, updating the edge index to match.
Parameters
----------
G : MultiDiGraph
The graph to remove an edge from.
edge_index : dict
A mapping from integers to the edges of the graph.
n : node
The node to remove from `G`.
"""
keys = set()
for keydict in G.pred[n].values():
keys.update(keydict)
for keydict in G.succ[n].values():
keys.update(keydict)
for key in keys:
del edge_index[key]
G.remove_node(n)
#######################
### Algorithm Setup ###
#######################
# Pick an attribute name that the original graph is unlikly to have
candidate_attr = "edmonds' secret candidate attribute"
new_node_base_name = "edmonds new node base name "
G_original = G
G = nx.MultiDiGraph()
G.__networkx_cache__ = None # Disable caching
# A dict to reliably track mutations to the edges using the key of the edge.
G_edge_index = {}
# Each edge is given an arbitrary numerical key
for key, (u, v, data) in enumerate(G_original.edges(data=True)):
d = {attr: data.get(attr, default)}
if data.get(partition) is not None:
d[partition] = data.get(partition)
if preserve_attrs:
for d_k, d_v in data.items():
if d_k != attr:
d[d_k] = d_v
edmonds_add_edge(G, G_edge_index, u, v, key, **d)
level = 0 # Stores the number of contracted nodes
# These are the buckets from the paper.
#
# In the paper, G^i are modified versions of the original graph.
# D^i and E^i are the nodes and edges of the maximal edges that are
# consistent with G^i. In this implementation, D^i and E^i are stored
# together as the graph B^i. We will have strictly more B^i then the
# paper will have.
#
# Note that the data in graphs and branchings are tuples with the graph as
# the first element and the edge index as the second.
B = nx.MultiDiGraph()
B_edge_index = {}
graphs = [] # G^i list
branchings = [] # B^i list
selected_nodes = set() # D^i bucket
uf = nx.utils.UnionFind()
# A list of lists of edge indices. Each list is a circuit for graph G^i.
# Note the edge list is not required to be a circuit in G^0.
circuits = []
# Stores the index of the minimum edge in the circuit found in G^i and B^i.
# The ordering of the edges seems to preserver the weight ordering from
# G^0. So even if the circuit does not form a circuit in G^0, it is still
# true that the minimum edges in circuit G^0 (despite their weights being
# different)
minedge_circuit = []
###########################
### Algorithm Structure ###
###########################
# Each step listed in the algorithm is an inner function. Thus, the overall
# loop structure is:
#
# while True:
# step_I1()
# if cycle detected:
# step_I2()
# elif every node of G is in D and E is a branching:
# break
##################################
### Algorithm Helper Functions ###
##################################
def edmonds_find_desired_edge(v):
"""
Find the edge directed towards v with maximal weight.
If an edge partition exists in this graph, return the included
edge if it exists and never return any excluded edge.
Note: There can only be one included edge for each vertex otherwise
the edge partition is empty.
Parameters
----------
v : node
The node to search for the maximal weight incoming edge.
"""
edge = None
max_weight = -INF
for u, _, key, data in G.in_edges(v, data=True, keys=True):
# Skip excluded edges
if data.get(partition) == nx.EdgePartition.EXCLUDED:
continue
new_weight = data[attr]
# Return the included edge
if data.get(partition) == nx.EdgePartition.INCLUDED:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
break
# Find the best open edge
if new_weight > max_weight:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
return edge, max_weight
def edmonds_step_I2(v, desired_edge, level):
"""
Perform step I2 from Edmonds' paper
First, check if the last step I1 created a cycle. If it did not, do nothing.
If it did, store the cycle for later reference and contract it.
Parameters
----------
v : node
The current node to consider
desired_edge : edge
The minimum desired edge to remove from the cycle.
level : int
The current level, i.e. the number of cycles that have already been removed.
"""
u = desired_edge[0]
Q_nodes = nx.shortest_path(B, v, u)
Q_edges = [
list(B[Q_nodes[i]][vv].keys())[0] for i, vv in enumerate(Q_nodes[1:])
]
Q_edges.append(desired_edge[2]) # Add the new edge key to complete the circuit
# Get the edge in the circuit with the minimum weight.
# Also, save the incoming weights for each node.
minweight = INF
minedge = None
Q_incoming_weight = {}
for edge_key in Q_edges:
u, v, data = B_edge_index[edge_key]
w = data[attr]
# We cannot remove an included edge, even if it is the
# minimum edge in the circuit
Q_incoming_weight[v] = w
if data.get(partition) == nx.EdgePartition.INCLUDED:
continue
if w < minweight:
minweight = w
minedge = edge_key
circuits.append(Q_edges)
minedge_circuit.append(minedge)
graphs.append((G.copy(), G_edge_index.copy()))
branchings.append((B.copy(), B_edge_index.copy()))
# Mutate the graph to contract the circuit
new_node = new_node_base_name + str(level)
G.add_node(new_node)
new_edges = []
for u, v, key, data in G.edges(data=True, keys=True):
if u in Q_incoming_weight:
if v in Q_incoming_weight:
# Circuit edge. For the moment do nothing,
# eventually it will be removed.
continue
else:
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|
30,895 | networkx.algorithms.tree.mst | maximum_spanning_edges | Generate edges in a maximum spanning forest of an undirected
weighted graph.
A maximum spanning tree is a subgraph of the graph (a tree)
with the maximum possible sum of edge weights. A spanning forest is a
union of the spanning trees for each connected component of the graph.
Parameters
----------
G : undirected Graph
An undirected graph. If `G` is connected, then the algorithm finds a
spanning tree. Otherwise, a spanning forest is found.
algorithm : string
The algorithm to use when finding a maximum spanning tree. Valid
choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'.
weight : string
Edge data key to use for weight (default 'weight').
keys : bool
Whether to yield edge key in multigraphs in addition to the edge.
If `G` is not a multigraph, this is ignored.
data : bool, optional
If True yield the edge data along with the edge.
ignore_nan : bool (default: False)
If a NaN is found as an edge weight normally an exception is raised.
If `ignore_nan is True` then that edge is ignored instead.
Returns
-------
edges : iterator
An iterator over edges in a maximum spanning tree of `G`.
Edges connecting nodes `u` and `v` are represented as tuples:
`(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)`
If `G` is a multigraph, `keys` indicates whether the edge key `k` will
be reported in the third position in the edge tuple. `data` indicates
whether the edge datadict `d` will appear at the end of the edge tuple.
If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True
or `(u, v)` if `data` is False.
Examples
--------
>>> from networkx.algorithms import tree
Find maximum spanning edges by Kruskal's algorithm
>>> G = nx.cycle_graph(4)
>>> G.add_edge(0, 3, weight=2)
>>> mst = tree.maximum_spanning_edges(G, algorithm="kruskal", data=False)
>>> edgelist = list(mst)
>>> sorted(sorted(e) for e in edgelist)
[[0, 1], [0, 3], [1, 2]]
Find maximum spanning edges by Prim's algorithm
>>> G = nx.cycle_graph(4)
>>> G.add_edge(0, 3, weight=2) # assign weight 2 to edge 0-3
>>> mst = tree.maximum_spanning_edges(G, algorithm="prim", data=False)
>>> edgelist = list(mst)
>>> sorted(sorted(e) for e in edgelist)
[[0, 1], [0, 3], [2, 3]]
Notes
-----
For Borůvka's algorithm, each edge must have a weight attribute, and
each edge weight must be distinct.
For the other algorithms, if the graph edges do not have a weight
attribute a default weight of 1 will be used.
Modified code from David Eppstein, April 2006
http://www.ics.uci.edu/~eppstein/PADS/
| def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None):
"""
Sample a random spanning tree using the edges weights of `G`.
This function supports two different methods for determining the
probability of the graph. If ``multiplicative=True``, the probability
is based on the product of edge weights, and if ``multiplicative=False``
it is based on the sum of the edge weight. However, since it is
easier to determine the total weight of all spanning trees for the
multiplicative version, that is significantly faster and should be used if
possible. Additionally, setting `weight` to `None` will cause a spanning tree
to be selected with uniform probability.
The function uses algorithm A8 in [1]_ .
Parameters
----------
G : nx.Graph
An undirected version of the original graph.
weight : string
The edge key for the edge attribute holding edge weight.
multiplicative : bool, default=True
If `True`, the probability of each tree is the product of its edge weight
over the sum of the product of all the spanning trees in the graph. If
`False`, the probability is the sum of its edge weight over the sum of
the sum of weights for all spanning trees in the graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
nx.Graph
A spanning tree using the distribution defined by the weight of the tree.
References
----------
.. [1] V. Kulkarni, Generating random combinatorial objects, Journal of
Algorithms, 11 (1990), pp. 185–207
"""
def find_node(merged_nodes, node):
"""
We can think of clusters of contracted nodes as having one
representative in the graph. Each node which is not in merged_nodes
is still its own representative. Since a representative can be later
contracted, we need to recursively search though the dict to find
the final representative, but once we know it we can use path
compression to speed up the access of the representative for next time.
This cannot be replaced by the standard NetworkX union_find since that
data structure will merge nodes with less representing nodes into the
one with more representing nodes but this function requires we merge
them using the order that contract_edges contracts using.
Parameters
----------
merged_nodes : dict
The dict storing the mapping from node to representative
node
The node whose representative we seek
Returns
-------
The representative of the `node`
"""
if node not in merged_nodes:
return node
else:
rep = find_node(merged_nodes, merged_nodes[node])
merged_nodes[node] = rep
return rep
def prepare_graph():
"""
For the graph `G`, remove all edges not in the set `V` and then
contract all edges in the set `U`.
Returns
-------
A copy of `G` which has had all edges not in `V` removed and all edges
in `U` contracted.
"""
# The result is a MultiGraph version of G so that parallel edges are
# allowed during edge contraction
result = nx.MultiGraph(incoming_graph_data=G)
# Remove all edges not in V
edges_to_remove = set(result.edges()).difference(V)
result.remove_edges_from(edges_to_remove)
# Contract all edges in U
#
# Imagine that you have two edges to contract and they share an
# endpoint like this:
# [0] ----- [1] ----- [2]
# If we contract (0, 1) first, the contraction function will always
# delete the second node it is passed so the resulting graph would be
# [0] ----- [2]
# and edge (1, 2) no longer exists but (0, 2) would need to be contracted
# in its place now. That is why I use the below dict as a merge-find
# data structure with path compression to track how the nodes are merged.
merged_nodes = {}
for u, v in U:
u_rep = find_node(merged_nodes, u)
v_rep = find_node(merged_nodes, v)
# We cannot contract a node with itself
if u_rep == v_rep:
continue
nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False)
merged_nodes[v_rep] = u_rep
return merged_nodes, result
def spanning_tree_total_weight(G, weight):
"""
Find the sum of weights of the spanning trees of `G` using the
appropriate `method`.
This is easy if the chosen method is 'multiplicative', since we can
use Kirchhoff's Tree Matrix Theorem directly. However, with the
'additive' method, this process is slightly more complex and less
computationally efficient as we have to find the number of spanning
trees which contain each possible edge in the graph.
Parameters
----------
G : NetworkX Graph
The graph to find the total weight of all spanning trees on.
weight : string
The key for the weight edge attribute of the graph.
Returns
-------
float
The sum of either the multiplicative or additive weight for all
spanning trees in the graph.
"""
if multiplicative:
return nx.total_spanning_tree_weight(G, weight)
else:
# There are two cases for the total spanning tree additive weight.
# 1. There is one edge in the graph. Then the only spanning tree is
# that edge itself, which will have a total weight of that edge
# itself.
if G.number_of_edges() == 1:
return G.edges(data=weight).__iter__().__next__()[2]
# 2. There are no edges or two or more edges in the graph. Then, we find the
# total weight of the spanning trees using the formula in the
# reference paper: take the weight of each edge and multiply it by
# the number of spanning trees which include that edge. This
# can be accomplished by contracting the edge and finding the
# multiplicative total spanning tree weight if the weight of each edge
# is assumed to be 1, which is conveniently built into networkx already,
# by calling total_spanning_tree_weight with weight=None.
# Note that with no edges the returned value is just zero.
else:
total = 0
for u, v, w in G.edges(data=weight):
total += w * nx.total_spanning_tree_weight(
nx.contracted_edge(G, edge=(u, v), self_loops=False), None
)
return total
if G.number_of_nodes() < 2:
# no edges in the spanning tree
return nx.empty_graph(G.nodes)
U = set()
st_cached_value = 0
V = set(G.edges())
shuffled_edges = list(G.edges())
seed.shuffle(shuffled_edges)
for u, v in shuffled_edges:
e_weight = G[u][v][weight] if weight is not None else 1
node_map, prepared_G = prepare_graph()
G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight)
# Add the edge to U so that we can compute the total tree weight
# assuming we include that edge
# Now, if (u, v) cannot exist in G because it is fully contracted out
# of existence, then it by definition cannot influence G_e's Kirchhoff
# value. But, we also cannot pick it.
rep_edge = (find_node(node_map, u), find_node(node_map, v))
# Check to see if the 'representative edge' for the current edge is
# in prepared_G. If so, then we can pick it.
if rep_edge in prepared_G.edges:
prepared_G_e = nx.contracted_edge(
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|
30,896 | networkx.algorithms.tree.mst | maximum_spanning_tree | Returns a maximum spanning tree or forest on an undirected graph `G`.
Parameters
----------
G : undirected graph
An undirected graph. If `G` is connected, then the algorithm finds a
spanning tree. Otherwise, a spanning forest is found.
weight : str
Data key to use for edge weights.
algorithm : string
The algorithm to use when finding a maximum spanning tree. Valid
choices are 'kruskal', 'prim', or 'boruvka'. The default is
'kruskal'.
ignore_nan : bool (default: False)
If a NaN is found as an edge weight normally an exception is raised.
If `ignore_nan is True` then that edge is ignored instead.
Returns
-------
G : NetworkX Graph
A maximum spanning tree or forest.
Examples
--------
>>> G = nx.cycle_graph(4)
>>> G.add_edge(0, 3, weight=2)
>>> T = nx.maximum_spanning_tree(G)
>>> sorted(T.edges(data=True))
[(0, 1, {}), (0, 3, {'weight': 2}), (1, 2, {})]
Notes
-----
For Borůvka's algorithm, each edge must have a weight attribute, and
each edge weight must be distinct.
For the other algorithms, if the graph edges do not have a weight
attribute a default weight of 1 will be used.
There may be more than one tree with the same minimum or maximum weight.
See :mod:`networkx.tree.recognition` for more detailed definitions.
Isolated nodes with self-loops are in the tree as edgeless isolated nodes.
| def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None):
"""
Sample a random spanning tree using the edges weights of `G`.
This function supports two different methods for determining the
probability of the graph. If ``multiplicative=True``, the probability
is based on the product of edge weights, and if ``multiplicative=False``
it is based on the sum of the edge weight. However, since it is
easier to determine the total weight of all spanning trees for the
multiplicative version, that is significantly faster and should be used if
possible. Additionally, setting `weight` to `None` will cause a spanning tree
to be selected with uniform probability.
The function uses algorithm A8 in [1]_ .
Parameters
----------
G : nx.Graph
An undirected version of the original graph.
weight : string
The edge key for the edge attribute holding edge weight.
multiplicative : bool, default=True
If `True`, the probability of each tree is the product of its edge weight
over the sum of the product of all the spanning trees in the graph. If
`False`, the probability is the sum of its edge weight over the sum of
the sum of weights for all spanning trees in the graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
nx.Graph
A spanning tree using the distribution defined by the weight of the tree.
References
----------
.. [1] V. Kulkarni, Generating random combinatorial objects, Journal of
Algorithms, 11 (1990), pp. 185–207
"""
def find_node(merged_nodes, node):
"""
We can think of clusters of contracted nodes as having one
representative in the graph. Each node which is not in merged_nodes
is still its own representative. Since a representative can be later
contracted, we need to recursively search though the dict to find
the final representative, but once we know it we can use path
compression to speed up the access of the representative for next time.
This cannot be replaced by the standard NetworkX union_find since that
data structure will merge nodes with less representing nodes into the
one with more representing nodes but this function requires we merge
them using the order that contract_edges contracts using.
Parameters
----------
merged_nodes : dict
The dict storing the mapping from node to representative
node
The node whose representative we seek
Returns
-------
The representative of the `node`
"""
if node not in merged_nodes:
return node
else:
rep = find_node(merged_nodes, merged_nodes[node])
merged_nodes[node] = rep
return rep
def prepare_graph():
"""
For the graph `G`, remove all edges not in the set `V` and then
contract all edges in the set `U`.
Returns
-------
A copy of `G` which has had all edges not in `V` removed and all edges
in `U` contracted.
"""
# The result is a MultiGraph version of G so that parallel edges are
# allowed during edge contraction
result = nx.MultiGraph(incoming_graph_data=G)
# Remove all edges not in V
edges_to_remove = set(result.edges()).difference(V)
result.remove_edges_from(edges_to_remove)
# Contract all edges in U
#
# Imagine that you have two edges to contract and they share an
# endpoint like this:
# [0] ----- [1] ----- [2]
# If we contract (0, 1) first, the contraction function will always
# delete the second node it is passed so the resulting graph would be
# [0] ----- [2]
# and edge (1, 2) no longer exists but (0, 2) would need to be contracted
# in its place now. That is why I use the below dict as a merge-find
# data structure with path compression to track how the nodes are merged.
merged_nodes = {}
for u, v in U:
u_rep = find_node(merged_nodes, u)
v_rep = find_node(merged_nodes, v)
# We cannot contract a node with itself
if u_rep == v_rep:
continue
nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False)
merged_nodes[v_rep] = u_rep
return merged_nodes, result
def spanning_tree_total_weight(G, weight):
"""
Find the sum of weights of the spanning trees of `G` using the
appropriate `method`.
This is easy if the chosen method is 'multiplicative', since we can
use Kirchhoff's Tree Matrix Theorem directly. However, with the
'additive' method, this process is slightly more complex and less
computationally efficient as we have to find the number of spanning
trees which contain each possible edge in the graph.
Parameters
----------
G : NetworkX Graph
The graph to find the total weight of all spanning trees on.
weight : string
The key for the weight edge attribute of the graph.
