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ckpts/universal/global_step80/zero/5.attention.query_key_value.weight/exp_avg_sq.pt

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venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/__init__.py

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+from networkx.algorithms.assortativity.connectivity import *
+from networkx.algorithms.assortativity.correlation import *
+from networkx.algorithms.assortativity.mixing import *
+from networkx.algorithms.assortativity.neighbor_degree import *
+from networkx.algorithms.assortativity.pairs import *

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/connectivity.py

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+from collections import defaultdict
+
+import networkx as nx
+
+__all__ = ["average_degree_connectivity"]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def average_degree_connectivity(
+    G, source="in+out", target="in+out", nodes=None, weight=None
+):
+    r"""Compute the average degree connectivity of graph.
+
+    The average degree connectivity is the average nearest neighbor degree of
+    nodes with degree k. For weighted graphs, an analogous measure can
+    be computed using the weighted average neighbors degree defined in
+    [1]_, for a node `i`, as
+
+    .. math::
+
+        k_{nn,i}^{w} = \frac{1}{s_i} \sum_{j \in N(i)} w_{ij} k_j
+
+    where `s_i` is the weighted degree of node `i`,
+    `w_{ij}` is the weight of the edge that links `i` and `j`,
+    and `N(i)` are the neighbors of node `i`.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    source :  "in"|"out"|"in+out" (default:"in+out")
+       Directed graphs only. Use "in"- or "out"-degree for source node.
+
+    target : "in"|"out"|"in+out" (default:"in+out"
+       Directed graphs only. Use "in"- or "out"-degree for target node.
+
+    nodes : list or iterable (optional)
+        Compute neighbor connectivity for these nodes. The default is all
+        nodes.
+
+    weight : string or None, optional (default=None)
+       The edge attribute that holds the numerical value used as a weight.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    d : dict
+       A dictionary keyed by degree k with the value of average connectivity.
+
+    Raises
+    ------
+    NetworkXError
+        If either `source` or `target` are not one of 'in',
+        'out', or 'in+out'.
+        If either `source` or `target` is passed for an undirected graph.
+
+    Examples
+    --------
+    >>> G = nx.path_graph(4)
+    >>> G.edges[1, 2]["weight"] = 3
+    >>> nx.average_degree_connectivity(G)
+    {1: 2.0, 2: 1.5}
+    >>> nx.average_degree_connectivity(G, weight="weight")
+    {1: 2.0, 2: 1.75}
+
+    See Also
+    --------
+    average_neighbor_degree
+
+    References
+    ----------
+    .. [1] A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani,
+       "The architecture of complex weighted networks".
+       PNAS 101 (11): 3747–3752 (2004).
+    """
+    # First, determine the type of neighbors and the type of degree to use.
+    if G.is_directed():
+        if source not in ("in", "out", "in+out"):
+            raise nx.NetworkXError('source must be one of "in", "out", or "in+out"')
+        if target not in ("in", "out", "in+out"):
+            raise nx.NetworkXError('target must be one of "in", "out", or "in+out"')
+        direction = {"out": G.out_degree, "in": G.in_degree, "in+out": G.degree}
+        neighbor_funcs = {
+            "out": G.successors,
+            "in": G.predecessors,
+            "in+out": G.neighbors,
+        }
+        source_degree = direction[source]
+        target_degree = direction[target]
+        neighbors = neighbor_funcs[source]
+        # `reverse` indicates whether to look at the in-edge when
+        # computing the weight of an edge.
+        reverse = source == "in"
+    else:
+        if source != "in+out" or target != "in+out":
+            raise nx.NetworkXError(
+                f"source and target arguments are only supported for directed graphs"
+            )
+        source_degree = G.degree
+        target_degree = G.degree
+        neighbors = G.neighbors
+        reverse = False
+    dsum = defaultdict(int)
+    dnorm = defaultdict(int)
+    # Check if `source_nodes` is actually a single node in the graph.
+    source_nodes = source_degree(nodes)
+    if nodes in G:
+        source_nodes = [(nodes, source_degree(nodes))]
+    for n, k in source_nodes:
+        nbrdeg = target_degree(neighbors(n))
+        if weight is None:
+            s = sum(d for n, d in nbrdeg)
+        else:  # weight nbr degree by weight of (n,nbr) edge
+            if reverse:
+                s = sum(G[nbr][n].get(weight, 1) * d for nbr, d in nbrdeg)
+            else:
+                s = sum(G[n][nbr].get(weight, 1) * d for nbr, d in nbrdeg)
+        dnorm[k] += source_degree(n, weight=weight)
+        dsum[k] += s
+
+    # normalize
+    return {k: avg if dnorm[k] == 0 else avg / dnorm[k] for k, avg in dsum.items()}

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/correlation.py

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+"""Node assortativity coefficients and correlation measures.
+"""
+import networkx as nx
+from networkx.algorithms.assortativity.mixing import (
+    attribute_mixing_matrix,
+    degree_mixing_matrix,
+)
+from networkx.algorithms.assortativity.pairs import node_degree_xy
+
+__all__ = [
+    "degree_pearson_correlation_coefficient",
+    "degree_assortativity_coefficient",
+    "attribute_assortativity_coefficient",
+    "numeric_assortativity_coefficient",
+]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def degree_assortativity_coefficient(G, x="out", y="in", weight=None, nodes=None):
+    """Compute degree assortativity of graph.
+
+    Assortativity measures the similarity of connections
+    in the graph with respect to the node degree.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    x: string ('in','out')
+       The degree type for source node (directed graphs only).
+
+    y: string ('in','out')
+       The degree type for target node (directed graphs only).
+
+    weight: string or None, optional (default=None)
+       The edge attribute that holds the numerical value used
+       as a weight.  If None, then each edge has weight 1.
+       The degree is the sum of the edge weights adjacent to the node.
+
+    nodes: list or iterable (optional)
+        Compute degree assortativity only for nodes in container.
+        The default is all nodes.
+
+    Returns
+    -------
+    r : float
+       Assortativity of graph by degree.
+
+    Examples
+    --------
+    >>> G = nx.path_graph(4)
+    >>> r = nx.degree_assortativity_coefficient(G)
+    >>> print(f"{r:3.1f}")
+    -0.5
+
+    See Also
+    --------
+    attribute_assortativity_coefficient
+    numeric_assortativity_coefficient
+    degree_mixing_dict
+    degree_mixing_matrix
+
+    Notes
+    -----
+    This computes Eq. (21) in Ref. [1]_ , where e is the joint
+    probability distribution (mixing matrix) of the degrees.  If G is
+    directed than the matrix e is the joint probability of the
+    user-specified degree type for the source and target.
+
+    References
+    ----------
+    .. [1] M. E. J. Newman, Mixing patterns in networks,
+       Physical Review E, 67 026126, 2003
+    .. [2] Foster, J.G., Foster, D.V., Grassberger, P. & Paczuski, M.
+       Edge direction and the structure of networks, PNAS 107, 10815-20 (2010).
+    """
+    if nodes is None:
+        nodes = G.nodes
+
+    degrees = None
+
+    if G.is_directed():
+        indeg = (
+            {d for _, d in G.in_degree(nodes, weight=weight)}
+            if "in" in (x, y)
+            else set()
+        )
+        outdeg = (
+            {d for _, d in G.out_degree(nodes, weight=weight)}
+            if "out" in (x, y)
+            else set()
+        )
+        degrees = set.union(indeg, outdeg)
+    else:
+        degrees = {d for _, d in G.degree(nodes, weight=weight)}
+
+    mapping = {d: i for i, d in enumerate(degrees)}
+    M = degree_mixing_matrix(G, x=x, y=y, nodes=nodes, weight=weight, mapping=mapping)
+
+    return _numeric_ac(M, mapping=mapping)
+
+
+@nx._dispatchable(edge_attrs="weight")
+def degree_pearson_correlation_coefficient(G, x="out", y="in", weight=None, nodes=None):
+    """Compute degree assortativity of graph.
+
+    Assortativity measures the similarity of connections
+    in the graph with respect to the node degree.
+
+    This is the same as degree_assortativity_coefficient but uses the
+    potentially faster scipy.stats.pearsonr function.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    x: string ('in','out')
+       The degree type for source node (directed graphs only).
+
+    y: string ('in','out')
+       The degree type for target node (directed graphs only).
+
+    weight: string or None, optional (default=None)
+       The edge attribute that holds the numerical value used
+       as a weight.  If None, then each edge has weight 1.
+       The degree is the sum of the edge weights adjacent to the node.
+
+    nodes: list or iterable (optional)
+        Compute pearson correlation of degrees only for specified nodes.
+        The default is all nodes.
+
+    Returns
+    -------
+    r : float
+       Assortativity of graph by degree.
+
+    Examples
+    --------
+    >>> G = nx.path_graph(4)
+    >>> r = nx.degree_pearson_correlation_coefficient(G)
+    >>> print(f"{r:3.1f}")
+    -0.5
+
+    Notes
+    -----
+    This calls scipy.stats.pearsonr.
+
+    References
+    ----------
+    .. [1] M. E. J. Newman, Mixing patterns in networks
+           Physical Review E, 67 026126, 2003
+    .. [2] Foster, J.G., Foster, D.V., Grassberger, P. & Paczuski, M.
+       Edge direction and the structure of networks, PNAS 107, 10815-20 (2010).
+    """
+    import scipy as sp
+
+    xy = node_degree_xy(G, x=x, y=y, nodes=nodes, weight=weight)
+    x, y = zip(*xy)
+    return float(sp.stats.pearsonr(x, y)[0])
+
+
+@nx._dispatchable(node_attrs="attribute")
+def attribute_assortativity_coefficient(G, attribute, nodes=None):
+    """Compute assortativity for node attributes.
+
+    Assortativity measures the similarity of connections
+    in the graph with respect to the given attribute.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    attribute : string
+        Node attribute key
+
+    nodes: list or iterable (optional)
+        Compute attribute assortativity for nodes in container.
+        The default is all nodes.
+
+    Returns
+    -------
+    r: float
+       Assortativity of graph for given attribute
+
+    Examples
+    --------
+    >>> G = nx.Graph()
+    >>> G.add_nodes_from([0, 1], color="red")
+    >>> G.add_nodes_from([2, 3], color="blue")
+    >>> G.add_edges_from([(0, 1), (2, 3)])
+    >>> print(nx.attribute_assortativity_coefficient(G, "color"))
+    1.0
+
+    Notes
+    -----
+    This computes Eq. (2) in Ref. [1]_ , (trace(M)-sum(M^2))/(1-sum(M^2)),
+    where M is the joint probability distribution (mixing matrix)
+    of the specified attribute.
+
+    References
+    ----------
+    .. [1] M. E. J. Newman, Mixing patterns in networks,
+       Physical Review E, 67 026126, 2003
+    """
+    M = attribute_mixing_matrix(G, attribute, nodes)
+    return attribute_ac(M)
+
+
+@nx._dispatchable(node_attrs="attribute")
+def numeric_assortativity_coefficient(G, attribute, nodes=None):
+    """Compute assortativity for numerical node attributes.
+
+    Assortativity measures the similarity of connections
+    in the graph with respect to the given numeric attribute.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    attribute : string
+        Node attribute key.
+
+    nodes: list or iterable (optional)
+        Compute numeric assortativity only for attributes of nodes in
+        container. The default is all nodes.
+
+    Returns
+    -------
+    r: float
+       Assortativity of graph for given attribute
+
+    Examples
+    --------
+    >>> G = nx.Graph()
+    >>> G.add_nodes_from([0, 1], size=2)
+    >>> G.add_nodes_from([2, 3], size=3)
+    >>> G.add_edges_from([(0, 1), (2, 3)])
+    >>> print(nx.numeric_assortativity_coefficient(G, "size"))
+    1.0
+
+    Notes
+    -----
+    This computes Eq. (21) in Ref. [1]_ , which is the Pearson correlation
+    coefficient of the specified (scalar valued) attribute across edges.
+
+    References
+    ----------
+    .. [1] M. E. J. Newman, Mixing patterns in networks
+           Physical Review E, 67 026126, 2003
+    """
+    if nodes is None:
+        nodes = G.nodes
+    vals = {G.nodes[n][attribute] for n in nodes}
+    mapping = {d: i for i, d in enumerate(vals)}
+    M = attribute_mixing_matrix(G, attribute, nodes, mapping)
+    return _numeric_ac(M, mapping)
+
+
+def attribute_ac(M):
+    """Compute assortativity for attribute matrix M.
+
+    Parameters
+    ----------
+    M : numpy.ndarray
+        2D ndarray representing the attribute mixing matrix.
+
+    Notes
+    -----
+    This computes Eq. (2) in Ref. [1]_ , (trace(e)-sum(e^2))/(1-sum(e^2)),
+    where e is the joint probability distribution (mixing matrix)
+    of the specified attribute.
+
+    References
+    ----------
+    .. [1] M. E. J. Newman, Mixing patterns in networks,
+       Physical Review E, 67 026126, 2003
+    """
+    if M.sum() != 1.0:
+        M = M / M.sum()
+    s = (M @ M).sum()
+    t = M.trace()
+    r = (t - s) / (1 - s)
+    return float(r)
+
+
+def _numeric_ac(M, mapping):
+    # M is a 2D numpy array
+    # numeric assortativity coefficient, pearsonr
+    import numpy as np
+
+    if M.sum() != 1.0:
+        M = M / M.sum()
+    x = np.array(list(mapping.keys()))
+    y = x  # x and y have the same support
+    idx = list(mapping.values())
+    a = M.sum(axis=0)
+    b = M.sum(axis=1)
+    vara = (a[idx] * x**2).sum() - ((a[idx] * x).sum()) ** 2
+    varb = (b[idx] * y**2).sum() - ((b[idx] * y).sum()) ** 2
+    xy = np.outer(x, y)
+    ab = np.outer(a[idx], b[idx])
+    return float((xy * (M - ab)).sum() / np.sqrt(vara * varb))