Returns
-------
float
The sum of either the multiplicative or additive weight for all
spanning trees in the graph.
"""
if multiplicative:
return nx.total_spanning_tree_weight(G, weight)
else:
# There are two cases for the total spanning tree additive weight.
# 1. There is one edge in the graph. Then the only spanning tree is
# that edge itself, which will have a total weight of that edge
# itself.
if G.number_of_edges() == 1:
return G.edges(data=weight).__iter__().__next__()[2]
# 2. There are no edges or two or more edges in the graph. Then, we find the
# total weight of the spanning trees using the formula in the
# reference paper: take the weight of each edge and multiply it by
# the number of spanning trees which include that edge. This
# can be accomplished by contracting the edge and finding the
# multiplicative total spanning tree weight if the weight of each edge
# is assumed to be 1, which is conveniently built into networkx already,
# by calling total_spanning_tree_weight with weight=None.
# Note that with no edges the returned value is just zero.
else:
total = 0
for u, v, w in G.edges(data=weight):
total += w * nx.total_spanning_tree_weight(
nx.contracted_edge(G, edge=(u, v), self_loops=False), None
)
return total
if G.number_of_nodes() < 2:
# no edges in the spanning tree
return nx.empty_graph(G.nodes)
U = set()
st_cached_value = 0
V = set(G.edges())
shuffled_edges = list(G.edges())
seed.shuffle(shuffled_edges)
for u, v in shuffled_edges:
e_weight = G[u][v][weight] if weight is not None else 1
node_map, prepared_G = prepare_graph()
G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight)
# Add the edge to U so that we can compute the total tree weight
# assuming we include that edge
# Now, if (u, v) cannot exist in G because it is fully contracted out
# of existence, then it by definition cannot influence G_e's Kirchhoff
# value. But, we also cannot pick it.
rep_edge = (find_node(node_map, u), find_node(node_map, v))
# Check to see if the 'representative edge' for the current edge is
# in prepared_G. If so, then we can pick it.
if rep_edge in prepared_G.edges:
prepared_G_e = nx.contracted_edge(
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|
30,901 | networkx.algorithms.matching | min_weight_matching | Computing a minimum-weight maximal matching of G.
Use the maximum-weight algorithm with edge weights subtracted
from the maximum weight of all edges.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
This method replaces the edge weights with 1 plus the maximum edge weight
minus the original edge weight.
new_weight = (max_weight + 1) - edge_weight
then runs :func:`max_weight_matching` with the new weights.
The max weight matching with these new weights corresponds
to the min weight matching using the original weights.
Adding 1 to the max edge weight keeps all edge weights positive
and as integers if they started as integers.
You might worry that adding 1 to each weight would make the algorithm
favor matchings with more edges. But we use the parameter
`maxcardinality=True` in `max_weight_matching` to ensure that the
number of edges in the competing matchings are the same and thus
the optimum does not change due to changes in the number of edges.
Read the documentation of `max_weight_matching` for more information.
Parameters
----------
G : NetworkX graph
Undirected graph
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A minimal weight matching of the graph.
See Also
--------
max_weight_matching
| @not_implemented_for("multigraph")
@not_implemented_for("directed")
@nx._dispatchable(edge_attrs="weight")
def max_weight_matching(G, maxcardinality=False, weight="weight"):
"""Compute a maximum-weighted matching of G.
A matching is a subset of edges in which no node occurs more than once.
The weight of a matching is the sum of the weights of its edges.
A maximal matching cannot add more edges and still be a matching.
The cardinality of a matching is the number of matched edges.
Parameters
----------
G : NetworkX graph
Undirected graph
maxcardinality: bool, optional (default=False)
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
weight: string, optional (default='weight')
Edge data key corresponding to the edge weight.
If key not found, uses 1 as weight.
Returns
-------
matching : set
A maximal matching of the graph.
Examples
--------
>>> G = nx.Graph()
>>> edges = [(1, 2, 6), (1, 3, 2), (2, 3, 1), (2, 4, 7), (3, 5, 9), (4, 5, 3)]
>>> G.add_weighted_edges_from(edges)
>>> sorted(nx.max_weight_matching(G))
[(2, 4), (5, 3)]
Notes
-----
If G has edges with weight attributes the edge data are used as
weight values else the weights are assumed to be 1.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm
could return a slightly suboptimal matching due to numeric
precision errors.
This method is based on the "blossom" method for finding augmenting
paths and the "primal-dual" method for finding a matching of maximum
weight, both methods invented by Jack Edmonds [1]_.
Bipartite graphs can also be matched using the functions present in
:mod:`networkx.algorithms.bipartite.matching`.
References
----------
.. [1] "Efficient Algorithms for Finding Maximum Matching in Graphs",
Zvi Galil, ACM Computing Surveys, 1986.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primal-dual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
class Blossom:
"""Representation of a non-trivial blossom or sub-blossom."""
__slots__ = ["childs", "edges", "mybestedges"]
# b.childs is an ordered list of b's sub-blossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a top-level S-blossom,
# b.mybestedges is a list of least-slack edges to neighboring
# S-blossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
stack = [*self.childs]
while stack:
t = stack.pop()
if isinstance(t, Blossom):
stack.extend(t.childs)
else:
yield t
# Get a list of vertices.
gnodes = list(G)
if not gnodes:
return set() # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for i, j, d in G.edges(data=True):
wt = d.get(weight, 1)
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (str(type(wt)).split("'")[1] in ("int", "long"))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = {}
# If b is a top-level blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an S-blossom,
# 2 if b is a T-blossom.
# The label of a vertex is found by looking at the label of its top-level
# containing blossom.
# If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
# from an S-vertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = {}
# If b is a labeled top-level blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a T-blossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = {}
# If v is a vertex, inblossom[v] is the top-level blossom to which v
# belongs.
# If v is a top-level vertex, inblossom[v] == v since v is itself
# a (trivial) top-level blossom.
# Initially all vertices are top-level trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a sub-blossom,
# blossomparent[b] is its immediate parent (sub-)blossom.
# If b is a top-level blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub-)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a T-blossom),
# bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
# or None if there is no such edge.
# If b is a (possibly trivial) top-level S-blossom,
# bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = {}
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a non-trivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = {}
# If (v, w) in allowedge or (w, v) in allowedg, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = {}
# Queue of newly discovered S-vertices.
queue = []
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
def slack(v, w):
return dualvar[v] + dualvar[w] - 2 * G[v][w].get(weight, 1)
# Assign label t to the top-level blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an S-vertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a T-vertex/blossom; assign label S to its mate.
# (If b is a non-trivial blos | (G, weight='weight', *, backend=None, **backend_kwargs) | [
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|
30,902 | networkx.algorithms.d_separation | minimal_d_separator | Returns a minimal_d-separating set between `x` and `y` if possible
.. deprecated:: 3.3
minimal_d_separator is deprecated and will be removed in NetworkX v3.5.
Please use `find_minimal_d_separator(G, x, y)`.
| def minimal_d_separator(G, u, v):
"""Returns a minimal_d-separating set between `x` and `y` if possible
.. deprecated:: 3.3
minimal_d_separator is deprecated and will be removed in NetworkX v3.5.
Please use `find_minimal_d_separator(G, x, y)`.
"""
import warnings
warnings.warn(
(
"This function is deprecated and will be removed in NetworkX v3.5."
"Please use `is_d_separator(G, x, y)`."
),
category=DeprecationWarning,
stacklevel=2,
)
return nx.find_minimal_d_separator(G, u, v)
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|
30,903 | networkx.algorithms.tree.branchings | minimum_branching |
Returns a minimum branching from G.
Parameters
----------
G : (multi)digraph-like
The graph to be searched.
attr : str
The edge attribute used to in determining optimality.
default : float
The value of the edge attribute used if an edge does not have
the attribute `attr`.
preserve_attrs : bool
If True, preserve the other attributes of the original graph (that are not
passed to `attr`)
partition : str
The key for the edge attribute containing the partition
data on the graph. Edges can be included, excluded or open using the
`EdgePartition` enum.
Returns
-------
B : (multi)digraph-like
A minimum branching.
See Also
--------
minimal_branching
| @nx._dispatchable(preserve_edge_attrs=True, returns_graph=True)
def maximum_branching(
G,
attr="weight",
default=1,
preserve_attrs=False,
partition=None,
):
#######################################
### Data Structure Helper Functions ###
#######################################
def edmonds_add_edge(G, edge_index, u, v, key, **d):
"""
Adds an edge to `G` while also updating the edge index.
This algorithm requires the use of an external dictionary to track
the edge keys since it is possible that the source or destination
node of an edge will be changed and the default key-handling
capabilities of the MultiDiGraph class do not account for this.
Parameters
----------
G : MultiDiGraph
The graph to insert an edge into.
edge_index : dict
A mapping from integers to the edges of the graph.
u : node
The source node of the new edge.
v : node
The destination node of the new edge.
key : int
The key to use from `edge_index`.
d : keyword arguments, optional
Other attributes to store on the new edge.
"""
if key in edge_index:
uu, vv, _ = edge_index[key]
if (u != uu) or (v != vv):
raise Exception(f"Key {key!r} is already in use.")
G.add_edge(u, v, key, **d)
edge_index[key] = (u, v, G.succ[u][v][key])
def edmonds_remove_node(G, edge_index, n):
"""
Remove a node from the graph, updating the edge index to match.
Parameters
----------
G : MultiDiGraph
The graph to remove an edge from.
edge_index : dict
A mapping from integers to the edges of the graph.
n : node
The node to remove from `G`.
"""
keys = set()
for keydict in G.pred[n].values():
keys.update(keydict)
for keydict in G.succ[n].values():
keys.update(keydict)
for key in keys:
del edge_index[key]
G.remove_node(n)
#######################
### Algorithm Setup ###
#######################
# Pick an attribute name that the original graph is unlikly to have
candidate_attr = "edmonds' secret candidate attribute"
new_node_base_name = "edmonds new node base name "
G_original = G
G = nx.MultiDiGraph()
G.__networkx_cache__ = None # Disable caching
# A dict to reliably track mutations to the edges using the key of the edge.
G_edge_index = {}
# Each edge is given an arbitrary numerical key
for key, (u, v, data) in enumerate(G_original.edges(data=True)):
d = {attr: data.get(attr, default)}
if data.get(partition) is not None:
d[partition] = data.get(partition)
if preserve_attrs:
for d_k, d_v in data.items():
if d_k != attr:
d[d_k] = d_v
edmonds_add_edge(G, G_edge_index, u, v, key, **d)
level = 0 # Stores the number of contracted nodes
# These are the buckets from the paper.
#
# In the paper, G^i are modified versions of the original graph.
# D^i and E^i are the nodes and edges of the maximal edges that are
# consistent with G^i. In this implementation, D^i and E^i are stored
# together as the graph B^i. We will have strictly more B^i then the
# paper will have.
#
# Note that the data in graphs and branchings are tuples with the graph as
# the first element and the edge index as the second.
B = nx.MultiDiGraph()
B_edge_index = {}
graphs = [] # G^i list
branchings = [] # B^i list
selected_nodes = set() # D^i bucket
uf = nx.utils.UnionFind()
# A list of lists of edge indices. Each list is a circuit for graph G^i.
# Note the edge list is not required to be a circuit in G^0.
circuits = []
# Stores the index of the minimum edge in the circuit found in G^i and B^i.
# The ordering of the edges seems to preserver the weight ordering from
# G^0. So even if the circuit does not form a circuit in G^0, it is still
# true that the minimum edges in circuit G^0 (despite their weights being
# different)
minedge_circuit = []
###########################
### Algorithm Structure ###
###########################
# Each step listed in the algorithm is an inner function. Thus, the overall
# loop structure is:
#
# while True:
# step_I1()
# if cycle detected:
# step_I2()
# elif every node of G is in D and E is a branching:
# break
##################################
### Algorithm Helper Functions ###
##################################
def edmonds_find_desired_edge(v):
"""
Find the edge directed towards v with maximal weight.
If an edge partition exists in this graph, return the included
edge if it exists and never return any excluded edge.
Note: There can only be one included edge for each vertex otherwise
the edge partition is empty.
Parameters
----------
v : node
The node to search for the maximal weight incoming edge.
"""
edge = None
max_weight = -INF
for u, _, key, data in G.in_edges(v, data=True, keys=True):
# Skip excluded edges
if data.get(partition) == nx.EdgePartition.EXCLUDED:
continue
new_weight = data[attr]
# Return the included edge
if data.get(partition) == nx.EdgePartition.INCLUDED:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
break
# Find the best open edge
if new_weight > max_weight:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
return edge, max_weight
def edmonds_step_I2(v, desired_edge, level):
"""
Perform step I2 from Edmonds' paper
First, check if the last step I1 created a cycle. If it did not, do nothing.
If it did, store the cycle for later reference and contract it.
Parameters
----------
v : node
The current node to consider
desired_edge : edge
The minimum desired edge to remove from the cycle.
level : int
The current level, i.e. the number of cycles that have already been removed.
"""
u = desired_edge[0]
Q_nodes = nx.shortest_path(B, v, u)
Q_edges = [
list(B[Q_nodes[i]][vv].keys())[0] for i, vv in enumerate(Q_nodes[1:])
]
Q_edges.append(desired_edge[2]) # Add the new edge key to complete the circuit
# Get the edge in the circuit with the minimum weight.
# Also, save the incoming weights for each node.
minweight = INF
minedge = None
Q_incoming_weight = {}
for edge_key in Q_edges:
u, v, data = B_edge_index[edge_key]
w = data[attr]
# We cannot remove an included edge, even if it is the
# minimum edge in the circuit
Q_incoming_weight[v] = w
if data.get(partition) == nx.EdgePartition.INCLUDED:
continue
if w < minweight:
minweight = w
minedge = edge_key
circuits.append(Q_edges)
minedge_circuit.append(minedge)
graphs.append((G.copy(), G_edge_index.copy()))
branchings.append((B.copy(), B_edge_index.copy()))
# Mutate the graph to contract the circuit
new_node = new_node_base_name + str(level)
G.add_node(new_node)
new_edges = []
for u, v, key, data in G.edges(data=True, keys=True):
if u in Q_incoming_weight:
if v in Q_incoming_weight:
# Circuit edge. For the moment do nothing,
# eventually it will be removed.
continue
else:
# | (G, attr='weight', default=1, preserve_attrs=False, partition=None, *, backend=None, **backend_kwargs) | [
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|
30,906 | networkx.algorithms.cycles | minimum_cycle_basis | Returns a minimum weight cycle basis for G
Minimum weight means a cycle basis for which the total weight
(length for unweighted graphs) of all the cycles is minimum.
Parameters
----------
G : NetworkX Graph
weight: string
name of the edge attribute to use for edge weights
Returns
-------
A list of cycle lists. Each cycle list is a list of nodes
which forms a cycle (loop) in G. Note that the nodes are not
necessarily returned in a order by which they appear in the cycle
Examples
--------
>>> G = nx.Graph()
>>> nx.add_cycle(G, [0, 1, 2, 3])
>>> nx.add_cycle(G, [0, 3, 4, 5])
>>> nx.minimum_cycle_basis(G)
[[5, 4, 3, 0], [3, 2, 1, 0]]
References:
[1] Kavitha, Telikepalli, et al. "An O(m^2n) Algorithm for
Minimum Cycle Basis of Graphs."
http://link.springer.com/article/10.1007/s00453-007-9064-z
[2] de Pina, J. 1995. Applications of shortest path methods.
Ph.D. thesis, University of Amsterdam, Netherlands
See Also
--------
simple_cycles, cycle_basis
| def recursive_simple_cycles(G):
"""Find simple cycles (elementary circuits) of a directed graph.
A `simple cycle`, or `elementary circuit`, is a closed path where
no node appears twice. Two elementary circuits are distinct if they
are not cyclic permutations of each other.
This version uses a recursive algorithm to build a list of cycles.
You should probably use the iterator version called simple_cycles().
Warning: This recursive version uses lots of RAM!
It appears in NetworkX for pedagogical value.
Parameters
----------
G : NetworkX DiGraph
A directed graph
Returns
-------
A list of cycles, where each cycle is represented by a list of nodes
along the cycle.
Example:
>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
>>> G = nx.DiGraph(edges)
>>> nx.recursive_simple_cycles(G)
[[0], [2], [0, 1, 2], [0, 2], [1, 2]]
Notes
-----
The implementation follows pp. 79-80 in [1]_.
The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
elementary circuits.
References
----------
.. [1] Finding all the elementary circuits of a directed graph.
D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
https://doi.org/10.1137/0204007
See Also
--------
simple_cycles, cycle_basis
"""
# Jon Olav Vik, 2010-08-09
def _unblock(thisnode):
"""Recursively unblock and remove nodes from B[thisnode]."""
if blocked[thisnode]:
blocked[thisnode] = False
while B[thisnode]:
_unblock(B[thisnode].pop())
def circuit(thisnode, startnode, component):
closed = False # set to True if elementary path is closed
path.append(thisnode)
blocked[thisnode] = True
for nextnode in component[thisnode]: # direct successors of thisnode
if nextnode == startnode:
result.append(path[:])
closed = True
elif not blocked[nextnode]:
if circuit(nextnode, startnode, component):
closed = True
if closed:
_unblock(thisnode)
else:
for nextnode in component[thisnode]:
if thisnode not in B[nextnode]: # TODO: use set for speedup?
B[nextnode].append(thisnode)
path.pop() # remove thisnode from path
return closed
path = [] # stack of nodes in current path
blocked = defaultdict(bool) # vertex: blocked from search?
B = defaultdict(list) # graph portions that yield no elementary circuit
result = [] # list to accumulate the circuits found
# Johnson's algorithm exclude self cycle edges like (v, v)
# To be backward compatible, we record those cycles in advance
# and then remove from subG
for v in G:
if G.has_edge(v, v):
result.append([v])
G.remove_edge(v, v)
# Johnson's algorithm requires some ordering of the nodes.
# They might not be sortable so we assign an arbitrary ordering.
ordering = dict(zip(G, range(len(G))))
for s in ordering:
# Build the subgraph induced by s and following nodes in the ordering
subgraph = G.subgraph(node for node in G if ordering[node] >= ordering[s])
# Find the strongly connected component in the subgraph
# that contains the least node according to the ordering
strongcomp = nx.strongly_connected_components(subgraph)
mincomp = min(strongcomp, key=lambda ns: min(ordering[n] for n in ns))
component = G.subgraph(mincomp)
if len(component) > 1:
# smallest node in the component according to the ordering
startnode = min(component, key=ordering.__getitem__)
for node in component:
blocked[node] = False
B[node][:] = []
dummy = circuit(startnode, startnode, component)
return result
| (G, weight=None, *, backend=None, **backend_kwargs) | [
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|
30,909 | networkx.algorithms.tree.branchings | minimum_spanning_arborescence |
Returns a minimum spanning arborescence from G.
Parameters
----------
G : (multi)digraph-like
The graph to be searched.
attr : str
The edge attribute used to in determining optimality.
default : float
The value of the edge attribute used if an edge does not have
the attribute `attr`.
preserve_attrs : bool
If True, preserve the other attributes of the original graph (that are not
passed to `attr`)
partition : str
The key for the edge attribute containing the partition
data on the graph. Edges can be included, excluded or open using the
`EdgePartition` enum.
Returns
-------
B : (multi)digraph-like
A minimum spanning arborescence.
Raises
------
NetworkXException
If the graph does not contain a minimum spanning arborescence.
| @nx._dispatchable(preserve_edge_attrs=True, returns_graph=True)
def maximum_branching(
G,
attr="weight",
default=1,
preserve_attrs=False,
partition=None,
):
#######################################
### Data Structure Helper Functions ###
#######################################
def edmonds_add_edge(G, edge_index, u, v, key, **d):
"""
Adds an edge to `G` while also updating the edge index.
This algorithm requires the use of an external dictionary to track
the edge keys since it is possible that the source or destination
node of an edge will be changed and the default key-handling
capabilities of the MultiDiGraph class do not account for this.
Parameters
----------
G : MultiDiGraph
The graph to insert an edge into.
edge_index : dict
A mapping from integers to the edges of the graph.
u : node
The source node of the new edge.
v : node
The destination node of the new edge.
key : int
The key to use from `edge_index`.
d : keyword arguments, optional
Other attributes to store on the new edge.
"""
if key in edge_index:
uu, vv, _ = edge_index[key]
if (u != uu) or (v != vv):
raise Exception(f"Key {key!r} is already in use.")
G.add_edge(u, v, key, **d)
edge_index[key] = (u, v, G.succ[u][v][key])
def edmonds_remove_node(G, edge_index, n):
"""
Remove a node from the graph, updating the edge index to match.
Parameters
----------
G : MultiDiGraph
The graph to remove an edge from.
edge_index : dict
A mapping from integers to the edges of the graph.
n : node
The node to remove from `G`.
"""
keys = set()
for keydict in G.pred[n].values():
keys.update(keydict)
for keydict in G.succ[n].values():
keys.update(keydict)
for key in keys:
del edge_index[key]
G.remove_node(n)
#######################
### Algorithm Setup ###
#######################
# Pick an attribute name that the original graph is unlikly to have
candidate_attr = "edmonds' secret candidate attribute"
new_node_base_name = "edmonds new node base name "
G_original = G
G = nx.MultiDiGraph()
G.__networkx_cache__ = None # Disable caching
# A dict to reliably track mutations to the edges using the key of the edge.
G_edge_index = {}
# Each edge is given an arbitrary numerical key
for key, (u, v, data) in enumerate(G_original.edges(data=True)):
d = {attr: data.get(attr, default)}
if data.get(partition) is not None:
d[partition] = data.get(partition)
if preserve_attrs:
for d_k, d_v in data.items():
if d_k != attr:
d[d_k] = d_v
edmonds_add_edge(G, G_edge_index, u, v, key, **d)
level = 0 # Stores the number of contracted nodes
# These are the buckets from the paper.
#
# In the paper, G^i are modified versions of the original graph.
# D^i and E^i are the nodes and edges of the maximal edges that are
# consistent with G^i. In this implementation, D^i and E^i are stored
# together as the graph B^i. We will have strictly more B^i then the
# paper will have.
#
# Note that the data in graphs and branchings are tuples with the graph as
# the first element and the edge index as the second.
B = nx.MultiDiGraph()
B_edge_index = {}
graphs = [] # G^i list
branchings = [] # B^i list
selected_nodes = set() # D^i bucket
uf = nx.utils.UnionFind()
# A list of lists of edge indices. Each list is a circuit for graph G^i.
# Note the edge list is not required to be a circuit in G^0.
circuits = []
# Stores the index of the minimum edge in the circuit found in G^i and B^i.
# The ordering of the edges seems to preserver the weight ordering from
# G^0. So even if the circuit does not form a circuit in G^0, it is still
# true that the minimum edges in circuit G^0 (despite their weights being
# different)
minedge_circuit = []
###########################
### Algorithm Structure ###
###########################
# Each step listed in the algorithm is an inner function. Thus, the overall
# loop structure is:
#
# while True:
# step_I1()
# if cycle detected:
# step_I2()
# elif every node of G is in D and E is a branching:
# break
##################################
### Algorithm Helper Functions ###
##################################
def edmonds_find_desired_edge(v):
"""
Find the edge directed towards v with maximal weight.
If an edge partition exists in this graph, return the included
edge if it exists and never return any excluded edge.
Note: There can only be one included edge for each vertex otherwise
the edge partition is empty.
Parameters
----------
v : node
The node to search for the maximal weight incoming edge.
"""
edge = None
max_weight = -INF
for u, _, key, data in G.in_edges(v, data=True, keys=True):
# Skip excluded edges
if data.get(partition) == nx.EdgePartition.EXCLUDED:
continue
new_weight = data[attr]
# Return the included edge
if data.get(partition) == nx.EdgePartition.INCLUDED:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
break
# Find the best open edge
if new_weight > max_weight:
max_weight = new_weight
edge = (u, v, key, new_weight, data)
return edge, max_weight
def edmonds_step_I2(v, desired_edge, level):
"""
Perform step I2 from Edmonds' paper
First, check if the last step I1 created a cycle. If it did not, do nothing.
If it did, store the cycle for later reference and contract it.
Parameters
----------
v : node
The current node to consider
desired_edge : edge
The minimum desired edge to remove from the cycle.
level : int
The current level, i.e. the number of cycles that have already been removed.
"""
u = desired_edge[0]
Q_nodes = nx.shortest_path(B, v, u)
Q_edges = [
list(B[Q_nodes[i]][vv].keys())[0] for i, vv in enumerate(Q_nodes[1:])
]
Q_edges.append(desired_edge[2]) # Add the new edge key to complete the circuit
# Get the edge in the circuit with the minimum weight.
# Also, save the incoming weights for each node.
minweight = INF
minedge = None
Q_incoming_weight = {}
for edge_key in Q_edges:
u, v, data = B_edge_index[edge_key]
w = data[attr]
# We cannot remove an included edge, even if it is the
# minimum edge in the circuit
Q_incoming_weight[v] = w
if data.get(partition) == nx.EdgePartition.INCLUDED:
continue
if w < minweight:
minweight = w
minedge = edge_key
circuits.append(Q_edges)
minedge_circuit.append(minedge)
graphs.append((G.copy(), G_edge_index.copy()))
branchings.append((B.copy(), B_edge_index.copy()))
# Mutate the graph to contract the circuit
new_node = new_node_base_name + str(level)
G.add_node(new_node)
new_edges = []
for u, v, key, data in G.edges(data=True, keys=True):
if u in Q_incoming_weight:
if v in Q_incoming_weight:
# Circuit edge. For the moment do nothing,
# eventually it will be removed.
continue
else:
# | (G, attr='weight', default=1, preserve_attrs=False, partition=None, *, backend=None, **backend_kwargs) | [
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|
30,910 | networkx.algorithms.tree.mst | minimum_spanning_edges | Generate edges in a minimum spanning forest of an undirected
weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree)
with the minimum sum of edge weights. A spanning forest is a
union of the spanning trees for each connected component of the graph.
Parameters
----------
G : undirected Graph
An undirected graph. If `G` is connected, then the algorithm finds a
spanning tree. Otherwise, a spanning forest is found.
algorithm : string
The algorithm to use when finding a minimum spanning tree. Valid
choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'.
weight : string
Edge data key to use for weight (default 'weight').
keys : bool
Whether to yield edge key in multigraphs in addition to the edge.
If `G` is not a multigraph, this is ignored.
data : bool, optional
If True yield the edge data along with the edge.
ignore_nan : bool (default: False)
If a NaN is found as an edge weight normally an exception is raised.
If `ignore_nan is True` then that edge is ignored instead.
Returns
-------
edges : iterator
An iterator over edges in a maximum spanning tree of `G`.
Edges connecting nodes `u` and `v` are represented as tuples:
`(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)`
If `G` is a multigraph, `keys` indicates whether the edge key `k` will
be reported in the third position in the edge tuple. `data` indicates
whether the edge datadict `d` will appear at the end of the edge tuple.
If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True
or `(u, v)` if `data` is False.
Examples
--------
>>> from networkx.algorithms import tree
Find minimum spanning edges by Kruskal's algorithm
>>> G = nx.cycle_graph(4)
>>> G.add_edge(0, 3, weight=2)
>>> mst = tree.minimum_spanning_edges(G, algorithm="kruskal", data=False)
>>> edgelist = list(mst)
>>> sorted(sorted(e) for e in edgelist)
[[0, 1], [1, 2], [2, 3]]
Find minimum spanning edges by Prim's algorithm
>>> G = nx.cycle_graph(4)
>>> G.add_edge(0, 3, weight=2)
>>> mst = tree.minimum_spanning_edges(G, algorithm="prim", data=False)
>>> edgelist = list(mst)
>>> sorted(sorted(e) for e in edgelist)
[[0, 1], [1, 2], [2, 3]]
Notes
-----
For Borůvka's algorithm, each edge must have a weight attribute, and
each edge weight must be distinct.
For the other algorithms, if the graph edges do not have a weight
attribute a default weight of 1 will be used.
Modified code from David Eppstein, April 2006
http://www.ics.uci.edu/~eppstein/PADS/
| def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None):
"""
Sample a random spanning tree using the edges weights of `G`.
This function supports two different methods for determining the
probability of the graph. If ``multiplicative=True``, the probability
is based on the product of edge weights, and if ``multiplicative=False``
it is based on the sum of the edge weight. However, since it is
easier to determine the total weight of all spanning trees for the
multiplicative version, that is significantly faster and should be used if
possible. Additionally, setting `weight` to `None` will cause a spanning tree
to be selected with uniform probability.
The function uses algorithm A8 in [1]_ .
Parameters
----------
G : nx.Graph
An undirected version of the original graph.
weight : string
The edge key for the edge attribute holding edge weight.
multiplicative : bool, default=True
If `True`, the probability of each tree is the product of its edge weight
over the sum of the product of all the spanning trees in the graph. If
`False`, the probability is the sum of its edge weight over the sum of
the sum of weights for all spanning trees in the graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
nx.Graph
A spanning tree using the distribution defined by the weight of the tree.
References
----------
.. [1] V. Kulkarni, Generating random combinatorial objects, Journal of
Algorithms, 11 (1990), pp. 185–207
"""
def find_node(merged_nodes, node):
"""
We can think of clusters of contracted nodes as having one
representative in the graph. Each node which is not in merged_nodes
is still its own representative. Since a representative can be later
contracted, we need to recursively search though the dict to find
the final representative, but once we know it we can use path
compression to speed up the access of the representative for next time.
This cannot be replaced by the standard NetworkX union_find since that
data structure will merge nodes with less representing nodes into the
one with more representing nodes but this function requires we merge
them using the order that contract_edges contracts using.
Parameters
----------
merged_nodes : dict
The dict storing the mapping from node to representative
node
The node whose representative we seek
Returns
-------
The representative of the `node`
"""
if node not in merged_nodes:
return node
else:
rep = find_node(merged_nodes, merged_nodes[node])
merged_nodes[node] = rep
return rep
def prepare_graph():
"""
For the graph `G`, remove all edges not in the set `V` and then
contract all edges in the set `U`.
Returns
-------
A copy of `G` which has had all edges not in `V` removed and all edges
in `U` contracted.
"""
# The result is a MultiGraph version of G so that parallel edges are
# allowed during edge contraction
result = nx.MultiGraph(incoming_graph_data=G)
# Remove all edges not in V
edges_to_remove = set(result.edges()).difference(V)
result.remove_edges_from(edges_to_remove)
# Contract all edges in U
#
# Imagine that you have two edges to contract and they share an
# endpoint like this:
# [0] ----- [1] ----- [2]
# If we contract (0, 1) first, the contraction function will always
# delete the second node it is passed so the resulting graph would be
# [0] ----- [2]
# and edge (1, 2) no longer exists but (0, 2) would need to be contracted
# in its place now. That is why I use the below dict as a merge-find
# data structure with path compression to track how the nodes are merged.
merged_nodes = {}
for u, v in U:
u_rep = find_node(merged_nodes, u)
v_rep = find_node(merged_nodes, v)
# We cannot contract a node with itself
if u_rep == v_rep:
continue
nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False)
merged_nodes[v_rep] = u_rep
return merged_nodes, result
def spanning_tree_total_weight(G, weight):
"""
Find the sum of weights of the spanning trees of `G` using the
appropriate `method`.
This is easy if the chosen method is 'multiplicative', since we can
use Kirchhoff's Tree Matrix Theorem directly. However, with the
'additive' method, this process is slightly more complex and less
computationally efficient as we have to find the number of spanning
trees which contain each possible edge in the graph.
Parameters
----------
G : NetworkX Graph
The graph to find the total weight of all spanning trees on.
weight : string
The key for the weight edge attribute of the graph.
Returns
-------
float
The sum of either the multiplicative or additive weight for all
spanning trees in the graph.
"""
if multiplicative:
return nx.total_spanning_tree_weight(G, weight)
else:
# There are two cases for the total spanning tree additive weight.
# 1. There is one edge in the graph. Then the only spanning tree is
# that edge itself, which will have a total weight of that edge
# itself.
if G.number_of_edges() == 1:
return G.edges(data=weight).__iter__().__next__()[2]
# 2. There are no edges or two or more edges in the graph. Then, we find the
# total weight of the spanning trees using the formula in the
# reference paper: take the weight of each edge and multiply it by
# the number of spanning trees which include that edge. This
# can be accomplished by contracting the edge and finding the
# multiplicative total spanning tree weight if the weight of each edge
# is assumed to be 1, which is conveniently built into networkx already,
# by calling total_spanning_tree_weight with weight=None.
# Note that with no edges the returned value is just zero.
else:
total = 0
for u, v, w in G.edges(data=weight):
total += w * nx.total_spanning_tree_weight(
nx.contracted_edge(G, edge=(u, v), self_loops=False), None
)
return total
if G.number_of_nodes() < 2:
# no edges in the spanning tree
return nx.empty_graph(G.nodes)
U = set()
st_cached_value = 0
V = set(G.edges())
shuffled_edges = list(G.edges())
seed.shuffle(shuffled_edges)
for u, v in shuffled_edges:
e_weight = G[u][v][weight] if weight is not None else 1
node_map, prepared_G = prepare_graph()
G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight)
# Add the edge to U so that we can compute the total tree weight
# assuming we include that edge
# Now, if (u, v) cannot exist in G because it is fully contracted out
# of existence, then it by definition cannot influence G_e's Kirchhoff
# value. But, we also cannot pick it.
rep_edge = (find_node(node_map, u), find_node(node_map, v))
# Check to see if the 'representative edge' for the current edge is
# in prepared_G. If so, then we can pick it.
if rep_edge in prepared_G.edges:
prepared_G_e = nx.contracted_edge(
prepa | (G, algorithm='kruskal', weight='weight', keys=True, data=True, ignore_nan=False, *, backend=None, **backend_kwargs) | [
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|
30,911 | networkx.algorithms.tree.mst | minimum_spanning_tree | Returns a minimum spanning tree or forest on an undirected graph `G`.
Parameters
----------
G : undirected graph
An undirected graph. If `G` is connected, then the algorithm finds a
spanning tree. Otherwise, a spanning forest is found.
weight : str
Data key to use for edge weights.
algorithm : string
The algorithm to use when finding a minimum spanning tree. Valid
choices are 'kruskal', 'prim', or 'boruvka'. The default is
'kruskal'.
ignore_nan : bool (default: False)
If a NaN is found as an edge weight normally an exception is raised.
If `ignore_nan is True` then that edge is ignored instead.
Returns
-------
G : NetworkX Graph
A minimum spanning tree or forest.
Examples
--------
>>> G = nx.cycle_graph(4)
>>> G.add_edge(0, 3, weight=2)
>>> T = nx.minimum_spanning_tree(G)
>>> sorted(T.edges(data=True))
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
Notes
-----
For Borůvka's algorithm, each edge must have a weight attribute, and
each edge weight must be distinct.
For the other algorithms, if the graph edges do not have a weight
attribute a default weight of 1 will be used.
There may be more than one tree with the same minimum or maximum weight.
See :mod:`networkx.tree.recognition` for more detailed definitions.
Isolated nodes with self-loops are in the tree as edgeless isolated nodes.
| def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None):
"""
Sample a random spanning tree using the edges weights of `G`.
This function supports two different methods for determining the
probability of the graph. If ``multiplicative=True``, the probability
is based on the product of edge weights, and if ``multiplicative=False``
it is based on the sum of the edge weight. However, since it is
easier to determine the total weight of all spanning trees for the
multiplicative version, that is significantly faster and should be used if
possible. Additionally, setting `weight` to `None` will cause a spanning tree
to be selected with uniform probability.
The function uses algorithm A8 in [1]_ .
Parameters
----------
G : nx.Graph
An undirected version of the original graph.
weight : string
The edge key for the edge attribute holding edge weight.
multiplicative : bool, default=True
If `True`, the probability of each tree is the product of its edge weight
over the sum of the product of all the spanning trees in the graph. If
`False`, the probability is the sum of its edge weight over the sum of
the sum of weights for all spanning trees in the graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
nx.Graph
A spanning tree using the distribution defined by the weight of the tree.