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/mixing.py

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+"""
+Mixing matrices for node attributes and degree.
+"""
+import networkx as nx
+from networkx.algorithms.assortativity.pairs import node_attribute_xy, node_degree_xy
+from networkx.utils import dict_to_numpy_array
+
+__all__ = [
+    "attribute_mixing_matrix",
+    "attribute_mixing_dict",
+    "degree_mixing_matrix",
+    "degree_mixing_dict",
+    "mixing_dict",
+]
+
+
+@nx._dispatchable(node_attrs="attribute")
+def attribute_mixing_dict(G, attribute, nodes=None, normalized=False):
+    """Returns dictionary representation of mixing matrix for attribute.
+
+    Parameters
+    ----------
+    G : graph
+       NetworkX graph object.
+
+    attribute : string
+       Node attribute key.
+
+    nodes: list or iterable (optional)
+        Unse nodes in container to build the dict. The default is all nodes.
+
+    normalized : bool (default=False)
+       Return counts if False or probabilities if True.
+
+    Examples
+    --------
+    >>> G = nx.Graph()
+    >>> G.add_nodes_from([0, 1], color="red")
+    >>> G.add_nodes_from([2, 3], color="blue")
+    >>> G.add_edge(1, 3)
+    >>> d = nx.attribute_mixing_dict(G, "color")
+    >>> print(d["red"]["blue"])
+    1
+    >>> print(d["blue"]["red"])  # d symmetric for undirected graphs
+    1
+
+    Returns
+    -------
+    d : dictionary
+       Counts or joint probability of occurrence of attribute pairs.
+    """
+    xy_iter = node_attribute_xy(G, attribute, nodes)
+    return mixing_dict(xy_iter, normalized=normalized)
+
+
+@nx._dispatchable(node_attrs="attribute")
+def attribute_mixing_matrix(G, attribute, nodes=None, mapping=None, normalized=True):
+    """Returns mixing matrix for attribute.
+
+    Parameters
+    ----------
+    G : graph
+       NetworkX graph object.
+
+    attribute : string
+       Node attribute key.
+
+    nodes: list or iterable (optional)
+        Use only nodes in container to build the matrix. The default is
+        all nodes.
+
+    mapping : dictionary, optional
+       Mapping from node attribute to integer index in matrix.
+       If not specified, an arbitrary ordering will be used.
+
+    normalized : bool (default=True)
+       Return counts if False or probabilities if True.
+
+    Returns
+    -------
+    m: numpy array
+       Counts or joint probability of occurrence of attribute pairs.
+
+    Notes
+    -----
+    If each node has a unique attribute value, the unnormalized mixing matrix
+    will be equal to the adjacency matrix. To get a denser mixing matrix,
+    the rounding can be performed to form groups of nodes with equal values.
+    For example, the exact height of persons in cm (180.79155222, 163.9080892,
+    163.30095355, 167.99016217, 168.21590163, ...) can be rounded to (180, 163,
+    163, 168, 168, ...).
+
+    Definitions of attribute mixing matrix vary on whether the matrix
+    should include rows for attribute values that don't arise. Here we
+    do not include such empty-rows. But you can force them to appear
+    by inputting a `mapping` that includes those values.
+
+    Examples
+    --------
+    >>> G = nx.path_graph(3)
+    >>> gender = {0: "male", 1: "female", 2: "female"}
+    >>> nx.set_node_attributes(G, gender, "gender")
+    >>> mapping = {"male": 0, "female": 1}
+    >>> mix_mat = nx.attribute_mixing_matrix(G, "gender", mapping=mapping)
+    >>> mix_mat
+    array([[0.  , 0.25],
+           [0.25, 0.5 ]])
+    """
+    d = attribute_mixing_dict(G, attribute, nodes)
+    a = dict_to_numpy_array(d, mapping=mapping)
+    if normalized:
+        a = a / a.sum()
+    return a
+
+
+@nx._dispatchable(edge_attrs="weight")
+def degree_mixing_dict(G, x="out", y="in", weight=None, nodes=None, normalized=False):
+    """Returns dictionary representation of mixing matrix for degree.
+
+    Parameters
+    ----------
+    G : graph
+        NetworkX graph object.
+
+    x: string ('in','out')
+       The degree type for source node (directed graphs only).
+
+    y: string ('in','out')
+       The degree type for target node (directed graphs only).
+
+    weight: string or None, optional (default=None)
+       The edge attribute that holds the numerical value used
+       as a weight.  If None, then each edge has weight 1.
+       The degree is the sum of the edge weights adjacent to the node.
+
+    normalized : bool (default=False)
+        Return counts if False or probabilities if True.
+
+    Returns
+    -------
+    d: dictionary
+       Counts or joint probability of occurrence of degree pairs.
+    """
+    xy_iter = node_degree_xy(G, x=x, y=y, nodes=nodes, weight=weight)
+    return mixing_dict(xy_iter, normalized=normalized)
+
+
+@nx._dispatchable(edge_attrs="weight")
+def degree_mixing_matrix(
+    G, x="out", y="in", weight=None, nodes=None, normalized=True, mapping=None
+):
+    """Returns mixing matrix for attribute.
+
+    Parameters
+    ----------
+    G : graph
+       NetworkX graph object.
+
+    x: string ('in','out')
+       The degree type for source node (directed graphs only).
+
+    y: string ('in','out')
+       The degree type for target node (directed graphs only).
+
+    nodes: list or iterable (optional)
+        Build the matrix using only nodes in container.
+        The default is all nodes.
+
+    weight: string or None, optional (default=None)
+       The edge attribute that holds the numerical value used
+       as a weight.  If None, then each edge has weight 1.
+       The degree is the sum of the edge weights adjacent to the node.
+
+    normalized : bool (default=True)
+       Return counts if False or probabilities if True.
+
+    mapping : dictionary, optional
+       Mapping from node degree to integer index in matrix.
+       If not specified, an arbitrary ordering will be used.
+
+    Returns
+    -------
+    m: numpy array
+       Counts, or joint probability, of occurrence of node degree.
+
+    Notes
+    -----
+    Definitions of degree mixing matrix vary on whether the matrix
+    should include rows for degree values that don't arise. Here we
+    do not include such empty-rows. But you can force them to appear
+    by inputting a `mapping` that includes those values. See examples.
+
+    Examples
+    --------
+    >>> G = nx.star_graph(3)
+    >>> mix_mat = nx.degree_mixing_matrix(G)
+    >>> mix_mat
+    array([[0. , 0.5],
+           [0.5, 0. ]])
+
+    If you want every possible degree to appear as a row, even if no nodes
+    have that degree, use `mapping` as follows,
+
+    >>> max_degree = max(deg for n, deg in G.degree)
+    >>> mapping = {x: x for x in range(max_degree + 1)}  # identity mapping
+    >>> mix_mat = nx.degree_mixing_matrix(G, mapping=mapping)
+    >>> mix_mat
+    array([[0. , 0. , 0. , 0. ],
+           [0. , 0. , 0. , 0.5],
+           [0. , 0. , 0. , 0. ],
+           [0. , 0.5, 0. , 0. ]])
+    """
+    d = degree_mixing_dict(G, x=x, y=y, nodes=nodes, weight=weight)
+    a = dict_to_numpy_array(d, mapping=mapping)
+    if normalized:
+        a = a / a.sum()
+    return a
+
+
+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

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/neighbor_degree.py

@@ -0,0 +1,160 @@
+import networkx as nx
+
+__all__ = ["average_neighbor_degree"]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def average_neighbor_degree(G, source="out", target="out", nodes=None, weight=None):
+    r"""Returns the average degree of the neighborhood of each node.
+
+    In an undirected graph, the neighborhood `N(i)` of node `i` contains the
+    nodes that are connected to `i` by an edge.
+
+    For directed graphs, `N(i)` is defined according to the parameter `source`:
+
+        - if source is 'in', then `N(i)` consists of predecessors of node `i`.
+        - if source is 'out', then `N(i)` consists of successors of node `i`.
+        - if source is 'in+out', then `N(i)` is both predecessors and successors.
+
+    The average neighborhood degree of a node `i` is
+
+    .. math::
+
+        k_{nn,i} = \frac{1}{|N(i)|} \sum_{j \in N(i)} k_j
+
+    where `N(i)` are the neighbors of node `i` and `k_j` is
+    the degree of node `j` which belongs to `N(i)`. For weighted
+    graphs, an analogous measure can be defined [1]_,
+
+    .. math::
+
+        k_{nn,i}^{w} = \frac{1}{s_i} \sum_{j \in N(i)} w_{ij} k_j
+
+    where `s_i` is the weighted degree of node `i`, `w_{ij}`
+    is the weight of the edge that links `i` and `j` and
+    `N(i)` are the neighbors of node `i`.
+
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    source : string ("in"|"out"|"in+out"), optional (default="out")
+       Directed graphs only.
+       Use "in"- or "out"-neighbors of source node.
+
+    target : string ("in"|"out"|"in+out"), optional (default="out")
+       Directed graphs only.
+       Use "in"- or "out"-degree for target node.
+
+    nodes : list or iterable, optional (default=G.nodes)
+        Compute neighbor degree only for specified nodes.
+
+    weight : string or None, optional (default=None)
+       The edge attribute that holds the numerical value used as a weight.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    d: dict
+       A dictionary keyed by node to the average degree of its neighbors.
+
+    Raises
+    ------
+    NetworkXError
+        If either `source` or `target` are not one of 'in', 'out', or 'in+out'.
+        If either `source` or `target` is passed for an undirected graph.
+
+    Examples
+    --------
+    >>> G = nx.path_graph(4)
+    >>> G.edges[0, 1]["weight"] = 5
+    >>> G.edges[2, 3]["weight"] = 3
+
+    >>> nx.average_neighbor_degree(G)
+    {0: 2.0, 1: 1.5, 2: 1.5, 3: 2.0}
+    >>> nx.average_neighbor_degree(G, weight="weight")
+    {0: 2.0, 1: 1.1666666666666667, 2: 1.25, 3: 2.0}
+
+    >>> G = nx.DiGraph()
+    >>> nx.add_path(G, [0, 1, 2, 3])
+    >>> nx.average_neighbor_degree(G, source="in", target="in")
+    {0: 0.0, 1: 0.0, 2: 1.0, 3: 1.0}
+
+    >>> nx.average_neighbor_degree(G, source="out", target="out")
+    {0: 1.0, 1: 1.0, 2: 0.0, 3: 0.0}
+
+    See Also
+    --------
+    average_degree_connectivity
+
+    References
+    ----------
+    .. [1] A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani,
+       "The architecture of complex weighted networks".
+       PNAS 101 (11): 3747–3752 (2004).
+    """
+    if G.is_directed():
+        if source == "in":
+            source_degree = G.in_degree
+        elif source == "out":
+            source_degree = G.out_degree
+        elif source == "in+out":
+            source_degree = G.degree
+        else:
+            raise nx.NetworkXError(
+                f"source argument {source} must be 'in', 'out' or 'in+out'"
+            )
+
+        if target == "in":
+            target_degree = G.in_degree
+        elif target == "out":
+            target_degree = G.out_degree
+        elif target == "in+out":
+            target_degree = G.degree
+        else:
+            raise nx.NetworkXError(
+                f"target argument {target} must be 'in', 'out' or 'in+out'"
+            )
+    else:
+        if source != "out" or target != "out":
+            raise nx.NetworkXError(
+                f"source and target arguments are only supported for directed graphs"
+            )
+        source_degree = target_degree = G.degree
+
+    # precompute target degrees -- should *not* be weighted degree
+    t_deg = dict(target_degree())
+
+    # Set up both predecessor and successor neighbor dicts leaving empty if not needed
+    G_P = G_S = {n: {} for n in G}
+    if G.is_directed():
+        # "in" or "in+out" cases: G_P contains predecessors
+        if "in" in source:
+            G_P = G.pred
+        # "out" or "in+out" cases: G_S contains successors
+        if "out" in source:
+            G_S = G.succ
+    else:
+        # undirected leave G_P empty but G_S is the adjacency
+        G_S = G.adj
+
+    # Main loop: Compute average degree of neighbors
+    avg = {}
+    for n, deg in source_degree(nodes, weight=weight):
+        # handle degree zero average
+        if deg == 0:
+            avg[n] = 0.0
+            continue
+
+        # we sum over both G_P and G_S, but one of the two is usually empty.
+        if weight is None:
+            avg[n] = (
+                sum(t_deg[nbr] for nbr in G_S[n]) + sum(t_deg[nbr] for nbr in G_P[n])
+            ) / deg
+        else:
+            avg[n] = (
+                sum(dd.get(weight, 1) * t_deg[nbr] for nbr, dd in G_S[n].items())
+                + sum(dd.get(weight, 1) * t_deg[nbr] for nbr, dd in G_P[n].items())
+            ) / deg
+    return avg

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/pairs.py

@@ -0,0 +1,118 @@
+"""Generators of  x-y pairs of node data."""
+import networkx as nx
+
+__all__ = ["node_attribute_xy", "node_degree_xy"]
+
+
+@nx._dispatchable(node_attrs="attribute")
+def node_attribute_xy(G, attribute, nodes=None):
+    """Returns iterator of node-attribute pairs for all edges in G.
+
+    Parameters
+    ----------
+    G: NetworkX graph
+
+    attribute: key
+       The node attribute key.
+
+    nodes: list or iterable (optional)
+        Use only edges that are incident to specified nodes.
+        The default is all nodes.
+
+    Returns
+    -------
+    (x, y): 2-tuple
+        Generates 2-tuple of (attribute, attribute) values.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph()
+    >>> G.add_node(1, color="red")
+    >>> G.add_node(2, color="blue")
+    >>> G.add_edge(1, 2)
+    >>> list(nx.node_attribute_xy(G, "color"))
+    [('red', 'blue')]
+
+    Notes
+    -----
+    For undirected graphs each edge is produced twice, once for each edge
+    representation (u, v) and (v, u), with the exception of self-loop edges
+    which only appear once.
+    """
+    if nodes is None:
+        nodes = set(G)
+    else:
+        nodes = set(nodes)
+    Gnodes = G.nodes
+    for u, nbrsdict in G.adjacency():
+        if u not in nodes:
+            continue
+        uattr = Gnodes[u].get(attribute, None)
+        if G.is_multigraph():
+            for v, keys in nbrsdict.items():
+                vattr = Gnodes[v].get(attribute, None)
+                for _ in keys:
+                    yield (uattr, vattr)
+        else:
+            for v in nbrsdict:
+                vattr = Gnodes[v].get(attribute, None)
+                yield (uattr, vattr)
+
+
+@nx._dispatchable(edge_attrs="weight")
+def node_degree_xy(G, x="out", y="in", weight=None, nodes=None):
+    """Generate node degree-degree pairs for edges in G.
+
+    Parameters
+    ----------
+    G: NetworkX graph
+
+    x: string ('in','out')
+       The degree type for source node (directed graphs only).
+
+    y: string ('in','out')
+       The degree type for target node (directed graphs only).
+
+    weight: string or None, optional (default=None)
+       The edge attribute that holds the numerical value used
+       as a weight.  If None, then each edge has weight 1.
+       The degree is the sum of the edge weights adjacent to the node.
+
+    nodes: list or iterable (optional)
+        Use only edges that are adjacency to specified nodes.
+        The default is all nodes.
+
+    Returns
+    -------
+    (x, y): 2-tuple
+        Generates 2-tuple of (degree, degree) values.
+
+
+    Examples
+    --------
+    >>> G = nx.DiGraph()
+    >>> G.add_edge(1, 2)
+    >>> list(nx.node_degree_xy(G, x="out", y="in"))
+    [(1, 1)]
+    >>> list(nx.node_degree_xy(G, x="in", y="out"))
+    [(0, 0)]
+
+    Notes
+    -----
+    For undirected graphs each edge is produced twice, once for each edge
+    representation (u, v) and (v, u), with the exception of self-loop edges
+    which only appear once.
+    """
+    nodes = set(G) if nodes is None else set(nodes)
+    if G.is_directed():
+        direction = {"out": G.out_degree, "in": G.in_degree}
+        xdeg = direction[x]
+        ydeg = direction[y]
+    else:
+        xdeg = ydeg = G.degree
+
+    for u, degu in xdeg(nodes, weight=weight):
+        # use G.edges to treat multigraphs correctly
+        neighbors = (nbr for _, nbr in G.edges(u) if nbr in nodes)
+        for _, degv in ydeg(neighbors, weight=weight):
+            yield degu, degv

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/tests/base_test.py

@@ -0,0 +1,81 @@
+import networkx as nx
+
+
+class BaseTestAttributeMixing:
+    @classmethod
+    def setup_class(cls):
+        G = nx.Graph()
+        G.add_nodes_from([0, 1], fish="one")
+        G.add_nodes_from([2, 3], fish="two")
+        G.add_nodes_from([4], fish="red")
+        G.add_nodes_from([5], fish="blue")
+        G.add_edges_from([(0, 1), (2, 3), (0, 4), (2, 5)])
+        cls.G = G
+
+        D = nx.DiGraph()
+        D.add_nodes_from([0, 1], fish="one")
+        D.add_nodes_from([2, 3], fish="two")
+        D.add_nodes_from([4], fish="red")
+        D.add_nodes_from([5], fish="blue")
+        D.add_edges_from([(0, 1), (2, 3), (0, 4), (2, 5)])
+        cls.D = D
+
+        M = nx.MultiGraph()
+        M.add_nodes_from([0, 1], fish="one")
+        M.add_nodes_from([2, 3], fish="two")
+        M.add_nodes_from([4], fish="red")
+        M.add_nodes_from([5], fish="blue")
+        M.add_edges_from([(0, 1), (0, 1), (2, 3)])
+        cls.M = M
+
+        S = nx.Graph()
+        S.add_nodes_from([0, 1], fish="one")
+        S.add_nodes_from([2, 3], fish="two")
+        S.add_nodes_from([4], fish="red")
+        S.add_nodes_from([5], fish="blue")
+        S.add_edge(0, 0)
+        S.add_edge(2, 2)
+        cls.S = S
+
+        N = nx.Graph()
+        N.add_nodes_from([0, 1], margin=-2)
+        N.add_nodes_from([2, 3], margin=-2)
+        N.add_nodes_from([4], margin=-3)
+        N.add_nodes_from([5], margin=-4)
+        N.add_edges_from([(0, 1), (2, 3), (0, 4), (2, 5)])
+        cls.N = N
+
+        F = nx.Graph()
+        F.add_edges_from([(0, 3), (1, 3), (2, 3)], weight=0.5)
+        F.add_edge(0, 2, weight=1)
+        nx.set_node_attributes(F, dict(F.degree(weight="weight")), "margin")
+        cls.F = F
+
+        K = nx.Graph()
+        K.add_nodes_from([1, 2], margin=-1)
+        K.add_nodes_from([3], margin=1)
+        K.add_nodes_from([4], margin=2)
+        K.add_edges_from([(3, 4), (1, 2), (1, 3)])
+        cls.K = K
+
+
+class BaseTestDegreeMixing:
+    @classmethod
+    def setup_class(cls):
+        cls.P4 = nx.path_graph(4)
+        cls.D = nx.DiGraph()
+        cls.D.add_edges_from([(0, 2), (0, 3), (1, 3), (2, 3)])
+        cls.D2 = nx.DiGraph()
+        cls.D2.add_edges_from([(0, 3), (1, 0), (1, 2), (2, 4), (4, 1), (4, 3), (4, 2)])
+        cls.M = nx.MultiGraph()
+        nx.add_path(cls.M, range(4))
+        cls.M.add_edge(0, 1)
+        cls.S = nx.Graph()
+        cls.S.add_edges_from([(0, 0), (1, 1)])
+        cls.W = nx.Graph()
+        cls.W.add_edges_from([(0, 3), (1, 3), (2, 3)], weight=0.5)
+        cls.W.add_edge(0, 2, weight=1)
+        S1 = nx.star_graph(4)
+        S2 = nx.star_graph(4)
+        cls.DS = nx.disjoint_union(S1, S2)
+        cls.DS.add_edge(4, 5)