References
----------
.. [1] V. Kulkarni, Generating random combinatorial objects, Journal of
Algorithms, 11 (1990), pp. 185–207
"""
def find_node(merged_nodes, node):
"""
We can think of clusters of contracted nodes as having one
representative in the graph. Each node which is not in merged_nodes
is still its own representative. Since a representative can be later
contracted, we need to recursively search though the dict to find
the final representative, but once we know it we can use path
compression to speed up the access of the representative for next time.
This cannot be replaced by the standard NetworkX union_find since that
data structure will merge nodes with less representing nodes into the
one with more representing nodes but this function requires we merge
them using the order that contract_edges contracts using.
Parameters
----------
merged_nodes : dict
The dict storing the mapping from node to representative
node
The node whose representative we seek
Returns
-------
The representative of the `node`
"""
if node not in merged_nodes:
return node
else:
rep = find_node(merged_nodes, merged_nodes[node])
merged_nodes[node] = rep
return rep
def prepare_graph():
"""
For the graph `G`, remove all edges not in the set `V` and then
contract all edges in the set `U`.
Returns
-------
A copy of `G` which has had all edges not in `V` removed and all edges
in `U` contracted.
"""
# The result is a MultiGraph version of G so that parallel edges are
# allowed during edge contraction
result = nx.MultiGraph(incoming_graph_data=G)
# Remove all edges not in V
edges_to_remove = set(result.edges()).difference(V)
result.remove_edges_from(edges_to_remove)
# Contract all edges in U
#
# Imagine that you have two edges to contract and they share an
# endpoint like this:
# [0] ----- [1] ----- [2]
# If we contract (0, 1) first, the contraction function will always
# delete the second node it is passed so the resulting graph would be
# [0] ----- [2]
# and edge (1, 2) no longer exists but (0, 2) would need to be contracted
# in its place now. That is why I use the below dict as a merge-find
# data structure with path compression to track how the nodes are merged.
merged_nodes = {}
for u, v in U:
u_rep = find_node(merged_nodes, u)
v_rep = find_node(merged_nodes, v)
# We cannot contract a node with itself
if u_rep == v_rep:
continue
nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False)
merged_nodes[v_rep] = u_rep
return merged_nodes, result
def spanning_tree_total_weight(G, weight):
"""
Find the sum of weights of the spanning trees of `G` using the
appropriate `method`.
This is easy if the chosen method is 'multiplicative', since we can
use Kirchhoff's Tree Matrix Theorem directly. However, with the
'additive' method, this process is slightly more complex and less
computationally efficient as we have to find the number of spanning
trees which contain each possible edge in the graph.
Parameters
----------
G : NetworkX Graph
The graph to find the total weight of all spanning trees on.
weight : string
The key for the weight edge attribute of the graph.
Returns
-------
float
The sum of either the multiplicative or additive weight for all
spanning trees in the graph.
"""
if multiplicative:
return nx.total_spanning_tree_weight(G, weight)
else:
# There are two cases for the total spanning tree additive weight.
# 1. There is one edge in the graph. Then the only spanning tree is
# that edge itself, which will have a total weight of that edge
# itself.
if G.number_of_edges() == 1:
return G.edges(data=weight).__iter__().__next__()[2]
# 2. There are no edges or two or more edges in the graph. Then, we find the
# total weight of the spanning trees using the formula in the
# reference paper: take the weight of each edge and multiply it by
# the number of spanning trees which include that edge. This
# can be accomplished by contracting the edge and finding the
# multiplicative total spanning tree weight if the weight of each edge
# is assumed to be 1, which is conveniently built into networkx already,
# by calling total_spanning_tree_weight with weight=None.
# Note that with no edges the returned value is just zero.
else:
total = 0
for u, v, w in G.edges(data=weight):
total += w * nx.total_spanning_tree_weight(
nx.contracted_edge(G, edge=(u, v), self_loops=False), None
)
return total
if G.number_of_nodes() < 2:
# no edges in the spanning tree
return nx.empty_graph(G.nodes)
U = set()
st_cached_value = 0
V = set(G.edges())
shuffled_edges = list(G.edges())
seed.shuffle(shuffled_edges)
for u, v in shuffled_edges:
e_weight = G[u][v][weight] if weight is not None else 1
node_map, prepared_G = prepare_graph()
G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight)
# Add the edge to U so that we can compute the total tree weight
# assuming we include that edge
# Now, if (u, v) cannot exist in G because it is fully contracted out
# of existence, then it by definition cannot influence G_e's Kirchhoff
# value. But, we also cannot pick it.
rep_edge = (find_node(node_map, u), find_node(node_map, v))
# Check to see if the 'representative edge' for the current edge is
# in prepared_G. If so, then we can pick it.
if rep_edge in prepared_G.edges:
prepared_G_e = nx.contracted_edge(
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|
30,915 | networkx.algorithms.assortativity.mixing | mixing_dict | Returns a dictionary representation of mixing matrix.
Parameters
----------
xy : list or container of two-tuples
Pairs of (x,y) items.
attribute : string
Node attribute key
normalized : bool (default=False)
Return counts if False or probabilities if True.
Returns
-------
d: dictionary
Counts or Joint probability of occurrence of values in xy.
| def mixing_dict(xy, normalized=False):
"""Returns a dictionary representation of mixing matrix.
Parameters
----------
xy : list or container of two-tuples
Pairs of (x,y) items.
attribute : string
Node attribute key
normalized : bool (default=False)
Return counts if False or probabilities if True.
Returns
-------
d: dictionary
Counts or Joint probability of occurrence of values in xy.
"""
d = {}
psum = 0.0
for x, y in xy:
if x not in d:
d[x] = {}
if y not in d:
d[y] = {}
v = d[x].get(y, 0)
d[x][y] = v + 1
psum += 1
if normalized:
for _, jdict in d.items():
for j in jdict:
jdict[j] /= psum
return d
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|
30,921 | networkx.generators.small | moebius_kantor_graph |
Returns the Moebius-Kantor graph.
The Möbius-Kantor graph is the cubic symmetric graph on 16 nodes.
Its LCF notation is [5,-5]^8, and it is isomorphic to the generalized
Petersen graph [1]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Moebius-Kantor graph
References
----------
.. [1] https://en.wikipedia.org/wiki/M%C3%B6bius%E2%80%93Kantor_graph
| def sedgewick_maze_graph(create_using=None):
"""
Return a small maze with a cycle.
This is the maze used in Sedgewick, 3rd Edition, Part 5, Graph
Algorithms, Chapter 18, e.g. Figure 18.2 and following [1]_.
Nodes are numbered 0,..,7
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Small maze with a cycle
References
----------
.. [1] Figure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick
"""
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
| (create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,924 | networkx.algorithms.shortest_paths.weighted | multi_source_dijkstra | Find shortest weighted paths and lengths from a given set of
source nodes.
Uses Dijkstra's algorithm to compute the shortest paths and lengths
between one of the source nodes and the given `target`, or all other
reachable nodes if not specified, for a weighted graph.
Parameters
----------
G : NetworkX graph
sources : non-empty set of nodes
Starting nodes for paths. If this is just a set containing a
single node, then all paths computed by this function will start
from that node. If there are two or more nodes in the set, the
computed paths may begin from any one of the start nodes.
target : node label, optional
Ending node for path
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number or None to indicate a hidden edge.
Returns
-------
distance, path : pair of dictionaries, or numeric and list
If target is None, returns a tuple of two dictionaries keyed by node.
The first dictionary stores distance from one of the source nodes.
The second stores the path from one of the sources to that node.
If target is not None, returns a tuple of (distance, path) where
distance is the distance from source to target and path is a list
representing the path from source to target.
Examples
--------
>>> G = nx.path_graph(5)
>>> length, path = nx.multi_source_dijkstra(G, {0, 4})
>>> for node in [0, 1, 2, 3, 4]:
... print(f"{node}: {length[node]}")
0: 0
1: 1
2: 2
3: 1
4: 0
>>> path[1]
[0, 1]
>>> path[3]
[4, 3]
>>> length, path = nx.multi_source_dijkstra(G, {0, 4}, 1)
>>> length
1
>>> path
[0, 1]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The weight function can be used to hide edges by returning None.
So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``
will find the shortest red path.
Based on the Python cookbook recipe (119466) at
https://code.activestate.com/recipes/119466/
This algorithm is not guaranteed to work if edge weights
are negative or are floating point numbers
(overflows and roundoff errors can cause problems).
Raises
------
ValueError
If `sources` is empty.
NodeNotFound
If any of `sources` is not in `G`.
See Also
--------
multi_source_dijkstra_path
multi_source_dijkstra_path_length
| def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths
Parameters
----------
G : NetworkX graph
sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.
weight: function
Function with (u, v, data) input that returns that edge's weight
or None to indicate a hidden edge
pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.
paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.
target : node label, optional
Ending node for path. Search is halted when target is found.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.
Raises
------
NodeNotFound
If any of `sources` is not in `G`.
Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.
"""
G_succ = G._adj # For speed-up (and works for both directed and undirected graphs)
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
seen[source] = 0
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist
| (G, sources, target=None, cutoff=None, weight='weight', *, backend=None, **backend_kwargs) | [
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|
30,925 | networkx.algorithms.shortest_paths.weighted | multi_source_dijkstra_path | Find shortest weighted paths in G from a given set of source
nodes.
Compute shortest path between any of the source nodes and all other
reachable nodes for a weighted graph.
Parameters
----------
G : NetworkX graph
sources : non-empty set of nodes
Starting nodes for paths. If this is just a set containing a
single node, then all paths computed by this function will start
from that node. If there are two or more nodes in the set, the
computed paths may begin from any one of the start nodes.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number or None to indicate a hidden edge.
Returns
-------
paths : dictionary
Dictionary of shortest paths keyed by target.
Examples
--------
>>> G = nx.path_graph(5)
>>> path = nx.multi_source_dijkstra_path(G, {0, 4})
>>> path[1]
[0, 1]
>>> path[3]
[4, 3]
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The weight function can be used to hide edges by returning None.
So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``
will find the shortest red path.
Raises
------
ValueError
If `sources` is empty.
NodeNotFound
If any of `sources` is not in `G`.
See Also
--------
multi_source_dijkstra, multi_source_bellman_ford
| def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths
Parameters
----------
G : NetworkX graph
sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.
weight: function
Function with (u, v, data) input that returns that edge's weight
or None to indicate a hidden edge
pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.
paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.
target : node label, optional
Ending node for path. Search is halted when target is found.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.
Raises
------
NodeNotFound
If any of `sources` is not in `G`.
Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.
"""
G_succ = G._adj # For speed-up (and works for both directed and undirected graphs)
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
seen[source] = 0
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist
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|
30,926 | networkx.algorithms.shortest_paths.weighted | multi_source_dijkstra_path_length | Find shortest weighted path lengths in G from a given set of
source nodes.
Compute the shortest path length between any of the source nodes and
all other reachable nodes for a weighted graph.
Parameters
----------
G : NetworkX graph
sources : non-empty set of nodes
Starting nodes for paths. If this is just a set containing a
single node, then all paths computed by this function will start
from that node. If there are two or more nodes in the set, the
computed paths may begin from any one of the start nodes.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number or None to indicate a hidden edge.
Returns
-------
length : dict
Dict keyed by node to shortest path length to nearest source.
Examples
--------
>>> G = nx.path_graph(5)
>>> length = nx.multi_source_dijkstra_path_length(G, {0, 4})
>>> for node in [0, 1, 2, 3, 4]:
... print(f"{node}: {length[node]}")
0: 0
1: 1
2: 2
3: 1
4: 0
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
The weight function can be used to hide edges by returning None.
So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``
will find the shortest red path.
Raises
------
ValueError
If `sources` is empty.
NodeNotFound
If any of `sources` is not in `G`.
See Also
--------
multi_source_dijkstra
| def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths
Parameters
----------
G : NetworkX graph
sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.
weight: function
Function with (u, v, data) input that returns that edge's weight
or None to indicate a hidden edge
pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.
paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.
target : node label, optional
Ending node for path. Search is halted when target is found.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.
Raises
------
NodeNotFound
If any of `sources` is not in `G`.
Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.
"""
G_succ = G._adj # For speed-up (and works for both directed and undirected graphs)
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
seen[source] = 0
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist
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|
30,930 | networkx.drawing.layout | multipartite_layout | Position nodes in layers of straight lines.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
subset_key : string or dict (default='subset')
If a string, the key of node data in G that holds the node subset.
If a dict, keyed by layer number to the nodes in that layer/subset.
align : string (default='vertical')
The alignment of nodes. Vertical or horizontal.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node.
Examples
--------
>>> G = nx.complete_multipartite_graph(28, 16, 10)
>>> pos = nx.multipartite_layout(G)
or use a dict to provide the layers of the layout
>>> G = nx.Graph([(0, 1), (1, 2), (1, 3), (3, 4)])
>>> layers = {"a": [0], "b": [1], "c": [2, 3], "d": [4]}
>>> pos = nx.multipartite_layout(G, subset_key=layers)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
Network does not need to be a complete multipartite graph. As long as nodes
have subset_key data, they will be placed in the corresponding layers.
| def multipartite_layout(G, subset_key="subset", align="vertical", scale=1, center=None):
"""Position nodes in layers of straight lines.
Parameters
----------
G : NetworkX graph or list of nodes
A position will be assigned to every node in G.
subset_key : string or dict (default='subset')
If a string, the key of node data in G that holds the node subset.
If a dict, keyed by layer number to the nodes in that layer/subset.
align : string (default='vertical')
The alignment of nodes. Vertical or horizontal.
scale : number (default: 1)
Scale factor for positions.
center : array-like or None
Coordinate pair around which to center the layout.
Returns
-------
pos : dict
A dictionary of positions keyed by node.
Examples
--------
>>> G = nx.complete_multipartite_graph(28, 16, 10)
>>> pos = nx.multipartite_layout(G)
or use a dict to provide the layers of the layout
>>> G = nx.Graph([(0, 1), (1, 2), (1, 3), (3, 4)])
>>> layers = {"a": [0], "b": [1], "c": [2, 3], "d": [4]}
>>> pos = nx.multipartite_layout(G, subset_key=layers)
Notes
-----
This algorithm currently only works in two dimensions and does not
try to minimize edge crossings.
Network does not need to be a complete multipartite graph. As long as nodes
have subset_key data, they will be placed in the corresponding layers.
"""
import numpy as np
if align not in ("vertical", "horizontal"):
msg = "align must be either vertical or horizontal."
raise ValueError(msg)
G, center = _process_params(G, center=center, dim=2)
if len(G) == 0:
return {}
try:
# check if subset_key is dict-like
if len(G) != sum(len(nodes) for nodes in subset_key.values()):
raise nx.NetworkXError(
"all nodes must be in one subset of `subset_key` dict"
)
except AttributeError:
# subset_key is not a dict, hence a string
node_to_subset = nx.get_node_attributes(G, subset_key)
if len(node_to_subset) != len(G):
raise nx.NetworkXError(
f"all nodes need a subset_key attribute: {subset_key}"
)
subset_key = nx.utils.groups(node_to_subset)
# Sort by layer, if possible
try:
layers = dict(sorted(subset_key.items()))
except TypeError:
layers = subset_key
pos = None
nodes = []
width = len(layers)
for i, layer in enumerate(layers.values()):
height = len(layer)
xs = np.repeat(i, height)
ys = np.arange(0, height, dtype=float)
offset = ((width - 1) / 2, (height - 1) / 2)
layer_pos = np.column_stack([xs, ys]) - offset
if pos is None:
pos = layer_pos
else:
pos = np.concatenate([pos, layer_pos])
nodes.extend(layer)
pos = rescale_layout(pos, scale=scale) + center
if align == "horizontal":
pos = pos[:, ::-1] # swap x and y coords
pos = dict(zip(nodes, pos))
return pos
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|
30,934 | networkx.generators.geometric | navigable_small_world_graph | Returns a navigable small-world graph.
A navigable small-world graph is a directed grid with additional long-range
connections that are chosen randomly.
[...] we begin with a set of nodes [...] that are identified with the set
of lattice points in an $n \times n$ square,
$\{(i, j): i \in \{1, 2, \ldots, n\}, j \in \{1, 2, \ldots, n\}\}$,
and we define the *lattice distance* between two nodes $(i, j)$ and
$(k, l)$ to be the number of "lattice steps" separating them:
$d((i, j), (k, l)) = |k - i| + |l - j|$.
For a universal constant $p >= 1$, the node $u$ has a directed edge to
every other node within lattice distance $p$---these are its *local
contacts*. For universal constants $q >= 0$ and $r >= 0$ we also
construct directed edges from $u$ to $q$ other nodes (the *long-range
contacts*) using independent random trials; the $i$th directed edge from
$u$ has endpoint $v$ with probability proportional to $[d(u,v)]^{-r}$.
-- [1]_
Parameters
----------
n : int
The length of one side of the lattice; the number of nodes in
the graph is therefore $n^2$.
p : int
The diameter of short range connections. Each node is joined with every
other node within this lattice distance.
q : int
The number of long-range connections for each node.
r : float
Exponent for decaying probability of connections. The probability of
connecting to a node at lattice distance $d$ is $1/d^r$.
dim : int
Dimension of grid
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
References
----------
.. [1] J. Kleinberg. The small-world phenomenon: An algorithmic
perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000.
| def thresholded_random_geometric_graph(
n,
radius,
theta,
dim=2,
pos=None,
weight=None,
p=2,
seed=None,
*,
pos_name="pos",
weight_name="weight",
):
r"""Returns a thresholded random geometric graph in the unit cube.
The thresholded random geometric graph [1] model places `n` nodes
uniformly at random in the unit cube of dimensions `dim`. Each node
`u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are
joined by an edge if they are within the maximum connection distance,
`radius` computed by the `p`-Minkowski distance and the summation of
weights :math:`w_u` + :math:`w_v` is greater than or equal
to the threshold parameter `theta`.
Edges within `radius` of each other are determined using a KDTree when
SciPy is available. This reduces the time complexity from :math:`O(n^2)`
to :math:`O(n)`.