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/tests/test_connectivity.py

@@ -0,0 +1,143 @@
+from itertools import permutations
+
+import pytest
+
+import networkx as nx
+
+
+class TestNeighborConnectivity:
+    def test_degree_p4(self):
+        G = nx.path_graph(4)
+        answer = {1: 2.0, 2: 1.5}
+        nd = nx.average_degree_connectivity(G)
+        assert nd == answer
+
+        D = G.to_directed()
+        answer = {2: 2.0, 4: 1.5}
+        nd = nx.average_degree_connectivity(D)
+        assert nd == answer
+
+        answer = {1: 2.0, 2: 1.5}
+        D = G.to_directed()
+        nd = nx.average_degree_connectivity(D, source="in", target="in")
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_degree_connectivity(D, source="in", target="in")
+        assert nd == answer
+
+    def test_degree_p4_weighted(self):
+        G = nx.path_graph(4)
+        G[1][2]["weight"] = 4
+        answer = {1: 2.0, 2: 1.8}
+        nd = nx.average_degree_connectivity(G, weight="weight")
+        assert nd == answer
+        answer = {1: 2.0, 2: 1.5}
+        nd = nx.average_degree_connectivity(G)
+        assert nd == answer
+
+        D = G.to_directed()
+        answer = {2: 2.0, 4: 1.8}
+        nd = nx.average_degree_connectivity(D, weight="weight")
+        assert nd == answer
+
+        answer = {1: 2.0, 2: 1.8}
+        D = G.to_directed()
+        nd = nx.average_degree_connectivity(
+            D, weight="weight", source="in", target="in"
+        )
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_degree_connectivity(
+            D, source="in", target="out", weight="weight"
+        )
+        assert nd == answer
+
+    def test_weight_keyword(self):
+        G = nx.path_graph(4)
+        G[1][2]["other"] = 4
+        answer = {1: 2.0, 2: 1.8}
+        nd = nx.average_degree_connectivity(G, weight="other")
+        assert nd == answer
+        answer = {1: 2.0, 2: 1.5}
+        nd = nx.average_degree_connectivity(G, weight=None)
+        assert nd == answer
+
+        D = G.to_directed()
+        answer = {2: 2.0, 4: 1.8}
+        nd = nx.average_degree_connectivity(D, weight="other")
+        assert nd == answer
+
+        answer = {1: 2.0, 2: 1.8}
+        D = G.to_directed()
+        nd = nx.average_degree_connectivity(D, weight="other", source="in", target="in")
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_degree_connectivity(D, weight="other", source="in", target="in")
+        assert nd == answer
+
+    def test_degree_barrat(self):
+        G = nx.star_graph(5)
+        G.add_edges_from([(5, 6), (5, 7), (5, 8), (5, 9)])
+        G[0][5]["weight"] = 5
+        nd = nx.average_degree_connectivity(G)[5]
+        assert nd == 1.8
+        nd = nx.average_degree_connectivity(G, weight="weight")[5]
+        assert nd == pytest.approx(3.222222, abs=1e-5)
+
+    def test_zero_deg(self):
+        G = nx.DiGraph()
+        G.add_edge(1, 2)
+        G.add_edge(1, 3)
+        G.add_edge(1, 4)
+        c = nx.average_degree_connectivity(G)
+        assert c == {1: 0, 3: 1}
+        c = nx.average_degree_connectivity(G, source="in", target="in")
+        assert c == {0: 0, 1: 0}
+        c = nx.average_degree_connectivity(G, source="in", target="out")
+        assert c == {0: 0, 1: 3}
+        c = nx.average_degree_connectivity(G, source="in", target="in+out")
+        assert c == {0: 0, 1: 3}
+        c = nx.average_degree_connectivity(G, source="out", target="out")
+        assert c == {0: 0, 3: 0}
+        c = nx.average_degree_connectivity(G, source="out", target="in")
+        assert c == {0: 0, 3: 1}
+        c = nx.average_degree_connectivity(G, source="out", target="in+out")
+        assert c == {0: 0, 3: 1}
+
+    def test_in_out_weight(self):
+        G = nx.DiGraph()
+        G.add_edge(1, 2, weight=1)
+        G.add_edge(1, 3, weight=1)
+        G.add_edge(3, 1, weight=1)
+        for s, t in permutations(["in", "out", "in+out"], 2):
+            c = nx.average_degree_connectivity(G, source=s, target=t)
+            cw = nx.average_degree_connectivity(G, source=s, target=t, weight="weight")
+            assert c == cw
+
+    def test_invalid_source(self):
+        with pytest.raises(nx.NetworkXError):
+            G = nx.DiGraph()
+            nx.average_degree_connectivity(G, source="bogus")
+
+    def test_invalid_target(self):
+        with pytest.raises(nx.NetworkXError):
+            G = nx.DiGraph()
+            nx.average_degree_connectivity(G, target="bogus")
+
+    def test_invalid_undirected_graph(self):
+        G = nx.Graph()
+        with pytest.raises(nx.NetworkXError):
+            nx.average_degree_connectivity(G, target="bogus")
+        with pytest.raises(nx.NetworkXError):
+            nx.average_degree_connectivity(G, source="bogus")
+
+    def test_single_node(self):
+        # TODO Is this really the intended behavior for providing a
+        # single node as the argument `nodes`? Shouldn't the function
+        # just return the connectivity value itself?
+        G = nx.trivial_graph()
+        conn = nx.average_degree_connectivity(G, nodes=0)
+        assert conn == {0: 0}

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/tests/test_correlation.py

@@ -0,0 +1,123 @@
+import pytest
+
+np = pytest.importorskip("numpy")
+pytest.importorskip("scipy")
+
+
+import networkx as nx
+from networkx.algorithms.assortativity.correlation import attribute_ac
+
+from .base_test import BaseTestAttributeMixing, BaseTestDegreeMixing
+
+
+class TestDegreeMixingCorrelation(BaseTestDegreeMixing):
+    def test_degree_assortativity_undirected(self):
+        r = nx.degree_assortativity_coefficient(self.P4)
+        np.testing.assert_almost_equal(r, -1.0 / 2, decimal=4)
+
+    def test_degree_assortativity_node_kwargs(self):
+        G = nx.Graph()
+        edges = [(0, 1), (0, 3), (1, 2), (1, 3), (1, 4), (5, 9), (9, 0)]
+        G.add_edges_from(edges)
+        r = nx.degree_assortativity_coefficient(G, nodes=[1, 2, 4])
+        np.testing.assert_almost_equal(r, -1.0, decimal=4)
+
+    def test_degree_assortativity_directed(self):
+        r = nx.degree_assortativity_coefficient(self.D)
+        np.testing.assert_almost_equal(r, -0.57735, decimal=4)
+
+    def test_degree_assortativity_directed2(self):
+        """Test degree assortativity for a directed graph where the set of
+        in/out degree does not equal the total degree."""
+        r = nx.degree_assortativity_coefficient(self.D2)
+        np.testing.assert_almost_equal(r, 0.14852, decimal=4)
+
+    def test_degree_assortativity_multigraph(self):
+        r = nx.degree_assortativity_coefficient(self.M)
+        np.testing.assert_almost_equal(r, -1.0 / 7.0, decimal=4)
+
+    def test_degree_pearson_assortativity_undirected(self):
+        r = nx.degree_pearson_correlation_coefficient(self.P4)
+        np.testing.assert_almost_equal(r, -1.0 / 2, decimal=4)
+
+    def test_degree_pearson_assortativity_directed(self):
+        r = nx.degree_pearson_correlation_coefficient(self.D)
+        np.testing.assert_almost_equal(r, -0.57735, decimal=4)
+
+    def test_degree_pearson_assortativity_directed2(self):
+        """Test degree assortativity with Pearson for a directed graph where
+        the set of in/out degree does not equal the total degree."""
+        r = nx.degree_pearson_correlation_coefficient(self.D2)
+        np.testing.assert_almost_equal(r, 0.14852, decimal=4)
+
+    def test_degree_pearson_assortativity_multigraph(self):
+        r = nx.degree_pearson_correlation_coefficient(self.M)
+        np.testing.assert_almost_equal(r, -1.0 / 7.0, decimal=4)
+
+    def test_degree_assortativity_weighted(self):
+        r = nx.degree_assortativity_coefficient(self.W, weight="weight")
+        np.testing.assert_almost_equal(r, -0.1429, decimal=4)
+
+    def test_degree_assortativity_double_star(self):
+        r = nx.degree_assortativity_coefficient(self.DS)
+        np.testing.assert_almost_equal(r, -0.9339, decimal=4)
+
+
+class TestAttributeMixingCorrelation(BaseTestAttributeMixing):
+    def test_attribute_assortativity_undirected(self):
+        r = nx.attribute_assortativity_coefficient(self.G, "fish")
+        assert r == 6.0 / 22.0
+
+    def test_attribute_assortativity_directed(self):
+        r = nx.attribute_assortativity_coefficient(self.D, "fish")
+        assert r == 1.0 / 3.0
+
+    def test_attribute_assortativity_multigraph(self):
+        r = nx.attribute_assortativity_coefficient(self.M, "fish")
+        assert r == 1.0
+
+    def test_attribute_assortativity_coefficient(self):
+        # from "Mixing patterns in networks"
+        # fmt: off
+        a = np.array([[0.258, 0.016, 0.035, 0.013],
+                      [0.012, 0.157, 0.058, 0.019],
+                      [0.013, 0.023, 0.306, 0.035],
+                      [0.005, 0.007, 0.024, 0.016]])
+        # fmt: on
+        r = attribute_ac(a)
+        np.testing.assert_almost_equal(r, 0.623, decimal=3)
+
+    def test_attribute_assortativity_coefficient2(self):
+        # fmt: off
+        a = np.array([[0.18, 0.02, 0.01, 0.03],
+                      [0.02, 0.20, 0.03, 0.02],
+                      [0.01, 0.03, 0.16, 0.01],
+                      [0.03, 0.02, 0.01, 0.22]])
+        # fmt: on
+        r = attribute_ac(a)
+        np.testing.assert_almost_equal(r, 0.68, decimal=2)
+
+    def test_attribute_assortativity(self):
+        a = np.array([[50, 50, 0], [50, 50, 0], [0, 0, 2]])
+        r = attribute_ac(a)
+        np.testing.assert_almost_equal(r, 0.029, decimal=3)
+
+    def test_attribute_assortativity_negative(self):
+        r = nx.numeric_assortativity_coefficient(self.N, "margin")
+        np.testing.assert_almost_equal(r, -0.2903, decimal=4)
+
+    def test_assortativity_node_kwargs(self):
+        G = nx.Graph()
+        G.add_nodes_from([0, 1], size=2)
+        G.add_nodes_from([2, 3], size=3)
+        G.add_edges_from([(0, 1), (2, 3)])
+        r = nx.numeric_assortativity_coefficient(G, "size", nodes=[0, 3])
+        np.testing.assert_almost_equal(r, 1.0, decimal=4)
+
+    def test_attribute_assortativity_float(self):
+        r = nx.numeric_assortativity_coefficient(self.F, "margin")
+        np.testing.assert_almost_equal(r, -0.1429, decimal=4)
+
+    def test_attribute_assortativity_mixed(self):
+        r = nx.numeric_assortativity_coefficient(self.K, "margin")
+        np.testing.assert_almost_equal(r, 0.4340, decimal=4)

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/tests/test_mixing.py

@@ -0,0 +1,176 @@
+import pytest
+
+np = pytest.importorskip("numpy")
+
+
+import networkx as nx
+
+from .base_test import BaseTestAttributeMixing, BaseTestDegreeMixing
+
+
+class TestDegreeMixingDict(BaseTestDegreeMixing):
+    def test_degree_mixing_dict_undirected(self):
+        d = nx.degree_mixing_dict(self.P4)
+        d_result = {1: {2: 2}, 2: {1: 2, 2: 2}}
+        assert d == d_result
+
+    def test_degree_mixing_dict_undirected_normalized(self):
+        d = nx.degree_mixing_dict(self.P4, normalized=True)
+        d_result = {1: {2: 1.0 / 3}, 2: {1: 1.0 / 3, 2: 1.0 / 3}}
+        assert d == d_result
+
+    def test_degree_mixing_dict_directed(self):
+        d = nx.degree_mixing_dict(self.D)
+        print(d)
+        d_result = {1: {3: 2}, 2: {1: 1, 3: 1}, 3: {}}
+        assert d == d_result
+
+    def test_degree_mixing_dict_multigraph(self):
+        d = nx.degree_mixing_dict(self.M)
+        d_result = {1: {2: 1}, 2: {1: 1, 3: 3}, 3: {2: 3}}
+        assert d == d_result
+
+    def test_degree_mixing_dict_weighted(self):
+        d = nx.degree_mixing_dict(self.W, weight="weight")
+        d_result = {0.5: {1.5: 1}, 1.5: {1.5: 6, 0.5: 1}}
+        assert d == d_result
+
+
+class TestDegreeMixingMatrix(BaseTestDegreeMixing):
+    def test_degree_mixing_matrix_undirected(self):
+        # fmt: off
+        a_result = np.array([[0, 2],
+                             [2, 2]]
+                            )
+        # fmt: on
+        a = nx.degree_mixing_matrix(self.P4, normalized=False)
+        np.testing.assert_equal(a, a_result)
+        a = nx.degree_mixing_matrix(self.P4)
+        np.testing.assert_equal(a, a_result / a_result.sum())
+
+    def test_degree_mixing_matrix_directed(self):
+        # fmt: off
+        a_result = np.array([[0, 0, 2],
+                             [1, 0, 1],
+                             [0, 0, 0]]
+                            )
+        # fmt: on
+        a = nx.degree_mixing_matrix(self.D, normalized=False)
+        np.testing.assert_equal(a, a_result)
+        a = nx.degree_mixing_matrix(self.D)
+        np.testing.assert_equal(a, a_result / a_result.sum())
+
+    def test_degree_mixing_matrix_multigraph(self):
+        # fmt: off
+        a_result = np.array([[0, 1, 0],
+                             [1, 0, 3],
+                             [0, 3, 0]]
+                            )
+        # fmt: on
+        a = nx.degree_mixing_matrix(self.M, normalized=False)
+        np.testing.assert_equal(a, a_result)
+        a = nx.degree_mixing_matrix(self.M)
+        np.testing.assert_equal(a, a_result / a_result.sum())
+
+    def test_degree_mixing_matrix_selfloop(self):
+        # fmt: off
+        a_result = np.array([[2]])
+        # fmt: on
+        a = nx.degree_mixing_matrix(self.S, normalized=False)
+        np.testing.assert_equal(a, a_result)
+        a = nx.degree_mixing_matrix(self.S)
+        np.testing.assert_equal(a, a_result / a_result.sum())
+
+    def test_degree_mixing_matrix_weighted(self):
+        a_result = np.array([[0.0, 1.0], [1.0, 6.0]])
+        a = nx.degree_mixing_matrix(self.W, weight="weight", normalized=False)
+        np.testing.assert_equal(a, a_result)
+        a = nx.degree_mixing_matrix(self.W, weight="weight")
+        np.testing.assert_equal(a, a_result / float(a_result.sum()))
+
+    def test_degree_mixing_matrix_mapping(self):
+        a_result = np.array([[6.0, 1.0], [1.0, 0.0]])
+        mapping = {0.5: 1, 1.5: 0}
+        a = nx.degree_mixing_matrix(
+            self.W, weight="weight", normalized=False, mapping=mapping
+        )
+        np.testing.assert_equal(a, a_result)
+
+
+class TestAttributeMixingDict(BaseTestAttributeMixing):
+    def test_attribute_mixing_dict_undirected(self):
+        d = nx.attribute_mixing_dict(self.G, "fish")
+        d_result = {
+            "one": {"one": 2, "red": 1},
+            "two": {"two": 2, "blue": 1},
+            "red": {"one": 1},
+            "blue": {"two": 1},
+        }
+        assert d == d_result
+
+    def test_attribute_mixing_dict_directed(self):
+        d = nx.attribute_mixing_dict(self.D, "fish")
+        d_result = {
+            "one": {"one": 1, "red": 1},
+            "two": {"two": 1, "blue": 1},
+            "red": {},
+            "blue": {},
+        }
+        assert d == d_result
+
+    def test_attribute_mixing_dict_multigraph(self):
+        d = nx.attribute_mixing_dict(self.M, "fish")
+        d_result = {"one": {"one": 4}, "two": {"two": 2}}
+        assert d == d_result
+
+
+class TestAttributeMixingMatrix(BaseTestAttributeMixing):
+    def test_attribute_mixing_matrix_undirected(self):
+        mapping = {"one": 0, "two": 1, "red": 2, "blue": 3}
+        a_result = np.array([[2, 0, 1, 0], [0, 2, 0, 1], [1, 0, 0, 0], [0, 1, 0, 0]])
+        a = nx.attribute_mixing_matrix(
+            self.G, "fish", mapping=mapping, normalized=False
+        )
+        np.testing.assert_equal(a, a_result)
+        a = nx.attribute_mixing_matrix(self.G, "fish", mapping=mapping)
+        np.testing.assert_equal(a, a_result / a_result.sum())
+
+    def test_attribute_mixing_matrix_directed(self):
+        mapping = {"one": 0, "two": 1, "red": 2, "blue": 3}
+        a_result = np.array([[1, 0, 1, 0], [0, 1, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0]])
+        a = nx.attribute_mixing_matrix(
+            self.D, "fish", mapping=mapping, normalized=False
+        )
+        np.testing.assert_equal(a, a_result)
+        a = nx.attribute_mixing_matrix(self.D, "fish", mapping=mapping)
+        np.testing.assert_equal(a, a_result / a_result.sum())
+
+    def test_attribute_mixing_matrix_multigraph(self):
+        mapping = {"one": 0, "two": 1, "red": 2, "blue": 3}
+        a_result = np.array([[4, 0, 0, 0], [0, 2, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])
+        a = nx.attribute_mixing_matrix(
+            self.M, "fish", mapping=mapping, normalized=False
+        )
+        np.testing.assert_equal(a, a_result)
+        a = nx.attribute_mixing_matrix(self.M, "fish", mapping=mapping)
+        np.testing.assert_equal(a, a_result / a_result.sum())
+
+    def test_attribute_mixing_matrix_negative(self):
+        mapping = {-2: 0, -3: 1, -4: 2}
+        a_result = np.array([[4.0, 1.0, 1.0], [1.0, 0.0, 0.0], [1.0, 0.0, 0.0]])
+        a = nx.attribute_mixing_matrix(
+            self.N, "margin", mapping=mapping, normalized=False
+        )
+        np.testing.assert_equal(a, a_result)
+        a = nx.attribute_mixing_matrix(self.N, "margin", mapping=mapping)
+        np.testing.assert_equal(a, a_result / float(a_result.sum()))
+
+    def test_attribute_mixing_matrix_float(self):
+        mapping = {0.5: 1, 1.5: 0}
+        a_result = np.array([[6.0, 1.0], [1.0, 0.0]])
+        a = nx.attribute_mixing_matrix(
+            self.F, "margin", mapping=mapping, normalized=False
+        )
+        np.testing.assert_equal(a, a_result)
+        a = nx.attribute_mixing_matrix(self.F, "margin", mapping=mapping)
+        np.testing.assert_equal(a, a_result / a_result.sum())

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/tests/test_neighbor_degree.py

@@ -0,0 +1,108 @@
+import pytest
+
+import networkx as nx
+
+
+class TestAverageNeighbor:
+    def test_degree_p4(self):
+        G = nx.path_graph(4)
+        answer = {0: 2, 1: 1.5, 2: 1.5, 3: 2}
+        nd = nx.average_neighbor_degree(G)
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_neighbor_degree(D)
+        assert nd == answer
+
+        D = nx.DiGraph(G.edges(data=True))
+        nd = nx.average_neighbor_degree(D)
+        assert nd == {0: 1, 1: 1, 2: 0, 3: 0}
+        nd = nx.average_neighbor_degree(D, "in", "out")
+        assert nd == {0: 0, 1: 1, 2: 1, 3: 1}
+        nd = nx.average_neighbor_degree(D, "out", "in")
+        assert nd == {0: 1, 1: 1, 2: 1, 3: 0}
+        nd = nx.average_neighbor_degree(D, "in", "in")
+        assert nd == {0: 0, 1: 0, 2: 1, 3: 1}
+
+    def test_degree_p4_weighted(self):
+        G = nx.path_graph(4)
+        G[1][2]["weight"] = 4
+        answer = {0: 2, 1: 1.8, 2: 1.8, 3: 2}
+        nd = nx.average_neighbor_degree(G, weight="weight")
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_neighbor_degree(D, weight="weight")
+        assert nd == answer
+
+        D = nx.DiGraph(G.edges(data=True))
+        print(D.edges(data=True))
+        nd = nx.average_neighbor_degree(D, weight="weight")
+        assert nd == {0: 1, 1: 1, 2: 0, 3: 0}
+        nd = nx.average_neighbor_degree(D, "out", "out", weight="weight")
+        assert nd == {0: 1, 1: 1, 2: 0, 3: 0}
+        nd = nx.average_neighbor_degree(D, "in", "in", weight="weight")
+        assert nd == {0: 0, 1: 0, 2: 1, 3: 1}
+        nd = nx.average_neighbor_degree(D, "in", "out", weight="weight")
+        assert nd == {0: 0, 1: 1, 2: 1, 3: 1}
+        nd = nx.average_neighbor_degree(D, "out", "in", weight="weight")
+        assert nd == {0: 1, 1: 1, 2: 1, 3: 0}
+        nd = nx.average_neighbor_degree(D, source="in+out", weight="weight")
+        assert nd == {0: 1.0, 1: 1.0, 2: 0.8, 3: 1.0}
+        nd = nx.average_neighbor_degree(D, target="in+out", weight="weight")
+        assert nd == {0: 2.0, 1: 2.0, 2: 1.0, 3: 0.0}
+
+        D = G.to_directed()
+        nd = nx.average_neighbor_degree(D, weight="weight")
+        assert nd == answer
+        nd = nx.average_neighbor_degree(D, source="out", target="out", weight="weight")
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_neighbor_degree(D, source="in", target="in", weight="weight")
+        assert nd == answer
+
+    def test_degree_k4(self):
+        G = nx.complete_graph(4)
+        answer = {0: 3, 1: 3, 2: 3, 3: 3}
+        nd = nx.average_neighbor_degree(G)
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_neighbor_degree(D)
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_neighbor_degree(D)
+        assert nd == answer
+
+        D = G.to_directed()
+        nd = nx.average_neighbor_degree(D, source="in", target="in")
+        assert nd == answer
+
+    def test_degree_k4_nodes(self):
+        G = nx.complete_graph(4)
+        answer = {1: 3.0, 2: 3.0}
+        nd = nx.average_neighbor_degree(G, nodes=[1, 2])
+        assert nd == answer
+
+    def test_degree_barrat(self):
+        G = nx.star_graph(5)
+        G.add_edges_from([(5, 6), (5, 7), (5, 8), (5, 9)])
+        G[0][5]["weight"] = 5
+        nd = nx.average_neighbor_degree(G)[5]
+        assert nd == 1.8
+        nd = nx.average_neighbor_degree(G, weight="weight")[5]
+        assert nd == pytest.approx(3.222222, abs=1e-5)
+
+    def test_error_invalid_source_target(self):
+        G = nx.path_graph(4)
+        with pytest.raises(nx.NetworkXError):
+            nx.average_neighbor_degree(G, "error")
+        with pytest.raises(nx.NetworkXError):
+            nx.average_neighbor_degree(G, "in", "error")
+        G = G.to_directed()
+        with pytest.raises(nx.NetworkXError):
+            nx.average_neighbor_degree(G, "error")
+        with pytest.raises(nx.NetworkXError):
+            nx.average_neighbor_degree(G, "in", "error")

venv/lib/python3.10/site-packages/networkx/algorithms/assortativity/tests/test_pairs.py