Parameters
----------
n : int or iterable
Number of nodes or iterable of nodes
radius: float
Distance threshold value
theta: float
Threshold value
dim : int, optional
Dimension of graph
pos : dict, optional
A dictionary keyed by node with node positions as values.
weight : dict, optional
Node weights as a dictionary of numbers keyed by node.
p : float, optional (default 2)
Which Minkowski distance metric to use. `p` has to meet the condition
``1 <= p <= infinity``.
If this argument is not specified, the :math:`L^2` metric
(the Euclidean distance metric), p = 2 is used.
This should not be confused with the `p` of an Erdős-Rényi random
graph, which represents probability.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
pos_name : string, default="pos"
The name of the node attribute which represents the position
in 2D coordinates of the node in the returned graph.
weight_name : string, default="weight"
The name of the node attribute which represents the weight
of the node in the returned graph.
Returns
-------
Graph
A thresholded random geographic graph, undirected and without
self-loops.
Each node has a node attribute ``'pos'`` that stores the
position of that node in Euclidean space as provided by the
``pos`` keyword argument or, if ``pos`` was not provided, as
generated by this function. Similarly, each node has a nodethre
attribute ``'weight'`` that stores the weight of that node as
provided or as generated.
Examples
--------
Default Graph:
G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
Custom Graph:
Create a thresholded random geometric graph on 50 uniformly distributed
nodes where nodes are joined by an edge if their sum weights drawn from
a exponential distribution with rate = 5 are >= theta = 0.1 and their
Euclidean distance is at most 0.2.
Notes
-----
This uses a *k*-d tree to build the graph.
The `pos` keyword argument can be used to specify node positions so you
can create an arbitrary distribution and domain for positions.
For example, to use a 2D Gaussian distribution of node positions with mean
(0, 0) and standard deviation 2
If weights are not specified they are assigned to nodes by drawing randomly
from the exponential distribution with rate parameter :math:`\lambda=1`.
To specify weights from a different distribution, use the `weight` keyword
argument::
::
>>> import random
>>> import math
>>> n = 50
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)
References
----------
.. [1] http://cole-maclean.github.io/blog/files/thesis.pdf
"""
G = nx.empty_graph(n)
G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})"
# If no weights are provided, choose them from an exponential
# distribution.
if weight is None:
weight = {v: seed.expovariate(1) for v in G}
# If no positions are provided, choose uniformly random vectors in
# Euclidean space of the specified dimension.
if pos is None:
pos = {v: [seed.random() for i in range(dim)] for v in G}
# If no distance metric is provided, use Euclidean distance.
nx.set_node_attributes(G, weight, weight_name)
nx.set_node_attributes(G, pos, pos_name)
edges = (
(u, v)
for u, v in _geometric_edges(G, radius, p, pos_name)
if weight[u] + weight[v] >= theta
)
G.add_edges_from(edges)
return G
| (n, p=1, q=1, r=2, dim=2, seed=None, *, backend=None, **backend_kwargs) | [
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|
30,935 | networkx.algorithms.shortest_paths.weighted | negative_edge_cycle | Returns True if there exists a negative edge cycle anywhere in G.
Parameters
----------
G : NetworkX graph
weight : string or function
If this is a string, then edge weights will be accessed via the
edge attribute with this key (that is, the weight of the edge
joining `u` to `v` will be ``G.edges[u, v][weight]``). If no
such edge attribute exists, the weight of the edge is assumed to
be one.
If this is a function, the weight of an edge is the value
returned by the function. The function must accept exactly three
positional arguments: the two endpoints of an edge and the
dictionary of edge attributes for that edge. The function must
return a number.
heuristic : bool
Determines whether to use a heuristic to early detect negative
cycles at a negligible cost. In case of graphs with a negative cycle,
the performance of detection increases by at least an order of magnitude.
Returns
-------
negative_cycle : bool
True if a negative edge cycle exists, otherwise False.
Examples
--------
>>> G = nx.cycle_graph(5, create_using=nx.DiGraph())
>>> print(nx.negative_edge_cycle(G))
False
>>> G[1][2]["weight"] = -7
>>> print(nx.negative_edge_cycle(G))
True
Notes
-----
Edge weight attributes must be numerical.
Distances are calculated as sums of weighted edges traversed.
This algorithm uses bellman_ford_predecessor_and_distance() but finds
negative cycles on any component by first adding a new node connected to
every node, and starting bellman_ford_predecessor_and_distance on that
node. It then removes that extra node.
| def _dijkstra_multisource(
G, sources, weight, pred=None, paths=None, cutoff=None, target=None
):
"""Uses Dijkstra's algorithm to find shortest weighted paths
Parameters
----------
G : NetworkX graph
sources : non-empty iterable of nodes
Starting nodes for paths. If this is just an iterable containing
a single node, then all paths computed by this function will
start from that node. If there are two or more nodes in this
iterable, the computed paths may begin from any one of the start
nodes.
weight: function
Function with (u, v, data) input that returns that edge's weight
or None to indicate a hidden edge
pred: dict of lists, optional(default=None)
dict to store a list of predecessors keyed by that node
If None, predecessors are not stored.
paths: dict, optional (default=None)
dict to store the path list from source to each node, keyed by node.
If None, paths are not stored.
target : node label, optional
Ending node for path. Search is halted when target is found.
cutoff : integer or float, optional
Length (sum of edge weights) at which the search is stopped.
If cutoff is provided, only return paths with summed weight <= cutoff.
Returns
-------
distance : dictionary
A mapping from node to shortest distance to that node from one
of the source nodes.
Raises
------
NodeNotFound
If any of `sources` is not in `G`.
Notes
-----
The optional predecessor and path dictionaries can be accessed by
the caller through the original pred and paths objects passed
as arguments. No need to explicitly return pred or paths.
"""
G_succ = G._adj # For speed-up (and works for both directed and undirected graphs)
push = heappush
pop = heappop
dist = {} # dictionary of final distances
seen = {}
# fringe is heapq with 3-tuples (distance,c,node)
# use the count c to avoid comparing nodes (may not be able to)
c = count()
fringe = []
for source in sources:
seen[source] = 0
push(fringe, (0, next(c), source))
while fringe:
(d, _, v) = pop(fringe)
if v in dist:
continue # already searched this node.
dist[v] = d
if v == target:
break
for u, e in G_succ[v].items():
cost = weight(v, u, e)
if cost is None:
continue
vu_dist = dist[v] + cost
if cutoff is not None:
if vu_dist > cutoff:
continue
if u in dist:
u_dist = dist[u]
if vu_dist < u_dist:
raise ValueError("Contradictory paths found:", "negative weights?")
elif pred is not None and vu_dist == u_dist:
pred[u].append(v)
elif u not in seen or vu_dist < seen[u]:
seen[u] = vu_dist
push(fringe, (vu_dist, next(c), u))
if paths is not None:
paths[u] = paths[v] + [u]
if pred is not None:
pred[u] = [v]
elif vu_dist == seen[u]:
if pred is not None:
pred[u].append(v)
# The optional predecessor and path dictionaries can be accessed
# by the caller via the pred and paths objects passed as arguments.
return dist
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|
30,937 | networkx.classes.function | neighbors | Returns an iterator over all neighbors of node n.
This function wraps the :func:`G.neighbors <networkx.Graph.neighbors>` function.
| def neighbors(G, n):
"""Returns an iterator over all neighbors of node n.
This function wraps the :func:`G.neighbors <networkx.Graph.neighbors>` function.
"""
return G.neighbors(n)
| (G, n) | [
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|
30,939 | networkx.generators.random_graphs | newman_watts_strogatz_graph | Returns a Newman–Watts–Strogatz small-world graph.
Parameters
----------
n : int
The number of nodes.
k : int
Each node is joined with its `k` nearest neighbors in a ring
topology.
p : float
The probability of adding a new edge for each edge.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Notes
-----
First create a ring over $n$ nodes [1]_. Then each node in the ring is
connected with its $k$ nearest neighbors (or $k - 1$ neighbors if $k$
is odd). Then shortcuts are created by adding new edges as follows: for
each edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest
neighbors" with probability $p$ add a new edge $(u, w)$ with
randomly-chosen existing node $w$. In contrast with
:func:`watts_strogatz_graph`, no edges are removed.
See Also
--------
watts_strogatz_graph
References
----------
.. [1] M. E. J. Newman and D. J. Watts,
Renormalization group analysis of the small-world network model,
Physics Letters A, 263, 341, 1999.
https://doi.org/10.1016/S0375-9601(99)00757-4
| def dual_barabasi_albert_graph(n, m1, m2, p, seed=None, initial_graph=None):
"""Returns a random graph using dual Barabási–Albert preferential attachment
A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$
edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that
are preferentially attached to existing nodes with high degree.
Parameters
----------
n : int
Number of nodes
m1 : int
Number of edges to link each new node to existing nodes with probability $p$
m2 : int
Number of edges to link each new node to existing nodes with probability $1-p$
p : float
The probability of attaching $m_1$ edges (as opposed to $m_2$ edges)
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
initial_graph : Graph or None (default)
Initial network for Barabási–Albert algorithm.
A copy of `initial_graph` is used.
It should be connected for most use cases.
If None, starts from an star graph on max(m1, m2) + 1 nodes.
Returns
-------
G : Graph
Raises
------
NetworkXError
If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n``, or
`p` does not satisfy ``0 <= p <= 1``, or
the initial graph number of nodes m0 does not satisfy m1, m2 <= m0 <= n.
References
----------
.. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538.
"""
if m1 < 1 or m1 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}"
)
if m2 < 1 or m2 >= n:
raise nx.NetworkXError(
f"Dual Barabási–Albert must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}"
)
if p < 0 or p > 1:
raise nx.NetworkXError(
f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}"
)
# For simplicity, if p == 0 or 1, just return BA
if p == 1:
return barabasi_albert_graph(n, m1, seed)
elif p == 0:
return barabasi_albert_graph(n, m2, seed)
if initial_graph is None:
# Default initial graph : empty graph on max(m1, m2) nodes
G = star_graph(max(m1, m2))
else:
if len(initial_graph) < max(m1, m2) or len(initial_graph) > n:
raise nx.NetworkXError(
f"Barabási–Albert initial graph must have between "
f"max(m1, m2) = {max(m1, m2)} and n = {n} nodes"
)
G = initial_graph.copy()
# Target nodes for new edges
targets = list(G)
# List of existing nodes, with nodes repeated once for each adjacent edge
repeated_nodes = [n for n, d in G.degree() for _ in range(d)]
# Start adding the remaining nodes.
source = len(G)
while source < n:
# Pick which m to use (m1 or m2)
if seed.random() < p:
m = m1
else:
m = m2
# Now choose m unique nodes from the existing nodes
# Pick uniformly from repeated_nodes (preferential attachment)
targets = _random_subset(repeated_nodes, m, seed)
# Add edges to m nodes from the source.
G.add_edges_from(zip([source] * m, targets))
# Add one node to the list for each new edge just created.
repeated_nodes.extend(targets)
# And the new node "source" has m edges to add to the list.
repeated_nodes.extend([source] * m)
source += 1
return G
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|
30,945 | networkx.algorithms.connectivity.connectivity | node_connectivity | Returns node connectivity for a graph or digraph G.
Node connectivity is equal to the minimum number of nodes that
must be removed to disconnect G or render it trivial. If source
and target nodes are provided, this function returns the local node
connectivity: the minimum number of nodes that must be removed to break
all paths from source to target in G.
Parameters
----------
G : NetworkX graph
Undirected graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
Returns
-------
K : integer
Node connectivity of G, or local node connectivity if source
and target are provided.
Examples
--------
>>> # Platonic icosahedral graph is 5-node-connected
>>> G = nx.icosahedral_graph()
>>> nx.node_connectivity(G)
5
You can use alternative flow algorithms for the underlying maximum
flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions. Alternative
flow functions have to be explicitly imported from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.node_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local node connectivity.
>>> nx.node_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_node_connectivity` for details.
Notes
-----
This is a flow based implementation of node connectivity. The
algorithm works by solving $O((n-\delta-1+\delta(\delta-1)/2))$
maximum flow problems on an auxiliary digraph. Where $\delta$
is the minimum degree of G. For details about the auxiliary
digraph and the computation of local node connectivity see
:meth:`local_node_connectivity`. This implementation is based
on algorithm 11 in [1]_.
See also
--------
:meth:`local_node_connectivity`
:meth:`edge_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
| @nx._dispatchable
def edge_connectivity(G, s=None, t=None, flow_func=None, cutoff=None):
r"""Returns the edge connectivity of the graph or digraph G.
The edge connectivity is equal to the minimum number of edges that
must be removed to disconnect G or render it trivial. If source
and target nodes are provided, this function returns the local edge
connectivity: the minimum number of edges that must be removed to
break all paths from source to target in G.
Parameters
----------
G : NetworkX graph
Undirected or directed graph
s : node
Source node. Optional. Default value: None.
t : node
Target node. Optional. Default value: None.
flow_func : function
A function for computing the maximum flow among a pair of nodes.
The function has to accept at least three parameters: a Digraph,
a source node, and a target node. And return a residual network
that follows NetworkX conventions (see :meth:`maximum_flow` for
details). If flow_func is None, the default maximum flow function
(:meth:`edmonds_karp`) is used. See below for details. The
choice of the default function may change from version
to version and should not be relied on. Default value: None.
cutoff : integer, float, or None (default: None)
If specified, the maximum flow algorithm will terminate when the
flow value reaches or exceeds the cutoff. This only works for flows
that support the cutoff parameter (most do) and is ignored otherwise.
Returns
-------
K : integer
Edge connectivity for G, or local edge connectivity if source
and target were provided
Examples
--------
>>> # Platonic icosahedral graph is 5-edge-connected
>>> G = nx.icosahedral_graph()
>>> nx.edge_connectivity(G)
5
You can use alternative flow algorithms for the underlying
maximum flow computation. In dense networks the algorithm
:meth:`shortest_augmenting_path` will usually perform better
than the default :meth:`edmonds_karp`, which is faster for
sparse networks with highly skewed degree distributions.
Alternative flow functions have to be explicitly imported
from the flow package.
>>> from networkx.algorithms.flow import shortest_augmenting_path
>>> nx.edge_connectivity(G, flow_func=shortest_augmenting_path)
5
If you specify a pair of nodes (source and target) as parameters,
this function returns the value of local edge connectivity.
>>> nx.edge_connectivity(G, 3, 7)
5
If you need to perform several local computations among different
pairs of nodes on the same graph, it is recommended that you reuse
the data structures used in the maximum flow computations. See
:meth:`local_edge_connectivity` for details.
Notes
-----
This is a flow based implementation of global edge connectivity.
For undirected graphs the algorithm works by finding a 'small'
dominating set of nodes of G (see algorithm 7 in [1]_ ) and
computing local maximum flow (see :meth:`local_edge_connectivity`)
between an arbitrary node in the dominating set and the rest of
nodes in it. This is an implementation of algorithm 6 in [1]_ .
For directed graphs, the algorithm does n calls to the maximum
flow function. This is an implementation of algorithm 8 in [1]_ .
See also
--------
:meth:`local_edge_connectivity`
:meth:`local_node_connectivity`
:meth:`node_connectivity`
:meth:`maximum_flow`
:meth:`edmonds_karp`
:meth:`preflow_push`
:meth:`shortest_augmenting_path`
:meth:`k_edge_components`
:meth:`k_edge_subgraphs`
References
----------
.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.
http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
"""
if (s is not None and t is None) or (s is None and t is not None):
raise nx.NetworkXError("Both source and target must be specified.")
# Local edge connectivity
if s is not None and t is not None:
if s not in G:
raise nx.NetworkXError(f"node {s} not in graph")
if t not in G:
raise nx.NetworkXError(f"node {t} not in graph")
return local_edge_connectivity(G, s, t, flow_func=flow_func, cutoff=cutoff)
# Global edge connectivity
# reuse auxiliary digraph and residual network
H = build_auxiliary_edge_connectivity(G)
R = build_residual_network(H, "capacity")
kwargs = {"flow_func": flow_func, "auxiliary": H, "residual": R}
if G.is_directed():
# Algorithm 8 in [1]
if not nx.is_weakly_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(d for n, d in G.degree())
nodes = list(G)
n = len(nodes)
if cutoff is not None:
L = min(cutoff, L)
for i in range(n):
kwargs["cutoff"] = L
try:
L = min(L, local_edge_connectivity(G, nodes[i], nodes[i + 1], **kwargs))
except IndexError: # last node!
L = min(L, local_edge_connectivity(G, nodes[i], nodes[0], **kwargs))
return L
else: # undirected
# Algorithm 6 in [1]
if not nx.is_connected(G):
return 0
# initial value for \lambda is minimum degree
L = min(d for n, d in G.degree())
if cutoff is not None:
L = min(cutoff, L)
# A dominating set is \lambda-covering
# We need a dominating set with at least two nodes
for node in G:
D = nx.dominating_set(G, start_with=node)
v = D.pop()
if D:
break
else:
# in complete graphs the dominating sets will always be of one node
# thus we return min degree
return L
for w in D:
kwargs["cutoff"] = L
L = min(L, local_edge_connectivity(G, v, w, **kwargs))
return L
| (G, s=None, t=None, flow_func=None, *, backend=None, **backend_kwargs) | [
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|
30,950 | networkx.readwrite.json_graph.node_link | node_link_data | Returns data in node-link format that is suitable for JSON serialization
and use in JavaScript documents.
Parameters
----------
G : NetworkX graph
source : string
A string that provides the 'source' attribute name for storing NetworkX-internal graph data.
target : string
A string that provides the 'target' attribute name for storing NetworkX-internal graph data.
name : string
A string that provides the 'name' attribute name for storing NetworkX-internal graph data.
key : string
A string that provides the 'key' attribute name for storing NetworkX-internal graph data.
link : string
A string that provides the 'link' attribute name for storing NetworkX-internal graph data.
Returns
-------
data : dict
A dictionary with node-link formatted data.
Raises
------
NetworkXError
If the values of 'source', 'target' and 'key' are not unique.