@@ -0,0 +1,87 @@
+import networkx as nx
+
+from .base_test import BaseTestAttributeMixing, BaseTestDegreeMixing
+
+
+class TestAttributeMixingXY(BaseTestAttributeMixing):
+    def test_node_attribute_xy_undirected(self):
+        attrxy = sorted(nx.node_attribute_xy(self.G, "fish"))
+        attrxy_result = sorted(
+            [
+                ("one", "one"),
+                ("one", "one"),
+                ("two", "two"),
+                ("two", "two"),
+                ("one", "red"),
+                ("red", "one"),
+                ("blue", "two"),
+                ("two", "blue"),
+            ]
+        )
+        assert attrxy == attrxy_result
+
+    def test_node_attribute_xy_undirected_nodes(self):
+        attrxy = sorted(nx.node_attribute_xy(self.G, "fish", nodes=["one", "yellow"]))
+        attrxy_result = sorted([])
+        assert attrxy == attrxy_result
+
+    def test_node_attribute_xy_directed(self):
+        attrxy = sorted(nx.node_attribute_xy(self.D, "fish"))
+        attrxy_result = sorted(
+            [("one", "one"), ("two", "two"), ("one", "red"), ("two", "blue")]
+        )
+        assert attrxy == attrxy_result
+
+    def test_node_attribute_xy_multigraph(self):
+        attrxy = sorted(nx.node_attribute_xy(self.M, "fish"))
+        attrxy_result = [
+            ("one", "one"),
+            ("one", "one"),
+            ("one", "one"),
+            ("one", "one"),
+            ("two", "two"),
+            ("two", "two"),
+        ]
+        assert attrxy == attrxy_result
+
+    def test_node_attribute_xy_selfloop(self):
+        attrxy = sorted(nx.node_attribute_xy(self.S, "fish"))
+        attrxy_result = [("one", "one"), ("two", "two")]
+        assert attrxy == attrxy_result
+
+
+class TestDegreeMixingXY(BaseTestDegreeMixing):
+    def test_node_degree_xy_undirected(self):
+        xy = sorted(nx.node_degree_xy(self.P4))
+        xy_result = sorted([(1, 2), (2, 1), (2, 2), (2, 2), (1, 2), (2, 1)])
+        assert xy == xy_result
+
+    def test_node_degree_xy_undirected_nodes(self):
+        xy = sorted(nx.node_degree_xy(self.P4, nodes=[0, 1, -1]))
+        xy_result = sorted([(1, 2), (2, 1)])
+        assert xy == xy_result
+
+    def test_node_degree_xy_directed(self):
+        xy = sorted(nx.node_degree_xy(self.D))
+        xy_result = sorted([(2, 1), (2, 3), (1, 3), (1, 3)])
+        assert xy == xy_result
+
+    def test_node_degree_xy_multigraph(self):
+        xy = sorted(nx.node_degree_xy(self.M))
+        xy_result = sorted(
+            [(2, 3), (2, 3), (3, 2), (3, 2), (2, 3), (3, 2), (1, 2), (2, 1)]
+        )
+        assert xy == xy_result
+
+    def test_node_degree_xy_selfloop(self):
+        xy = sorted(nx.node_degree_xy(self.S))
+        xy_result = sorted([(2, 2), (2, 2)])
+        assert xy == xy_result
+
+    def test_node_degree_xy_weighted(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=7)
+        G.add_edge(2, 3, weight=10)
+        xy = sorted(nx.node_degree_xy(G, weight="weight"))
+        xy_result = sorted([(7, 17), (17, 10), (17, 7), (10, 17)])
+        assert xy == xy_result

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/__init__.py

@@ -0,0 +1,20 @@
+from .betweenness import *
+from .betweenness_subset import *
+from .closeness import *
+from .current_flow_betweenness import *
+from .current_flow_betweenness_subset import *
+from .current_flow_closeness import *
+from .degree_alg import *
+from .dispersion import *
+from .eigenvector import *
+from .group import *
+from .harmonic import *
+from .katz import *
+from .load import *
+from .percolation import *
+from .reaching import *
+from .second_order import *
+from .subgraph_alg import *
+from .trophic import *
+from .voterank_alg import *
+from .laplacian import *

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/betweenness_subset.py

@@ -0,0 +1,274 @@
+"""Betweenness centrality measures for subsets of nodes."""
+import networkx as nx
+from networkx.algorithms.centrality.betweenness import (
+    _add_edge_keys,
+)
+from networkx.algorithms.centrality.betweenness import (
+    _single_source_dijkstra_path_basic as dijkstra,
+)
+from networkx.algorithms.centrality.betweenness import (
+    _single_source_shortest_path_basic as shortest_path,
+)
+
+__all__ = [
+    "betweenness_centrality_subset",
+    "edge_betweenness_centrality_subset",
+]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def betweenness_centrality_subset(G, sources, targets, normalized=False, weight=None):
+    r"""Compute betweenness centrality for a subset of nodes.
+
+    .. math::
+
+       c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\sigma(s, t)}
+
+    where $S$ is the set of sources, $T$ is the set of targets,
+    $\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
+    and $\sigma(s, t|v)$ is the number of those paths
+    passing through some  node $v$ other than $s, t$.
+    If $s = t$, $\sigma(s, t) = 1$,
+    and if $v \in {s, t}$, $\sigma(s, t|v) = 0$ [2]_.
+
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph.
+
+    sources: list of nodes
+      Nodes to use as sources for shortest paths in betweenness
+
+    targets: list of nodes
+      Nodes to use as targets for shortest paths in betweenness
+
+    normalized : bool, optional
+      If True the betweenness values are normalized by $2/((n-1)(n-2))$
+      for graphs, and $1/((n-1)(n-2))$ for directed graphs where $n$
+      is the number of nodes in G.
+
+    weight : None or string, optional (default=None)
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+      Weights are used to calculate weighted shortest paths, so they are
+      interpreted as distances.
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with betweenness centrality as the value.
+
+    See Also
+    --------
+    edge_betweenness_centrality
+    load_centrality
+
+    Notes
+    -----
+    The basic algorithm is from [1]_.
+
+    For weighted graphs the edge weights must be greater than zero.
+    Zero edge weights can produce an infinite number of equal length
+    paths between pairs of nodes.
+
+    The normalization might seem a little strange but it is
+    designed to make betweenness_centrality(G) be the same as
+    betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
+
+    The total number of paths between source and target is counted
+    differently for directed and undirected graphs. Directed paths
+    are easy to count. Undirected paths are tricky: should a path
+    from "u" to "v" count as 1 undirected path or as 2 directed paths?
+
+    For betweenness_centrality we report the number of undirected
+    paths when G is undirected.
+
+    For betweenness_centrality_subset the reporting is different.
+    If the source and target subsets are the same, then we want
+    to count undirected paths. But if the source and target subsets
+    differ -- for example, if sources is {0} and targets is {1},
+    then we are only counting the paths in one direction. They are
+    undirected paths but we are counting them in a directed way.
+    To count them as undirected paths, each should count as half a path.
+
+    References
+    ----------
+    .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
+       Journal of Mathematical Sociology 25(2):163-177, 2001.
+       https://doi.org/10.1080/0022250X.2001.9990249
+    .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
+       Centrality and their Generic Computation.
+       Social Networks 30(2):136-145, 2008.
+       https://doi.org/10.1016/j.socnet.2007.11.001
+    """
+    b = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
+    for s in sources:
+        # single source shortest paths
+        if weight is None:  # use BFS
+            S, P, sigma, _ = shortest_path(G, s)
+        else:  # use Dijkstra's algorithm
+            S, P, sigma, _ = dijkstra(G, s, weight)
+        b = _accumulate_subset(b, S, P, sigma, s, targets)
+    b = _rescale(b, len(G), normalized=normalized, directed=G.is_directed())
+    return b
+
+
+@nx._dispatchable(edge_attrs="weight")
+def edge_betweenness_centrality_subset(
+    G, sources, targets, normalized=False, weight=None
+):
+    r"""Compute betweenness centrality for edges for a subset of nodes.
+
+    .. math::
+
+       c_B(v) =\sum_{s\in S,t \in T} \frac{\sigma(s, t|e)}{\sigma(s, t)}
+
+    where $S$ is the set of sources, $T$ is the set of targets,
+    $\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
+    and $\sigma(s, t|e)$ is the number of those paths
+    passing through edge $e$ [2]_.
+
+    Parameters
+    ----------
+    G : graph
+      A networkx graph.
+
+    sources: list of nodes
+      Nodes to use as sources for shortest paths in betweenness
+
+    targets: list of nodes
+      Nodes to use as targets for shortest paths in betweenness
+
+    normalized : bool, optional
+      If True the betweenness values are normalized by `2/(n(n-1))`
+      for graphs, and `1/(n(n-1))` for directed graphs where `n`
+      is the number of nodes in G.
+
+    weight : None or string, optional (default=None)
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+      Weights are used to calculate weighted shortest paths, so they are
+      interpreted as distances.
+
+    Returns
+    -------
+    edges : dictionary
+       Dictionary of edges with Betweenness centrality as the value.
+
+    See Also
+    --------
+    betweenness_centrality
+    edge_load
+
+    Notes
+    -----
+    The basic algorithm is from [1]_.
+
+    For weighted graphs the edge weights must be greater than zero.
+    Zero edge weights can produce an infinite number of equal length
+    paths between pairs of nodes.
+
+    The normalization might seem a little strange but it is the same
+    as in edge_betweenness_centrality() and is designed to make
+    edge_betweenness_centrality(G) be the same as
+    edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
+
+    References
+    ----------
+    .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
+       Journal of Mathematical Sociology 25(2):163-177, 2001.
+       https://doi.org/10.1080/0022250X.2001.9990249
+    .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
+       Centrality and their Generic Computation.
+       Social Networks 30(2):136-145, 2008.
+       https://doi.org/10.1016/j.socnet.2007.11.001
+    """
+    b = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
+    b.update(dict.fromkeys(G.edges(), 0.0))  # b[e] for e in G.edges()
+    for s in sources:
+        # single source shortest paths
+        if weight is None:  # use BFS
+            S, P, sigma, _ = shortest_path(G, s)
+        else:  # use Dijkstra's algorithm
+            S, P, sigma, _ = dijkstra(G, s, weight)
+        b = _accumulate_edges_subset(b, S, P, sigma, s, targets)
+    for n in G:  # remove nodes to only return edges
+        del b[n]
+    b = _rescale_e(b, len(G), normalized=normalized, directed=G.is_directed())
+    if G.is_multigraph():
+        b = _add_edge_keys(G, b, weight=weight)
+    return b
+
+
+def _accumulate_subset(betweenness, S, P, sigma, s, targets):
+    delta = dict.fromkeys(S, 0.0)
+    target_set = set(targets) - {s}
+    while S:
+        w = S.pop()
+        if w in target_set:
+            coeff = (delta[w] + 1.0) / sigma[w]
+        else:
+            coeff = delta[w] / sigma[w]
+        for v in P[w]:
+            delta[v] += sigma[v] * coeff
+        if w != s:
+            betweenness[w] += delta[w]
+    return betweenness
+
+
+def _accumulate_edges_subset(betweenness, S, P, sigma, s, targets):
+    """edge_betweenness_centrality_subset helper."""
+    delta = dict.fromkeys(S, 0)
+    target_set = set(targets)
+    while S:
+        w = S.pop()
+        for v in P[w]:
+            if w in target_set:
+                c = (sigma[v] / sigma[w]) * (1.0 + delta[w])
+            else:
+                c = delta[w] / len(P[w])
+            if (v, w) not in betweenness:
+                betweenness[(w, v)] += c
+            else:
+                betweenness[(v, w)] += c
+            delta[v] += c
+        if w != s:
+            betweenness[w] += delta[w]
+    return betweenness
+
+
+def _rescale(betweenness, n, normalized, directed=False):
+    """betweenness_centrality_subset helper."""
+    if normalized:
+        if n <= 2:
+            scale = None  # no normalization b=0 for all nodes
+        else:
+            scale = 1.0 / ((n - 1) * (n - 2))
+    else:  # rescale by 2 for undirected graphs
+        if not directed:
+            scale = 0.5
+        else:
+            scale = None
+    if scale is not None:
+        for v in betweenness:
+            betweenness[v] *= scale
+    return betweenness
+
+
+def _rescale_e(betweenness, n, normalized, directed=False):
+    """edge_betweenness_centrality_subset helper."""
+    if normalized:
+        if n <= 1:
+            scale = None  # no normalization b=0 for all nodes
+        else:
+            scale = 1.0 / (n * (n - 1))
+    else:  # rescale by 2 for undirected graphs
+        if not directed:
+            scale = 0.5
+        else:
+            scale = None
+    if scale is not None:
+        for v in betweenness:
+            betweenness[v] *= scale
+    return betweenness