Examples
--------
>>> G = nx.Graph([("A", "B")])
>>> data1 = nx.node_link_data(G)
>>> data1
{'directed': False, 'multigraph': False, 'graph': {}, 'nodes': [{'id': 'A'}, {'id': 'B'}], 'links': [{'source': 'A', 'target': 'B'}]}
To serialize with JSON
>>> import json
>>> s1 = json.dumps(data1)
>>> s1
'{"directed": false, "multigraph": false, "graph": {}, "nodes": [{"id": "A"}, {"id": "B"}], "links": [{"source": "A", "target": "B"}]}'
A graph can also be serialized by passing `node_link_data` as an encoder function. The two methods are equivalent.
>>> s1 = json.dumps(G, default=nx.node_link_data)
>>> s1
'{"directed": false, "multigraph": false, "graph": {}, "nodes": [{"id": "A"}, {"id": "B"}], "links": [{"source": "A", "target": "B"}]}'
The attribute names for storing NetworkX-internal graph data can
be specified as keyword options.
>>> H = nx.gn_graph(2)
>>> data2 = nx.node_link_data(H, link="edges", source="from", target="to")
>>> data2
{'directed': True, 'multigraph': False, 'graph': {}, 'nodes': [{'id': 0}, {'id': 1}], 'edges': [{'from': 1, 'to': 0}]}
Notes
-----
Graph, node, and link attributes are stored in this format. Note that
attribute keys will be converted to strings in order to comply with JSON.
Attribute 'key' is only used for multigraphs.
To use `node_link_data` in conjunction with `node_link_graph`,
the keyword names for the attributes must match.
See Also
--------
node_link_graph, adjacency_data, tree_data
| def node_link_data(
G,
*,
source="source",
target="target",
name="id",
key="key",
link="links",
):
"""Returns data in node-link format that is suitable for JSON serialization
and use in JavaScript documents.
Parameters
----------
G : NetworkX graph
source : string
A string that provides the 'source' attribute name for storing NetworkX-internal graph data.
target : string
A string that provides the 'target' attribute name for storing NetworkX-internal graph data.
name : string
A string that provides the 'name' attribute name for storing NetworkX-internal graph data.
key : string
A string that provides the 'key' attribute name for storing NetworkX-internal graph data.
link : string
A string that provides the 'link' attribute name for storing NetworkX-internal graph data.
Returns
-------
data : dict
A dictionary with node-link formatted data.
Raises
------
NetworkXError
If the values of 'source', 'target' and 'key' are not unique.
Examples
--------
>>> G = nx.Graph([("A", "B")])
>>> data1 = nx.node_link_data(G)
>>> data1
{'directed': False, 'multigraph': False, 'graph': {}, 'nodes': [{'id': 'A'}, {'id': 'B'}], 'links': [{'source': 'A', 'target': 'B'}]}
To serialize with JSON
>>> import json
>>> s1 = json.dumps(data1)
>>> s1
'{"directed": false, "multigraph": false, "graph": {}, "nodes": [{"id": "A"}, {"id": "B"}], "links": [{"source": "A", "target": "B"}]}'
A graph can also be serialized by passing `node_link_data` as an encoder function. The two methods are equivalent.
>>> s1 = json.dumps(G, default=nx.node_link_data)
>>> s1
'{"directed": false, "multigraph": false, "graph": {}, "nodes": [{"id": "A"}, {"id": "B"}], "links": [{"source": "A", "target": "B"}]}'
The attribute names for storing NetworkX-internal graph data can
be specified as keyword options.
>>> H = nx.gn_graph(2)
>>> data2 = nx.node_link_data(H, link="edges", source="from", target="to")
>>> data2
{'directed': True, 'multigraph': False, 'graph': {}, 'nodes': [{'id': 0}, {'id': 1}], 'edges': [{'from': 1, 'to': 0}]}
Notes
-----
Graph, node, and link attributes are stored in this format. Note that
attribute keys will be converted to strings in order to comply with JSON.
Attribute 'key' is only used for multigraphs.
To use `node_link_data` in conjunction with `node_link_graph`,
the keyword names for the attributes must match.
See Also
--------
node_link_graph, adjacency_data, tree_data
"""
multigraph = G.is_multigraph()
# Allow 'key' to be omitted from attrs if the graph is not a multigraph.
key = None if not multigraph else key
if len({source, target, key}) < 3:
raise nx.NetworkXError("Attribute names are not unique.")
data = {
"directed": G.is_directed(),
"multigraph": multigraph,
"graph": G.graph,
"nodes": [{**G.nodes[n], name: n} for n in G],
}
if multigraph:
data[link] = [
{**d, source: u, target: v, key: k}
for u, v, k, d in G.edges(keys=True, data=True)
]
else:
data[link] = [{**d, source: u, target: v} for u, v, d in G.edges(data=True)]
return data
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|
30,952 | networkx.classes.function | nodes | Returns a NodeView over the graph nodes.
This function wraps the :func:`G.nodes <networkx.Graph.nodes>` property.
| def nodes(G):
"""Returns a NodeView over the graph nodes.
This function wraps the :func:`G.nodes <networkx.Graph.nodes>` property.
"""
return G.nodes()
| (G) | [
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|
30,953 | networkx.classes.function | nodes_with_selfloops | Returns an iterator over nodes with self loops.
A node with a self loop has an edge with both ends adjacent
to that node.
Returns
-------
nodelist : iterator
A iterator over nodes with self loops.
See Also
--------
selfloop_edges, number_of_selfloops
Examples
--------
>>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge(1, 1)
>>> G.add_edge(1, 2)
>>> list(nx.nodes_with_selfloops(G))
[1]
| def nodes_with_selfloops(G):
"""Returns an iterator over nodes with self loops.
A node with a self loop has an edge with both ends adjacent
to that node.
Returns
-------
nodelist : iterator
A iterator over nodes with self loops.
See Also
--------
selfloop_edges, number_of_selfloops
Examples
--------
>>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge(1, 1)
>>> G.add_edge(1, 2)
>>> list(nx.nodes_with_selfloops(G))
[1]
"""
return (n for n, nbrs in G._adj.items() if n in nbrs)
| (G) | [
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|
30,954 | networkx.classes.function | non_edges | Returns the nonexistent edges in the graph.
Parameters
----------
graph : NetworkX graph.
Graph to find nonexistent edges.
Returns
-------
non_edges : iterator
Iterator of edges that are not in the graph.
| def non_edges(graph):
"""Returns the nonexistent edges in the graph.
Parameters
----------
graph : NetworkX graph.
Graph to find nonexistent edges.
Returns
-------
non_edges : iterator
Iterator of edges that are not in the graph.
"""
if graph.is_directed():
for u in graph:
for v in non_neighbors(graph, u):
yield (u, v)
else:
nodes = set(graph)
while nodes:
u = nodes.pop()
for v in nodes - set(graph[u]):
yield (u, v)
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|
30,955 | networkx.classes.function | non_neighbors | Returns the non-neighbors of the node in the graph.
Parameters
----------
graph : NetworkX graph
Graph to find neighbors.
node : node
The node whose neighbors will be returned.
Returns
-------
non_neighbors : set
Set of nodes in the graph that are not neighbors of the node.
| def non_neighbors(graph, node):
"""Returns the non-neighbors of the node in the graph.
Parameters
----------
graph : NetworkX graph
Graph to find neighbors.
node : node
The node whose neighbors will be returned.
Returns
-------
non_neighbors : set
Set of nodes in the graph that are not neighbors of the node.
"""
return graph._adj.keys() - graph._adj[node].keys() - {node}
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|
30,961 | networkx.generators.classic | null_graph | Returns the Null graph with no nodes or edges.
See empty_graph for the use of create_using.
| def star_graph(n, create_using=None):
"""Return the star graph
The star graph consists of one center node connected to n outer nodes.
.. plot::
>>> nx.draw(nx.star_graph(6))
Parameters
----------
n : int or iterable
If an integer, node labels are 0 to n with center 0.
If an iterable of nodes, the center is the first.
Warning: n is not checked for duplicates and if present the
resulting graph may not be as desired. Make sure you have no duplicates.
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Notes
-----
The graph has n+1 nodes for integer n.
So star_graph(3) is the same as star_graph(range(4)).
"""
n, nodes = n
if isinstance(n, numbers.Integral):
nodes.append(int(n)) # there should be n+1 nodes
G = empty_graph(nodes, create_using)
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
if len(nodes) > 1:
hub, *spokes = nodes
G.add_edges_from((hub, node) for node in spokes)
return G
| (create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,964 | networkx.algorithms.clique | number_of_cliques | Returns the number of maximal cliques for each node.
Returns a single or list depending on input nodes.
Optional list of cliques can be input if already computed.
| def number_of_cliques(G, nodes=None, cliques=None):
"""Returns the number of maximal cliques for each node.
Returns a single or list depending on input nodes.
Optional list of cliques can be input if already computed.
"""
if cliques is None:
cliques = list(find_cliques(G))
if nodes is None:
nodes = list(G.nodes()) # none, get entire graph
if not isinstance(nodes, list): # check for a list
v = nodes
# assume it is a single value
numcliq = len([1 for c in cliques if v in c])
else:
numcliq = {}
for v in nodes:
numcliq[v] = len([1 for c in cliques if v in c])
return numcliq
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|
30,965 | networkx.classes.function | number_of_edges | Returns the number of edges in the graph.
This function wraps the :func:`G.number_of_edges <networkx.Graph.number_of_edges>` function.
| def number_of_edges(G):
"""Returns the number of edges in the graph.
This function wraps the :func:`G.number_of_edges <networkx.Graph.number_of_edges>` function.
"""
return G.number_of_edges()
| (G) | [
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|
30,967 | networkx.classes.function | number_of_nodes | Returns the number of nodes in the graph.
This function wraps the :func:`G.number_of_nodes <networkx.Graph.number_of_nodes>` function.
| def number_of_nodes(G):
"""Returns the number of nodes in the graph.
This function wraps the :func:`G.number_of_nodes <networkx.Graph.number_of_nodes>` function.
"""
return G.number_of_nodes()
| (G) | [
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|
30,969 | networkx.classes.function | number_of_selfloops | Returns the number of selfloop edges.
A selfloop edge has the same node at both ends.
Returns
-------
nloops : int
The number of selfloops.
See Also
--------
nodes_with_selfloops, selfloop_edges
Examples
--------
>>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge(1, 1)
>>> G.add_edge(1, 2)
>>> nx.number_of_selfloops(G)
1
| def number_of_selfloops(G):
"""Returns the number of selfloop edges.
A selfloop edge has the same node at both ends.
Returns
-------
nloops : int
The number of selfloops.
See Also
--------
nodes_with_selfloops, selfloop_edges
Examples
--------
>>> G = nx.Graph() # or DiGraph, MultiGraph, MultiDiGraph, etc
>>> G.add_edge(1, 1)
>>> G.add_edge(1, 2)
>>> nx.number_of_selfloops(G)
1
"""
return sum(1 for _ in nx.selfloop_edges(G))
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|
30,970 | networkx.algorithms.tree.mst | number_of_spanning_trees | Returns the number of spanning trees in `G`.
A spanning tree for an undirected graph is a tree that connects
all nodes in the graph. For a directed graph, the analog of a
spanning tree is called a (spanning) arborescence. The arborescence
includes a unique directed path from the `root` node to each other node.
The graph must be weakly connected, and the root must be a node
that includes all nodes as successors [3]_. Note that to avoid
discussing sink-roots and reverse-arborescences, we have reversed
the edge orientation from [3]_ and use the in-degree laplacian.
This function (when `weight` is `None`) returns the number of
spanning trees for an undirected graph and the number of
arborescences from a single root node for a directed graph.
When `weight` is the name of an edge attribute which holds the
weight value of each edge, the function returns the sum over
all trees of the multiplicative weight of each tree. That is,
the weight of the tree is the product of its edge weights.
Kirchoff's Tree Matrix Theorem states that any cofactor of the
Laplacian matrix of a graph is the number of spanning trees in the
graph. (Here we use cofactors for a diagonal entry so that the
cofactor becomes the determinant of the matrix with one row
and its matching column removed.) For a weighted Laplacian matrix,
the cofactor is the sum across all spanning trees of the
multiplicative weight of each tree. That is, the weight of each
tree is the product of its edge weights. The theorem is also
known as Kirchhoff's theorem [1]_ and the Matrix-Tree theorem [2]_.
For directed graphs, a similar theorem (Tutte's Theorem) holds with
the cofactor chosen to be the one with row and column removed that
correspond to the root. The cofactor is the number of arborescences
with the specified node as root. And the weighted version gives the
sum of the arborescence weights with root `root`. The arborescence
weight is the product of its edge weights.
Parameters
----------
G : NetworkX graph
root : node
A node in the directed graph `G` that has all nodes as descendants.
(This is ignored for undirected graphs.)
weight : string or None, optional (default=None)
The name of the edge attribute holding the edge weight.
If `None`, then each edge is assumed to have a weight of 1.
Returns
-------
Number
Undirected graphs:
The number of spanning trees of the graph `G`.
Or the sum of all spanning tree weights of the graph `G`
where the weight of a tree is the product of its edge weights.
Directed graphs:
The number of arborescences of `G` rooted at node `root`.
Or the sum of all arborescence weights of the graph `G` with
specified root where the weight of an arborescence is the product
of its edge weights.
Raises
------
NetworkXPointlessConcept
If `G` does not contain any nodes.
NetworkXError
If the graph `G` is directed and the root node
is not specified or is not in G.
Examples
--------
>>> G = nx.complete_graph(5)
>>> round(nx.number_of_spanning_trees(G))
125
>>> G = nx.Graph()
>>> G.add_edge(1, 2, weight=2)
>>> G.add_edge(1, 3, weight=1)
>>> G.add_edge(2, 3, weight=1)
>>> round(nx.number_of_spanning_trees(G, weight="weight"))
5
Notes
-----
Self-loops are excluded. Multi-edges are contracted in one edge
equal to the sum of the weights.
References
----------
.. [1] Wikipedia
"Kirchhoff's theorem."
https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem
.. [2] Kirchhoff, G. R.
Über die Auflösung der Gleichungen, auf welche man
bei der Untersuchung der linearen Vertheilung
Galvanischer Ströme geführt wird
Annalen der Physik und Chemie, vol. 72, pp. 497-508, 1847.
.. [3] Margoliash, J.
"Matrix-Tree Theorem for Directed Graphs"
https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Margoliash.pdf
| def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None):
"""
Sample a random spanning tree using the edges weights of `G`.
This function supports two different methods for determining the
probability of the graph. If ``multiplicative=True``, the probability
is based on the product of edge weights, and if ``multiplicative=False``
it is based on the sum of the edge weight. However, since it is
easier to determine the total weight of all spanning trees for the
multiplicative version, that is significantly faster and should be used if
possible. Additionally, setting `weight` to `None` will cause a spanning tree
to be selected with uniform probability.
The function uses algorithm A8 in [1]_ .
Parameters
----------
G : nx.Graph
An undirected version of the original graph.
weight : string
The edge key for the edge attribute holding edge weight.
multiplicative : bool, default=True
If `True`, the probability of each tree is the product of its edge weight
over the sum of the product of all the spanning trees in the graph. If
`False`, the probability is the sum of its edge weight over the sum of
the sum of weights for all spanning trees in the graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
nx.Graph
A spanning tree using the distribution defined by the weight of the tree.
References
----------
.. [1] V. Kulkarni, Generating random combinatorial objects, Journal of
Algorithms, 11 (1990), pp. 185–207
"""
def find_node(merged_nodes, node):
"""
We can think of clusters of contracted nodes as having one
representative in the graph. Each node which is not in merged_nodes
is still its own representative. Since a representative can be later
contracted, we need to recursively search though the dict to find
the final representative, but once we know it we can use path
compression to speed up the access of the representative for next time.
This cannot be replaced by the standard NetworkX union_find since that
data structure will merge nodes with less representing nodes into the
one with more representing nodes but this function requires we merge
them using the order that contract_edges contracts using.
Parameters
----------
merged_nodes : dict
The dict storing the mapping from node to representative
node
The node whose representative we seek
Returns
-------
The representative of the `node`
"""
if node not in merged_nodes:
return node
else:
rep = find_node(merged_nodes, merged_nodes[node])
merged_nodes[node] = rep
return rep
def prepare_graph():
"""
For the graph `G`, remove all edges not in the set `V` and then
contract all edges in the set `U`.
Returns
-------
A copy of `G` which has had all edges not in `V` removed and all edges
in `U` contracted.
"""
# The result is a MultiGraph version of G so that parallel edges are
# allowed during edge contraction
result = nx.MultiGraph(incoming_graph_data=G)
# Remove all edges not in V
edges_to_remove = set(result.edges()).difference(V)
result.remove_edges_from(edges_to_remove)
# Contract all edges in U
#
# Imagine that you have two edges to contract and they share an
# endpoint like this:
# [0] ----- [1] ----- [2]
# If we contract (0, 1) first, the contraction function will always
# delete the second node it is passed so the resulting graph would be
# [0] ----- [2]
# and edge (1, 2) no longer exists but (0, 2) would need to be contracted
# in its place now. That is why I use the below dict as a merge-find
# data structure with path compression to track how the nodes are merged.
merged_nodes = {}
for u, v in U:
u_rep = find_node(merged_nodes, u)
v_rep = find_node(merged_nodes, v)
# We cannot contract a node with itself
if u_rep == v_rep:
continue
nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False)
merged_nodes[v_rep] = u_rep
return merged_nodes, result
def spanning_tree_total_weight(G, weight):
"""
Find the sum of weights of the spanning trees of `G` using the
appropriate `method`.
This is easy if the chosen method is 'multiplicative', since we can
use Kirchhoff's Tree Matrix Theorem directly. However, with the
'additive' method, this process is slightly more complex and less
computationally efficient as we have to find the number of spanning
trees which contain each possible edge in the graph.
Parameters
----------
G : NetworkX Graph
The graph to find the total weight of all spanning trees on.
weight : string
The key for the weight edge attribute of the graph.
Returns
-------
float
The sum of either the multiplicative or additive weight for all
spanning trees in the graph.