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/closeness.py

@@ -0,0 +1,281 @@
+"""
+Closeness centrality measures.
+"""
+import functools
+
+import networkx as nx
+from networkx.exception import NetworkXError
+from networkx.utils.decorators import not_implemented_for
+
+__all__ = ["closeness_centrality", "incremental_closeness_centrality"]
+
+
+@nx._dispatchable(edge_attrs="distance")
+def closeness_centrality(G, u=None, distance=None, wf_improved=True):
+    r"""Compute closeness centrality for nodes.
+
+    Closeness centrality [1]_ of a node `u` is the reciprocal of the
+    average shortest path distance to `u` over all `n-1` reachable nodes.
+
+    .. math::
+
+        C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},
+
+    where `d(v, u)` is the shortest-path distance between `v` and `u`,
+    and `n-1` is the number of nodes reachable from `u`. Notice that the
+    closeness distance function computes the incoming distance to `u`
+    for directed graphs. To use outward distance, act on `G.reverse()`.
+
+    Notice that higher values of closeness indicate higher centrality.
+
+    Wasserman and Faust propose an improved formula for graphs with
+    more than one connected component. The result is "a ratio of the
+    fraction of actors in the group who are reachable, to the average
+    distance" from the reachable actors [2]_. You might think this
+    scale factor is inverted but it is not. As is, nodes from small
+    components receive a smaller closeness value. Letting `N` denote
+    the number of nodes in the graph,
+
+    .. math::
+
+        C_{WF}(u) = \frac{n-1}{N-1} \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph
+
+    u : node, optional
+      Return only the value for node u
+
+    distance : edge attribute key, optional (default=None)
+      Use the specified edge attribute as the edge distance in shortest
+      path calculations.  If `None` (the default) all edges have a distance of 1.
+      Absent edge attributes are assigned a distance of 1. Note that no check
+      is performed to ensure that edges have the provided attribute.
+
+    wf_improved : bool, optional (default=True)
+      If True, scale by the fraction of nodes reachable. This gives the
+      Wasserman and Faust improved formula. For single component graphs
+      it is the same as the original formula.
+
+    Returns
+    -------
+    nodes : dictionary
+      Dictionary of nodes with closeness centrality as the value.
+
+    Examples
+    --------
+    >>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)])
+    >>> nx.closeness_centrality(G)
+    {0: 1.0, 1: 1.0, 2: 0.75, 3: 0.75}
+
+    See Also
+    --------
+    betweenness_centrality, load_centrality, eigenvector_centrality,
+    degree_centrality, incremental_closeness_centrality
+
+    Notes
+    -----
+    The closeness centrality is normalized to `(n-1)/(|G|-1)` where
+    `n` is the number of nodes in the connected part of graph
+    containing the node.  If the graph is not completely connected,
+    this algorithm computes the closeness centrality for each
+    connected part separately scaled by that parts size.
+
+    If the 'distance' keyword is set to an edge attribute key then the
+    shortest-path length will be computed using Dijkstra's algorithm with
+    that edge attribute as the edge weight.
+
+    The closeness centrality uses *inward* distance to a node, not outward.
+    If you want to use outword distances apply the function to `G.reverse()`
+
+    In NetworkX 2.2 and earlier a bug caused Dijkstra's algorithm to use the
+    outward distance rather than the inward distance. If you use a 'distance'
+    keyword and a DiGraph, your results will change between v2.2 and v2.3.
+
+    References
+    ----------
+    .. [1] Linton C. Freeman: Centrality in networks: I.
+       Conceptual clarification. Social Networks 1:215-239, 1979.
+       https://doi.org/10.1016/0378-8733(78)90021-7
+    .. [2] pg. 201 of Wasserman, S. and Faust, K.,
+       Social Network Analysis: Methods and Applications, 1994,
+       Cambridge University Press.
+    """
+    if G.is_directed():
+        G = G.reverse()  # create a reversed graph view
+
+    if distance is not None:
+        # use Dijkstra's algorithm with specified attribute as edge weight
+        path_length = functools.partial(
+            nx.single_source_dijkstra_path_length, weight=distance
+        )
+    else:
+        path_length = nx.single_source_shortest_path_length
+
+    if u is None:
+        nodes = G.nodes
+    else:
+        nodes = [u]
+    closeness_dict = {}
+    for n in nodes:
+        sp = path_length(G, n)
+        totsp = sum(sp.values())
+        len_G = len(G)
+        _closeness_centrality = 0.0
+        if totsp > 0.0 and len_G > 1:
+            _closeness_centrality = (len(sp) - 1.0) / totsp
+            # normalize to number of nodes-1 in connected part
+            if wf_improved:
+                s = (len(sp) - 1.0) / (len_G - 1)
+                _closeness_centrality *= s
+        closeness_dict[n] = _closeness_centrality
+    if u is not None:
+        return closeness_dict[u]
+    return closeness_dict
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(mutates_input=True)
+def incremental_closeness_centrality(
+    G, edge, prev_cc=None, insertion=True, wf_improved=True
+):
+    r"""Incremental closeness centrality for nodes.
+
+    Compute closeness centrality for nodes using level-based work filtering
+    as described in Incremental Algorithms for Closeness Centrality by Sariyuce et al.
+
+    Level-based work filtering detects unnecessary updates to the closeness
+    centrality and filters them out.
+
+    ---
+    From "Incremental Algorithms for Closeness Centrality":
+
+    Theorem 1: Let :math:`G = (V, E)` be a graph and u and v be two vertices in V
+    such that there is no edge (u, v) in E. Let :math:`G' = (V, E \cup uv)`
+    Then :math:`cc[s] = cc'[s]` if and only if :math:`\left|dG(s, u) - dG(s, v)\right| \leq 1`.
+
+    Where :math:`dG(u, v)` denotes the length of the shortest path between
+    two vertices u, v in a graph G, cc[s] is the closeness centrality for a
+    vertex s in V, and cc'[s] is the closeness centrality for a
+    vertex s in V, with the (u, v) edge added.
+    ---
+
+    We use Theorem 1 to filter out updates when adding or removing an edge.
+    When adding an edge (u, v), we compute the shortest path lengths from all
+    other nodes to u and to v before the node is added. When removing an edge,
+    we compute the shortest path lengths after the edge is removed. Then we
+    apply Theorem 1 to use previously computed closeness centrality for nodes
+    where :math:`\left|dG(s, u) - dG(s, v)\right| \leq 1`. This works only for
+    undirected, unweighted graphs; the distance argument is not supported.
+
+    Closeness centrality [1]_ of a node `u` is the reciprocal of the
+    sum of the shortest path distances from `u` to all `n-1` other nodes.
+    Since the sum of distances depends on the number of nodes in the
+    graph, closeness is normalized by the sum of minimum possible
+    distances `n-1`.
+
+    .. math::
+
+        C(u) = \frac{n - 1}{\sum_{v=1}^{n-1} d(v, u)},
+
+    where `d(v, u)` is the shortest-path distance between `v` and `u`,
+    and `n` is the number of nodes in the graph.
+
+    Notice that higher values of closeness indicate higher centrality.
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph
+
+    edge : tuple
+      The modified edge (u, v) in the graph.
+
+    prev_cc : dictionary
+      The previous closeness centrality for all nodes in the graph.
+
+    insertion : bool, optional
+      If True (default) the edge was inserted, otherwise it was deleted from the graph.
+
+    wf_improved : bool, optional (default=True)
+      If True, scale by the fraction of nodes reachable. This gives the
+      Wasserman and Faust improved formula. For single component graphs
+      it is the same as the original formula.
+
+    Returns
+    -------
+    nodes : dictionary
+      Dictionary of nodes with closeness centrality as the value.
+
+    See Also
+    --------
+    betweenness_centrality, load_centrality, eigenvector_centrality,
+    degree_centrality, closeness_centrality
+
+    Notes
+    -----
+    The closeness centrality is normalized to `(n-1)/(|G|-1)` where
+    `n` is the number of nodes in the connected part of graph
+    containing the node.  If the graph is not completely connected,
+    this algorithm computes the closeness centrality for each
+    connected part separately.
+
+    References
+    ----------
+    .. [1] Freeman, L.C., 1979. Centrality in networks: I.
+       Conceptual clarification.  Social Networks 1, 215--239.
+       https://doi.org/10.1016/0378-8733(78)90021-7
+    .. [2] Sariyuce, A.E. ; Kaya, K. ; Saule, E. ; Catalyiirek, U.V. Incremental
+       Algorithms for Closeness Centrality. 2013 IEEE International Conference on Big Data
+       http://sariyuce.com/papers/bigdata13.pdf
+    """
+    if prev_cc is not None and set(prev_cc.keys()) != set(G.nodes()):
+        raise NetworkXError("prev_cc and G do not have the same nodes")
+
+    # Unpack edge
+    (u, v) = edge
+    path_length = nx.single_source_shortest_path_length
+
+    if insertion:
+        # For edge insertion, we want shortest paths before the edge is inserted
+        du = path_length(G, u)
+        dv = path_length(G, v)
+
+        G.add_edge(u, v)
+    else:
+        G.remove_edge(u, v)
+
+        # For edge removal, we want shortest paths after the edge is removed
+        du = path_length(G, u)
+        dv = path_length(G, v)
+
+    if prev_cc is None:
+        return nx.closeness_centrality(G)
+
+    nodes = G.nodes()
+    closeness_dict = {}
+    for n in nodes:
+        if n in du and n in dv and abs(du[n] - dv[n]) <= 1:
+            closeness_dict[n] = prev_cc[n]
+        else:
+            sp = path_length(G, n)
+            totsp = sum(sp.values())
+            len_G = len(G)
+            _closeness_centrality = 0.0
+            if totsp > 0.0 and len_G > 1:
+                _closeness_centrality = (len(sp) - 1.0) / totsp
+                # normalize to number of nodes-1 in connected part
+                if wf_improved:
+                    s = (len(sp) - 1.0) / (len_G - 1)
+                    _closeness_centrality *= s
+            closeness_dict[n] = _closeness_centrality
+
+    # Leave the graph as we found it
+    if insertion:
+        G.remove_edge(u, v)
+    else:
+        G.add_edge(u, v)
+
+    return closeness_dict