"""
if multiplicative:
return nx.total_spanning_tree_weight(G, weight)
else:
# There are two cases for the total spanning tree additive weight.
# 1. There is one edge in the graph. Then the only spanning tree is
# that edge itself, which will have a total weight of that edge
# itself.
if G.number_of_edges() == 1:
return G.edges(data=weight).__iter__().__next__()[2]
# 2. There are no edges or two or more edges in the graph. Then, we find the
# total weight of the spanning trees using the formula in the
# reference paper: take the weight of each edge and multiply it by
# the number of spanning trees which include that edge. This
# can be accomplished by contracting the edge and finding the
# multiplicative total spanning tree weight if the weight of each edge
# is assumed to be 1, which is conveniently built into networkx already,
# by calling total_spanning_tree_weight with weight=None.
# Note that with no edges the returned value is just zero.
else:
total = 0
for u, v, w in G.edges(data=weight):
total += w * nx.total_spanning_tree_weight(
nx.contracted_edge(G, edge=(u, v), self_loops=False), None
)
return total
if G.number_of_nodes() < 2:
# no edges in the spanning tree
return nx.empty_graph(G.nodes)
U = set()
st_cached_value = 0
V = set(G.edges())
shuffled_edges = list(G.edges())
seed.shuffle(shuffled_edges)
for u, v in shuffled_edges:
e_weight = G[u][v][weight] if weight is not None else 1
node_map, prepared_G = prepare_graph()
G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight)
# Add the edge to U so that we can compute the total tree weight
# assuming we include that edge
# Now, if (u, v) cannot exist in G because it is fully contracted out
# of existence, then it by definition cannot influence G_e's Kirchhoff
# value. But, we also cannot pick it.
rep_edge = (find_node(node_map, u), find_node(node_map, v))
# Check to see if the 'representative edge' for the current edge is
# in prepared_G. If so, then we can pick it.
if rep_edge in prepared_G.edges:
prepared_G_e = nx.contracted_edge(
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|
30,979 | networkx.generators.small | octahedral_graph |
Returns the Platonic Octahedral graph.
The octahedral graph is the 6-node 12-edge Platonic graph having the
connectivity of the octahedron [1]_. If 6 couples go to a party,
and each person shakes hands with every person except his or her partner,
then this graph describes the set of handshakes that take place;
for this reason it is also called the cocktail party graph [2]_.
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Octahedral graph
References
----------
.. [1] https://mathworld.wolfram.com/OctahedralGraph.html
.. [2] https://en.wikipedia.org/wiki/Tur%C3%A1n_graph#Special_cases
| def _raise_on_directed(func):
"""
A decorator which inspects the `create_using` argument and raises a
NetworkX exception when `create_using` is a DiGraph (class or instance) for
graph generators that do not support directed outputs.
"""
@wraps(func)
def wrapper(*args, **kwargs):
if kwargs.get("create_using") is not None:
G = nx.empty_graph(create_using=kwargs["create_using"])
if G.is_directed():
raise NetworkXError("Directed Graph not supported")
return func(*args, **kwargs)
return wrapper
| (create_using=None, *, backend=None, **backend_kwargs) | [
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|
30,983 | networkx.algorithms.similarity | optimal_edit_paths | Returns all minimum-cost edit paths transforming G1 to G2.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
Returns
-------
edit_paths : list of tuples (node_edit_path, edge_edit_path)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
Optimal edit path cost (graph edit distance). When the cost
is zero, it indicates that `G1` and `G2` are isomorphic.
Examples
--------
>>> G1 = nx.cycle_graph(4)
>>> G2 = nx.wheel_graph(5)
>>> paths, cost = nx.optimal_edit_paths(G1, G2)
>>> len(paths)
40
>>> cost
5.0
Notes
-----
To transform `G1` into a graph isomorphic to `G2`, apply the node
and edge edits in the returned ``edit_paths``.
In the case of isomorphic graphs, the cost is zero, and the paths
represent different isomorphic mappings (isomorphisms). That is, the
edits involve renaming nodes and edges to match the structure of `G2`.
See Also
--------
graph_edit_distance, optimize_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
| def optimize_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
strictly_decreasing=True,
roots=None,
timeout=None,
):
"""GED (graph edit distance) calculation: advanced interface.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Graph edit distance is defined as minimum cost of edit path.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost. Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
See Also
--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
"""
# TODO: support DiGraph
import numpy as np
import scipy as sp
@dataclass
class CostMatrix:
C: ...
lsa_row_ind: ...
lsa_col_ind: ...
ls: ...
def make_CostMatrix(C, m, n):
# assert(C.shape == (m + n, m + n))
lsa_row_ind, lsa_col_ind = sp.optimize.linear_sum_assignment(C)
# Fixup dummy assignments:
# each substitution i<->j should have dummy assignment m+j<->n+i
# NOTE: fast reduce of Cv relies on it
# assert len(lsa_row_ind) == len(lsa_col_ind)
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
subst_ind = [k for k, i, j in indexes if i < m and j < n]
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
dummy_ind = [k for k, i, j in indexes if i >= m and j >= n]
# assert len(subst_ind) == len(dummy_ind)
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n
return CostMatrix(
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
)
def extract_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k in i or k - m in j for k in range(m + n)]
col_ind = [k in j or k - n in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k not in i and k - m not in j for k in range(m + n)]
col_ind = [k not in j and k - n not in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_ind(ind, i):
# assert set(ind) == set(range(len(ind)))
rind = ind[[k not in i for k in ind]]
for k in set(i):
rind[rind >= k] -= 1
return rind
def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=None):
"""
Parameters:
u, v: matched vertices, u=None or v=None for
deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_uv: partial vertex edit path
list of tuples (u, v) of previously matched vertex
mappings u<->v, u=None or v=None for
deletion/insertion
Returns:
list of (i, j): indices of edge mappings g<->h
localCe: local CostMatrix of edge mappings
(basically submatrix of Ce at cross of rows i, cols j)
"""
M = len(pending_g)
N = len(pending_h)
# assert Ce.C.shape == (M + N, M + N)
# only attempt to match edges after one node match has been made
# this will stop self-edges on the first node being automatically deleted
# even when a substitution is the better option
if matched_uv is None or len(matched_uv) == 0:
g_ind = []
h_ind = []
else:
| (G1, G2, node_match=None, edge_match=None, node_subst_cost=None, node_del_cost=None, node_ins_cost=None, edge_subst_cost=None, edge_del_cost=None, edge_ins_cost=None, upper_bound=None, *, backend=None, **backend_kwargs) | [
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|
30,984 | networkx.algorithms.similarity | optimize_edit_paths | GED (graph edit distance) calculation: advanced interface.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Graph edit distance is defined as minimum cost of edit path.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost. Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
See Also
--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
| def optimize_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
strictly_decreasing=True,
roots=None,
timeout=None,
):
"""GED (graph edit distance) calculation: advanced interface.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Graph edit distance is defined as minimum cost of edit path.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost. Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
See Also
--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
"""
# TODO: support DiGraph
import numpy as np
import scipy as sp
@dataclass
class CostMatrix:
C: ...
lsa_row_ind: ...
lsa_col_ind: ...
ls: ...
def make_CostMatrix(C, m, n):
# assert(C.shape == (m + n, m + n))
lsa_row_ind, lsa_col_ind = sp.optimize.linear_sum_assignment(C)
# Fixup dummy assignments:
# each substitution i<->j should have dummy assignment m+j<->n+i
# NOTE: fast reduce of Cv relies on it
# assert len(lsa_row_ind) == len(lsa_col_ind)
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
subst_ind = [k for k, i, j in indexes if i < m and j < n]
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
dummy_ind = [k for k, i, j in indexes if i >= m and j >= n]
# assert len(subst_ind) == len(dummy_ind)
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n
return CostMatrix(
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
)
def extract_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k in i or k - m in j for k in range(m + n)]
col_ind = [k in j or k - n in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k not in i and k - m not in j for k in range(m + n)]
col_ind = [k not in j and k - n not in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_ind(ind, i):
# assert set(ind) == set(range(len(ind)))
rind = ind[[k not in i for k in ind]]
for k in set(i):
rind[rind >= k] -= 1
return rind
def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=None):
"""
Parameters:
u, v: matched vertices, u=None or v=None for
deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_uv: partial vertex edit path
list of tuples (u, v) of previously matched vertex
mappings u<->v, u=None or v=None for
deletion/insertion
Returns:
list of (i, j): indices of edge mappings g<->h
localCe: local CostMatrix of edge mappings
(basically submatrix of Ce at cross of rows i, cols j)
"""
M = len(pending_g)
N = len(pending_h)
# assert Ce.C.shape == (M + N, M + N)
# only attempt to match edges after one node match has been made
# this will stop self-edges on the first node being automatically deleted
# even when a substitution is the better option
if matched_uv is None or len(matched_uv) == 0:
g_ind = []
h_ind = []
else:
| (G1, G2, node_match=None, edge_match=None, node_subst_cost=None, node_del_cost=None, node_ins_cost=None, edge_subst_cost=None, edge_del_cost=None, edge_ins_cost=None, upper_bound=None, strictly_decreasing=True, roots=None, timeout=None, *, backend=None, **backend_kwargs) | [
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|
30,985 | networkx.algorithms.similarity | optimize_graph_edit_distance | Returns consecutive approximations of GED (graph edit distance)
between graphs G1 and G2.
Graph edit distance is a graph similarity measure analogous to
Levenshtein distance for strings. It is defined as minimum cost
of edit path (sequence of node and edge edit operations)
transforming graph G1 to graph isomorphic to G2.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
Returns
-------
Generator of consecutive approximations of graph edit distance.
Examples
--------
>>> G1 = nx.cycle_graph(6)
>>> G2 = nx.wheel_graph(7)
>>> for v in nx.optimize_graph_edit_distance(G1, G2):
... minv = v
>>> minv
7.0
See Also
--------
graph_edit_distance, optimize_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
| def optimize_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
strictly_decreasing=True,
roots=None,
timeout=None,
):
"""GED (graph edit distance) calculation: advanced interface.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Graph edit distance is defined as minimum cost of edit path.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost. Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
See Also
--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
"""
# TODO: support DiGraph
import numpy as np
import scipy as sp
@dataclass
class CostMatrix:
C: ...
lsa_row_ind: ...
lsa_col_ind: ...
ls: ...
def make_CostMatrix(C, m, n):
# assert(C.shape == (m + n, m + n))
lsa_row_ind, lsa_col_ind = sp.optimize.linear_sum_assignment(C)
# Fixup dummy assignments:
# each substitution i<->j should have dummy assignment m+j<->n+i
# NOTE: fast reduce of Cv relies on it
# assert len(lsa_row_ind) == len(lsa_col_ind)
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
subst_ind = [k for k, i, j in indexes if i < m and j < n]
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
dummy_ind = [k for k, i, j in indexes if i >= m and j >= n]
# assert len(subst_ind) == len(dummy_ind)
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n
return CostMatrix(
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
)
def extract_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k in i or k - m in j for k in range(m + n)]
col_ind = [k in j or k - n in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k not in i and k - m not in j for k in range(m + n)]
col_ind = [k not in j and k - n not in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_ind(ind, i):
# assert set(ind) == set(range(len(ind)))
rind = ind[[k not in i for k in ind]]
for k in set(i):
rind[rind >= k] -= 1
return rind
def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=None):
"""
Parameters:
u, v: matched vertices, u=None or v=None for
deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_uv: partial vertex edit path
list of tuples (u, v) of previously matched vertex
mappings u<->v, u=None or v=None for
deletion/insertion
Returns:
list of (i, j): indices of edge mappings g<->h
localCe: local CostMatrix of edge mappings
(basically submatrix of Ce at cross of rows i, cols j)
"""
M = len(pending_g)
N = len(pending_h)
# assert Ce.C.shape == (M + N, M + N)
# only attempt to match edges after one node match has been made
# this will stop self-edges on the first node being automatically deleted
# even when a substitution is the better option
if matched_uv is None or len(matched_uv) == 0:
g_ind = []
h_ind = []
else:
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|
30,993 | networkx.algorithms.similarity | panther_similarity | Returns the Panther similarity of nodes in the graph `G` to node ``v``.
Panther is a similarity metric that says "two objects are considered
to be similar if they frequently appear on the same paths." [1]_.
Parameters
----------
G : NetworkX graph
A NetworkX graph
source : node
Source node for which to find the top `k` similar other nodes
k : int (default = 5)
The number of most similar nodes to return.
path_length : int (default = 5)
How long the randomly generated paths should be (``T`` in [1]_)
c : float (default = 0.5)
A universal positive constant used to scale the number
of sample random paths to generate.
delta : float (default = 0.1)
The probability that the similarity $S$ is not an epsilon-approximation to (R, phi),
where $R$ is the number of random paths and $\phi$ is the probability
that an element sampled from a set $A \subseteq D$, where $D$ is the domain.
eps : float or None (default = None)
The error bound. Per [1]_, a good value is ``sqrt(1/|E|)``. Therefore,
if no value is provided, the recommended computed value will be used.
weight : string or None, optional (default="weight")
The name of an edge attribute that holds the numerical value
used as a weight. If None then each edge has weight 1.
Returns
-------
similarity : dictionary
Dictionary of nodes to similarity scores (as floats). Note:
the self-similarity (i.e., ``v``) will not be included in
the returned dictionary. So, for ``k = 5``, a dictionary of
top 4 nodes and their similarity scores will be returned.
Raises
------
NetworkXUnfeasible
If `source` is an isolated node.
NodeNotFound
If `source` is not in `G`.
Notes
-----
The isolated nodes in `G` are ignored.
Examples
--------
>>> G = nx.star_graph(10)
>>> sim = nx.panther_similarity(G, 0)
References
----------
.. [1] Zhang, J., Tang, J., Ma, C., Tong, H., Jing, Y., & Li, J.
Panther: Fast top-k similarity search on large networks.
In Proceedings of the ACM SIGKDD International Conference
on Knowledge Discovery and Data Mining (Vol. 2015-August, pp. 1445–1454).
Association for Computing Machinery. https://doi.org/10.1145/2783258.2783267.
| def optimize_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
strictly_decreasing=True,
roots=None,
timeout=None,
):
"""GED (graph edit distance) calculation: advanced interface.
Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2. Edit operations
include substitutions, deletions, and insertions.
Graph edit distance is defined as minimum cost of edit path.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.
Ignored if node_subst_cost is specified. If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.
edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute
dictionaries of the edges under consideration.
Ignored if edge_subst_cost is specified. If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.
node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.
The functions will be called like
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).
That is, the functions will receive the node attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).
If node_del_cost is not specified then default node deletion
cost of 1 is used. If node_ins_cost is not specified then
default node insertion cost of 1 is used.
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.
The functions will be called like
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).
That is, the functions will receive the edge attribute
dictionaries as inputs. The functions are expected to return
positive numeric values.
Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).
If edge_del_cost is not specified then default edge deletion
cost of 1 is used. If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.
upper_bound : numeric
Maximum edit distance to consider.
strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost. Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.
roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.
timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.
Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric
See Also
--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
References
----------
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
"""
# TODO: support DiGraph
import numpy as np
import scipy as sp
@dataclass
class CostMatrix:
C: ...
lsa_row_ind: ...
lsa_col_ind: ...
ls: ...
def make_CostMatrix(C, m, n):
# assert(C.shape == (m + n, m + n))
lsa_row_ind, lsa_col_ind = sp.optimize.linear_sum_assignment(C)
# Fixup dummy assignments:
# each substitution i<->j should have dummy assignment m+j<->n+i
# NOTE: fast reduce of Cv relies on it
# assert len(lsa_row_ind) == len(lsa_col_ind)
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
subst_ind = [k for k, i, j in indexes if i < m and j < n]
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
dummy_ind = [k for k, i, j in indexes if i >= m and j >= n]
# assert len(subst_ind) == len(dummy_ind)
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n
return CostMatrix(
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
)
def extract_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k in i or k - m in j for k in range(m + n)]
col_ind = [k in j or k - n in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k not in i and k - m not in j for k in range(m + n)]
col_ind = [k not in j and k - n not in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]
def reduce_ind(ind, i):
# assert set(ind) == set(range(len(ind)))
rind = ind[[k not in i for k in ind]]
for k in set(i):
rind[rind >= k] -= 1
return rind
def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=None):
"""
Parameters:
u, v: matched vertices, u=None or v=None for
deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_uv: partial vertex edit path
list of tuples (u, v) of previously matched vertex
mappings u<->v, u=None or v=None for
deletion/insertion
Returns:
list of (i, j): indices of edge mappings g<->h
localCe: local CostMatrix of edge mappings
(basically submatrix of Ce at cross of rows i, cols j)
"""
M = len(pending_g)
N = len(pending_h)
# assert Ce.C.shape == (M + N, M + N)
# only attempt to match edges after one node match has been made
# this will stop self-edges on the first node being automatically deleted
# even when a substitution is the better option
if matched_uv is None or len(matched_uv) == 0:
g_ind = []
h_ind = []
else:
| (G, source, k=5, path_length=5, c=0.5, delta=0.1, eps=None, weight='weight', *, backend=None, **backend_kwargs) | [
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|
30,994 | networkx.generators.small | pappus_graph |
Returns the Pappus graph.
The Pappus graph is a cubic symmetric distance-regular graph with 18 nodes
and 27 edges. It is Hamiltonian and can be represented in LCF notation as
[5,7,-7,7,-7,-5]^3 [1]_.
Returns
-------
G : networkx Graph
Pappus graph
References
----------
.. [1] https://en.wikipedia.org/wiki/Pappus_graph
| def sedgewick_maze_graph(create_using=None):
"""
Return a small maze with a cycle.
This is the maze used in Sedgewick, 3rd Edition, Part 5, Graph
Algorithms, Chapter 18, e.g. Figure 18.2 and following [1]_.
Nodes are numbered 0,..,7
Parameters
----------
create_using : NetworkX graph constructor, optional (default=nx.Graph)
Graph type to create. If graph instance, then cleared before populated.
Returns
-------
G : networkx Graph
Small maze with a cycle
References
----------
.. [1] Figure 18.2, Chapter 18, Graph Algorithms (3rd Ed), Sedgewick
"""
G = empty_graph(0, create_using)
G.add_nodes_from(range(8))
G.add_edges_from([[0, 2], [0, 7], [0, 5]])
G.add_edges_from([[1, 7], [2, 6]])
G.add_edges_from([[3, 4], [3, 5]])
G.add_edges_from([[4, 5], [4, 7], [4, 6]])
G.name = "Sedgewick Maze"
return G
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|
30,997 | networkx.readwrite.gml | parse_gml | Parse GML graph from a string or iterable.
Parameters
----------
lines : string or iterable of strings
Data in GML format.
label : string, optional
If not None, the parsed nodes will be renamed according to node
attributes indicated by `label`. Default value: 'label'.
destringizer : callable, optional
A `destringizer` that recovers values stored as strings in GML. If it
cannot convert a string to a value, a `ValueError` is raised. Default
value : None.
Returns
-------
G : NetworkX graph
The parsed graph.
Raises
------
NetworkXError
If the input cannot be parsed.
See Also
--------
write_gml, read_gml
Notes
-----
This stores nested GML attributes as dictionaries in the NetworkX graph,
node, and edge attribute structures.
GML files are stored using a 7-bit ASCII encoding with any extended
ASCII characters (iso8859-1) appearing as HTML character entities.
Without specifying a `stringizer`/`destringizer`, the code is capable of
writing `int`/`float`/`str`/`dict`/`list` data as required by the GML
specification. For writing other data types, and for reading data other
than `str` you need to explicitly supply a `stringizer`/`destringizer`.
For additional documentation on the GML file format, please see the
`GML url <https://web.archive.org/web/20190207140002/http://www.fim.uni-passau.de/index.php?id=17297&L=1>`_.
See the module docstring :mod:`networkx.readwrite.gml` for more details.
| def generate_gml(G, stringizer=None):
r"""Generate a single entry of the graph `G` in GML format.
Parameters
----------
G : NetworkX graph
The graph to be converted to GML.
stringizer : callable, optional
A `stringizer` which converts non-int/non-float/non-dict values into
strings. If it cannot convert a value into a string, it should raise a
`ValueError` to indicate that. Default value: None.
Returns
-------
lines: generator of strings
Lines of GML data. Newlines are not appended.
Raises
------
NetworkXError
If `stringizer` cannot convert a value into a string, or the value to
convert is not a string while `stringizer` is None.
See Also
--------
literal_stringizer
Notes
-----
Graph attributes named 'directed', 'multigraph', 'node' or
'edge', node attributes named 'id' or 'label', edge attributes
named 'source' or 'target' (or 'key' if `G` is a multigraph)
are ignored because these attribute names are used to encode the graph
structure.
GML files are stored using a 7-bit ASCII encoding with any extended
ASCII characters (iso8859-1) appearing as HTML character entities.
Without specifying a `stringizer`/`destringizer`, the code is capable of
writing `int`/`float`/`str`/`dict`/`list` data as required by the GML
specification. For writing other data types, and for reading data other
than `str` you need to explicitly supply a `stringizer`/`destringizer`.
For additional documentation on the GML file format, please see the
`GML url <https://web.archive.org/web/20190207140002/http://www.fim.uni-passau.de/index.php?id=17297&L=1>`_.
See the module docstring :mod:`networkx.readwrite.gml` for more details.
Examples
--------
>>> G = nx.Graph()
>>> G.add_node("1")
>>> print("\n".join(nx.generate_gml(G)))
graph [
node [
id 0
label "1"
]
]
>>> G = nx.MultiGraph([("a", "b"), ("a", "b")])
>>> print("\n".join(nx.generate_gml(G)))
graph [
multigraph 1
node [
id 0
label "a"
]
node [
id 1
label "b"
]
edge [
source 0
target 1
key 0
]
edge [
source 0
target 1
key 1
]
]
"""
valid_keys = re.compile("^[A-Za-z][0-9A-Za-z_]*$")
def stringize(key, value, ignored_keys, indent, in_list=False):
if not isinstance(key, str):
raise NetworkXError(f"{key!r} is not a string")
if not valid_keys.match(key):
raise NetworkXError(f"{key!r} is not a valid key")
if not isinstance(key, str):
key = str(key)
if key not in ignored_keys:
if isinstance(value, int | bool):
if key == "label":
yield indent + key + ' "' + str(value) + '"'
elif value is True:
# python bool is an instance of int
yield indent + key + " 1"
elif value is False:
yield indent + key + " 0"
# GML only supports signed 32-bit integers
elif value < -(2**31) or value >= 2**31:
yield indent + key + ' "' + str(value) + '"'
else:
yield indent + key + " " + str(value)
elif isinstance(value, float):
text = repr(value).upper()
# GML matches INF to keys, so prepend + to INF. Use repr(float(*))
# instead of string literal to future proof against changes to repr.
if text == repr(float("inf")).upper():
text = "+" + text
else:
# GML requires that a real literal contain a decimal point, but
# repr may not output a decimal point when the mantissa is
# integral and hence needs fixing.
epos = text.rfind("E")
if epos != -1 and text.find(".", 0, epos) == -1:
text = text[:epos] + "." + text[epos:]
if key == "label":
yield indent + key + ' "' + text + '"'
else:
yield indent + key + " " + text
elif isinstance(value, dict):
yield indent + key + " ["
next_indent = indent + " "
for key, value in value.items():
yield from stringize(key, value, (), next_indent)
yield indent + "]"
elif isinstance(value, tuple) and key == "label":
yield indent + key + f" \"({','.join(repr(v) for v in value)})\""
elif isinstance(value, list | tuple) and key != "label" and not in_list:
if len(value) == 0:
yield indent + key + " " + f'"{value!r}"'
if len(value) == 1:
yield indent + key + " " + f'"{LIST_START_VALUE}"'
for val in value:
yield from stringize(key, val, (), indent, True)
else:
if stringizer:
try:
value = stringizer(value)
except ValueError as err:
raise NetworkXError(
f"{value!r} cannot be converted into a string"
) from err
if not isinstance(value, str):
raise NetworkXError(f"{value!r} is not a string")
yield indent + key + ' "' + escape(value) + '"'
multigraph = G.is_multigraph()
yield "graph ["
# Output graph attributes
if G.is_directed():
yield " directed 1"
if multigraph:
yield " multigraph 1"
ignored_keys = {"directed", "multigraph", "node", "edge"}
for attr, value in G.graph.items():
yield from stringize(attr, value, ignored_keys, " ")
# Output node data
node_id = dict(zip(G, range(len(G))))
ignored_keys = {"id", "label"}
for node, attrs in G.nodes.items():
yield " node ["
yield " id " + str(node_id[node])
yield from stringize("label", node, (), " ")
for attr, value in attrs.items():
yield from stringize(attr, value, ignored_keys, " ")
yield " ]"
# Output edge data
ignored_keys = {"source", "target"}
kwargs = {"data": True}
if multigraph:
ignored_keys.add("key")
kwargs["keys"] = True
for e in G.edges(**kwargs):
yield " edge ["
yield " source " + str(node_id[e[0]])
yield " target " + str(node_id[e[1]])
if multigraph:
yield from stringize("key", e[2], (), " ")
for attr, value in e[-1].items():
yield from stringize(attr, value, ignored_keys, " ")
yield " ]"
yield "]"
| (lines, label='label', destringizer=None, *, backend=None, **backend_kwargs) | [
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|
30,998 | networkx.readwrite.graphml | parse_graphml | Read graph in GraphML format from string.
Parameters
----------
graphml_string : string
String containing graphml information
(e.g., contents of a graphml file).
node_type: Python type (default: str)
Convert node ids to this type
edge_key_type: Python type (default: int)
Convert graphml edge ids to this type. Multigraphs use id as edge key.
Non-multigraphs add to edge attribute dict with name "id".
force_multigraph : bool (default: False)
If True, return a multigraph with edge keys. If False (the default)
return a multigraph when multiedges are in the graph.
Returns
-------
graph: NetworkX graph
If no parallel edges are found a Graph or DiGraph is returned.
Otherwise a MultiGraph or MultiDiGraph is returned.
Examples
--------
>>> G = nx.path_graph(4)
>>> linefeed = chr(10) # linefeed =
>>> s = linefeed.join(nx.generate_graphml(G))
>>> H = nx.parse_graphml(s)
Notes
-----
Default node and edge attributes are not propagated to each node and edge.
They can be obtained from `G.graph` and applied to node and edge attributes
if desired using something like this:
>>> default_color = G.graph["node_default"]["color"] # doctest: +SKIP
>>> for node, data in G.nodes(data=True): # doctest: +SKIP
... if "color" not in data:
... data["color"] = default_color
>>> default_color = G.graph["edge_default"]["color"] # doctest: +SKIP
>>> for u, v, data in G.edges(data=True): # doctest: +SKIP
... if "color" not in data:
... data["color"] = default_color
This implementation does not support mixed graphs (directed and unidirected
edges together), hypergraphs, nested graphs, or ports.
For multigraphs the GraphML edge "id" will be used as the edge
key. If not specified then they "key" attribute will be used. If
there is no "key" attribute a default NetworkX multigraph edge key
will be provided.
| def add_graph_element(self, G):
"""
Serialize graph G in GraphML to the stream.
"""
if G.is_directed():
default_edge_type = "directed"
else:
default_edge_type = "undirected"
graphid = G.graph.pop("id", None)
if graphid is None:
graph_element = self._xml.element("graph", edgedefault=default_edge_type)
else:
graph_element = self._xml.element(
"graph", edgedefault=default_edge_type, id=graphid
)
# gather attributes types for the whole graph
# to find the most general numeric format needed.
# Then pass through attributes to create key_id for each.
graphdata = {
k: v
for k, v in G.graph.items()
if k not in ("node_default", "edge_default")
}
node_default = G.graph.get("node_default", {})
edge_default = G.graph.get("edge_default", {})
# Graph attributes
for k, v in graphdata.items():
self.attribute_types[(str(k), "graph")].add(type(v))
for k, v in graphdata.items():
element_type = self.get_xml_type(self.attr_type(k, "graph", v))
self.get_key(str(k), element_type, "graph", None)
# Nodes and data
for node, d in G.nodes(data=True):
for k, v in d.items():
self.attribute_types[(str(k), "node")].add(type(v))
for node, d in G.nodes(data=True):
for k, v in d.items():
T = self.get_xml_type(self.attr_type(k, "node", v))
self.get_key(str(k), T, "node", node_default.get(k))
# Edges and data
if G.is_multigraph():
for u, v, ekey, d in G.edges(keys=True, data=True):
for k, v in d.items():
self.attribute_types[(str(k), "edge")].add(type(v))
for u, v, ekey, d in G.edges(keys=True, data=True):
for k, v in d.items():
T = self.get_xml_type(self.attr_type(k, "edge", v))
self.get_key(str(k), T, "edge", edge_default.get(k))
else:
for u, v, d in G.edges(data=True):
for k, v in d.items():
self.attribute_types[(str(k), "edge")].add(type(v))
for u, v, d in G.edges(data=True):
for k, v in d.items():
T = self.get_xml_type(self.attr_type(k, "edge", v))
self.get_key(str(k), T, "edge", edge_default.get(k))
# Now add attribute keys to the xml file
for key in self.xml:
self._xml.write(key, pretty_print=self._prettyprint)
# The incremental_writer writes each node/edge as it is created
incremental_writer = IncrementalElement(self._xml, self._prettyprint)
with graph_element:
self.add_attributes("graph", incremental_writer, graphdata, {})
self.add_nodes(G, incremental_writer) # adds attributes too
self.add_edges(G, incremental_writer) # adds attributes too
| (graphml_string, node_type=<class 'str'>, edge_key_type=<class 'int'>, force_multigraph=False, *, backend=None, **backend_kwargs) | [
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|
31,003 | networkx.algorithms.tree.mst | partition_spanning_tree |
Find a spanning tree while respecting a partition of edges.
Edges can be flagged as either `INCLUDED` which are required to be in the
returned tree, `EXCLUDED`, which cannot be in the returned tree and `OPEN`.
This is used in the SpanningTreeIterator to create new partitions following
the algorithm of Sörensen and Janssens [1]_.
Parameters
----------
G : undirected graph
An undirected graph.
minimum : bool (default: True)
Determines whether the returned tree is the minimum spanning tree of
the partition of the maximum one.
weight : str
Data key to use for edge weights.
partition : str
The key for the edge attribute containing the partition
data on the graph. Edges can be included, excluded or open using the
`EdgePartition` enum.
ignore_nan : bool (default: False)
If a NaN is found as an edge weight normally an exception is raised.
If `ignore_nan is True` then that edge is ignored instead.
Returns
-------
G : NetworkX Graph
A minimum spanning tree using all of the included edges in the graph and
none of the excluded edges.
References
----------
.. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning
trees in order of increasing cost, Pesquisa Operacional, 2005-08,
Vol. 25 (2), p. 219-229,
https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en
| def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None):
"""
Sample a random spanning tree using the edges weights of `G`.
This function supports two different methods for determining the
probability of the graph. If ``multiplicative=True``, the probability
is based on the product of edge weights, and if ``multiplicative=False``
it is based on the sum of the edge weight. However, since it is
easier to determine the total weight of all spanning trees for the
multiplicative version, that is significantly faster and should be used if
possible. Additionally, setting `weight` to `None` will cause a spanning tree
to be selected with uniform probability.
The function uses algorithm A8 in [1]_ .
Parameters
----------
G : nx.Graph
An undirected version of the original graph.
weight : string
The edge key for the edge attribute holding edge weight.
multiplicative : bool, default=True
If `True`, the probability of each tree is the product of its edge weight
over the sum of the product of all the spanning trees in the graph. If
`False`, the probability is the sum of its edge weight over the sum of
the sum of weights for all spanning trees in the graph.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
nx.Graph
A spanning tree using the distribution defined by the weight of the tree.
References
----------
.. [1] V. Kulkarni, Generating random combinatorial objects, Journal of
Algorithms, 11 (1990), pp. 185–207
"""
def find_node(merged_nodes, node):
"""
We can think of clusters of contracted nodes as having one
representative in the graph. Each node which is not in merged_nodes
is still its own representative. Since a representative can be later
contracted, we need to recursively search though the dict to find
the final representative, but once we know it we can use path
compression to speed up the access of the representative for next time.
This cannot be replaced by the standard NetworkX union_find since that
data structure will merge nodes with less representing nodes into the
one with more representing nodes but this function requires we merge
them using the order that contract_edges contracts using.
Parameters
----------
merged_nodes : dict
The dict storing the mapping from node to representative
node
The node whose representative we seek
Returns
-------
The representative of the `node`
"""
if node not in merged_nodes:
return node
else:
rep = find_node(merged_nodes, merged_nodes[node])
merged_nodes[node] = rep
return rep
def prepare_graph():
"""
For the graph `G`, remove all edges not in the set `V` and then
contract all edges in the set `U`.
Returns
-------
A copy of `G` which has had all edges not in `V` removed and all edges
in `U` contracted.
"""
# The result is a MultiGraph version of G so that parallel edges are
# allowed during edge contraction
result = nx.MultiGraph(incoming_graph_data=G)
# Remove all edges not in V
edges_to_remove = set(result.edges()).difference(V)
result.remove_edges_from(edges_to_remove)
# Contract all edges in U
#
# Imagine that you have two edges to contract and they share an
# endpoint like this:
# [0] ----- [1] ----- [2]
# If we contract (0, 1) first, the contraction function will always
# delete the second node it is passed so the resulting graph would be
# [0] ----- [2]
# and edge (1, 2) no longer exists but (0, 2) would need to be contracted
# in its place now. That is why I use the below dict as a merge-find
# data structure with path compression to track how the nodes are merged.
merged_nodes = {}
for u, v in U:
u_rep = find_node(merged_nodes, u)
v_rep = find_node(merged_nodes, v)
# We cannot contract a node with itself
if u_rep == v_rep:
continue
nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False)
merged_nodes[v_rep] = u_rep
return merged_nodes, result
def spanning_tree_total_weight(G, weight):
"""
Find the sum of weights of the spanning trees of `G` using the
appropriate `method`.
This is easy if the chosen method is 'multiplicative', since we can
use Kirchhoff's Tree Matrix Theorem directly. However, with the
'additive' method, this process is slightly more complex and less
computationally efficient as we have to find the number of spanning
trees which contain each possible edge in the graph.
Parameters
----------
G : NetworkX Graph
The graph to find the total weight of all spanning trees on.
weight : string
The key for the weight edge attribute of the graph.
Returns
-------
float
The sum of either the multiplicative or additive weight for all
spanning trees in the graph.
"""
if multiplicative:
return nx.total_spanning_tree_weight(G, weight)
else:
# There are two cases for the total spanning tree additive weight.
# 1. There is one edge in the graph. Then the only spanning tree is
# that edge itself, which will have a total weight of that edge
# itself.
if G.number_of_edges() == 1:
return G.edges(data=weight).__iter__().__next__()[2]
# 2. There are no edges or two or more edges in the graph. Then, we find the
# total weight of the spanning trees using the formula in the
# reference paper: take the weight of each edge and multiply it by
# the number of spanning trees which include that edge. This
# can be accomplished by contracting the edge and finding the
# multiplicative total spanning tree weight if the weight of each edge
# is assumed to be 1, which is conveniently built into networkx already,
# by calling total_spanning_tree_weight with weight=None.
# Note that with no edges the returned value is just zero.
else:
total = 0
for u, v, w in G.edges(data=weight):
total += w * nx.total_spanning_tree_weight(
nx.contracted_edge(G, edge=(u, v), self_loops=False), None
)
return total
if G.number_of_nodes() < 2:
# no edges in the spanning tree
return nx.empty_graph(G.nodes)
U = set()
st_cached_value = 0
V = set(G.edges())
shuffled_edges = list(G.edges())
seed.shuffle(shuffled_edges)
for u, v in shuffled_edges:
e_weight = G[u][v][weight] if weight is not None else 1
node_map, prepared_G = prepare_graph()
G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight)
# Add the edge to U so that we can compute the total tree weight
# assuming we include that edge
# Now, if (u, v) cannot exist in G because it is fully contracted out
# of existence, then it by definition cannot influence G_e's Kirchhoff
# value. But, we also cannot pick it.
rep_edge = (find_node(node_map, u), find_node(node_map, v))
# Check to see if the 'representative edge' for the current edge is
# in prepared_G. If so, then we can pick it.
if rep_edge in prepared_G.edges:
prepared_G_e = nx.contracted_edge(
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|
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