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/current_flow_betweenness.py

@@ -0,0 +1,341 @@
+"""Current-flow betweenness centrality measures."""
+import networkx as nx
+from networkx.algorithms.centrality.flow_matrix import (
+    CGInverseLaplacian,
+    FullInverseLaplacian,
+    SuperLUInverseLaplacian,
+    flow_matrix_row,
+)
+from networkx.utils import (
+    not_implemented_for,
+    py_random_state,
+    reverse_cuthill_mckee_ordering,
+)
+
+__all__ = [
+    "current_flow_betweenness_centrality",
+    "approximate_current_flow_betweenness_centrality",
+    "edge_current_flow_betweenness_centrality",
+]
+
+
+@not_implemented_for("directed")
+@py_random_state(7)
+@nx._dispatchable(edge_attrs="weight")
+def approximate_current_flow_betweenness_centrality(
+    G,
+    normalized=True,
+    weight=None,
+    dtype=float,
+    solver="full",
+    epsilon=0.5,
+    kmax=10000,
+    seed=None,
+):
+    r"""Compute the approximate current-flow betweenness centrality for nodes.
+
+    Approximates the current-flow betweenness centrality within absolute
+    error of epsilon with high probability [1]_.
+
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph
+
+    normalized : bool, optional (default=True)
+      If True the betweenness values are normalized by 2/[(n-1)(n-2)] where
+      n is the number of nodes in G.
+
+    weight : string or None, optional (default=None)
+      Key for edge data used as the edge weight.
+      If None, then use 1 as each edge weight.
+      The weight reflects the capacity or the strength of the
+      edge.
+
+    dtype : data type (float)
+      Default data type for internal matrices.
+      Set to np.float32 for lower memory consumption.
+
+    solver : string (default='full')
+       Type of linear solver to use for computing the flow matrix.
+       Options are "full" (uses most memory), "lu" (recommended), and
+       "cg" (uses least memory).
+
+    epsilon: float
+        Absolute error tolerance.
+
+    kmax: int
+       Maximum number of sample node pairs to use for approximation.
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with betweenness centrality as the value.
+
+    See Also
+    --------
+    current_flow_betweenness_centrality
+
+    Notes
+    -----
+    The running time is $O((1/\epsilon^2)m{\sqrt k} \log n)$
+    and the space required is $O(m)$ for $n$ nodes and $m$ edges.
+
+    If the edges have a 'weight' attribute they will be used as
+    weights in this algorithm.  Unspecified weights are set to 1.
+
+    References
+    ----------
+    .. [1] Ulrik Brandes and Daniel Fleischer:
+       Centrality Measures Based on Current Flow.
+       Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
+       LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
+       https://doi.org/10.1007/978-3-540-31856-9_44
+    """
+    import numpy as np
+
+    if not nx.is_connected(G):
+        raise nx.NetworkXError("Graph not connected.")
+    solvername = {
+        "full": FullInverseLaplacian,
+        "lu": SuperLUInverseLaplacian,
+        "cg": CGInverseLaplacian,
+    }
+    n = G.number_of_nodes()
+    ordering = list(reverse_cuthill_mckee_ordering(G))
+    # make a copy with integer labels according to rcm ordering
+    # this could be done without a copy if we really wanted to
+    H = nx.relabel_nodes(G, dict(zip(ordering, range(n))))
+    L = nx.laplacian_matrix(H, nodelist=range(n), weight=weight).asformat("csc")
+    L = L.astype(dtype)
+    C = solvername[solver](L, dtype=dtype)  # initialize solver
+    betweenness = dict.fromkeys(H, 0.0)
+    nb = (n - 1.0) * (n - 2.0)  # normalization factor
+    cstar = n * (n - 1) / nb
+    l = 1  # parameter in approximation, adjustable
+    k = l * int(np.ceil((cstar / epsilon) ** 2 * np.log(n)))
+    if k > kmax:
+        msg = f"Number random pairs k>kmax ({k}>{kmax}) "
+        raise nx.NetworkXError(msg, "Increase kmax or epsilon")
+    cstar2k = cstar / (2 * k)
+    for _ in range(k):
+        s, t = pair = seed.sample(range(n), 2)
+        b = np.zeros(n, dtype=dtype)
+        b[s] = 1
+        b[t] = -1
+        p = C.solve(b)
+        for v in H:
+            if v in pair:
+                continue
+            for nbr in H[v]:
+                w = H[v][nbr].get(weight, 1.0)
+                betweenness[v] += float(w * np.abs(p[v] - p[nbr]) * cstar2k)
+    if normalized:
+        factor = 1.0
+    else:
+        factor = nb / 2.0
+    # remap to original node names and "unnormalize" if required
+    return {ordering[k]: v * factor for k, v in betweenness.items()}
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(edge_attrs="weight")
+def current_flow_betweenness_centrality(
+    G, normalized=True, weight=None, dtype=float, solver="full"
+):
+    r"""Compute current-flow betweenness centrality for nodes.
+
+    Current-flow betweenness centrality uses an electrical current
+    model for information spreading in contrast to betweenness
+    centrality which uses shortest paths.
+
+    Current-flow betweenness centrality is also known as
+    random-walk betweenness centrality [2]_.
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph
+
+    normalized : bool, optional (default=True)
+      If True the betweenness values are normalized by 2/[(n-1)(n-2)] where
+      n is the number of nodes in G.
+
+    weight : string or None, optional (default=None)
+      Key for edge data used as the edge weight.
+      If None, then use 1 as each edge weight.
+      The weight reflects the capacity or the strength of the
+      edge.
+
+    dtype : data type (float)
+      Default data type for internal matrices.
+      Set to np.float32 for lower memory consumption.
+
+    solver : string (default='full')
+       Type of linear solver to use for computing the flow matrix.
+       Options are "full" (uses most memory), "lu" (recommended), and
+       "cg" (uses least memory).
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with betweenness centrality as the value.
+
+    See Also
+    --------
+    approximate_current_flow_betweenness_centrality
+    betweenness_centrality
+    edge_betweenness_centrality
+    edge_current_flow_betweenness_centrality
+
+    Notes
+    -----
+    Current-flow betweenness can be computed in  $O(I(n-1)+mn \log n)$
+    time [1]_, where $I(n-1)$ is the time needed to compute the
+    inverse Laplacian.  For a full matrix this is $O(n^3)$ but using
+    sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the
+    Laplacian matrix condition number.
+
+    The space required is $O(nw)$ where $w$ is the width of the sparse
+    Laplacian matrix.  Worse case is $w=n$ for $O(n^2)$.
+
+    If the edges have a 'weight' attribute they will be used as
+    weights in this algorithm.  Unspecified weights are set to 1.
+
+    References
+    ----------
+    .. [1] Centrality Measures Based on Current Flow.
+       Ulrik Brandes and Daniel Fleischer,
+       Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
+       LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
+       https://doi.org/10.1007/978-3-540-31856-9_44
+
+    .. [2] A measure of betweenness centrality based on random walks,
+       M. E. J. Newman, Social Networks 27, 39-54 (2005).
+    """
+    if not nx.is_connected(G):
+        raise nx.NetworkXError("Graph not connected.")
+    N = G.number_of_nodes()
+    ordering = list(reverse_cuthill_mckee_ordering(G))
+    # make a copy with integer labels according to rcm ordering
+    # this could be done without a copy if we really wanted to
+    H = nx.relabel_nodes(G, dict(zip(ordering, range(N))))
+    betweenness = dict.fromkeys(H, 0.0)  # b[n]=0 for n in H
+    for row, (s, t) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver):
+        pos = dict(zip(row.argsort()[::-1], range(N)))
+        for i in range(N):
+            betweenness[s] += (i - pos[i]) * row.item(i)
+            betweenness[t] += (N - i - 1 - pos[i]) * row.item(i)
+    if normalized:
+        nb = (N - 1.0) * (N - 2.0)  # normalization factor
+    else:
+        nb = 2.0
+    return {ordering[n]: (b - n) * 2.0 / nb for n, b in betweenness.items()}
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(edge_attrs="weight")
+def edge_current_flow_betweenness_centrality(
+    G, normalized=True, weight=None, dtype=float, solver="full"
+):
+    r"""Compute current-flow betweenness centrality for edges.
+
+    Current-flow betweenness centrality uses an electrical current
+    model for information spreading in contrast to betweenness
+    centrality which uses shortest paths.
+
+    Current-flow betweenness centrality is also known as
+    random-walk betweenness centrality [2]_.
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph
+
+    normalized : bool, optional (default=True)
+      If True the betweenness values are normalized by 2/[(n-1)(n-2)] where
+      n is the number of nodes in G.
+
+    weight : string or None, optional (default=None)
+      Key for edge data used as the edge weight.
+      If None, then use 1 as each edge weight.
+      The weight reflects the capacity or the strength of the
+      edge.
+
+    dtype : data type (default=float)
+      Default data type for internal matrices.
+      Set to np.float32 for lower memory consumption.
+
+    solver : string (default='full')
+       Type of linear solver to use for computing the flow matrix.
+       Options are "full" (uses most memory), "lu" (recommended), and
+       "cg" (uses least memory).
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of edge tuples with betweenness centrality as the value.
+
+    Raises
+    ------
+    NetworkXError
+        The algorithm does not support DiGraphs.
+        If the input graph is an instance of DiGraph class, NetworkXError
+        is raised.
+
+    See Also
+    --------
+    betweenness_centrality
+    edge_betweenness_centrality
+    current_flow_betweenness_centrality
+
+    Notes
+    -----
+    Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$
+    time [1]_, where $I(n-1)$ is the time needed to compute the
+    inverse Laplacian.  For a full matrix this is $O(n^3)$ but using
+    sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the
+    Laplacian matrix condition number.
+
+    The space required is $O(nw)$ where $w$ is the width of the sparse
+    Laplacian matrix.  Worse case is $w=n$ for $O(n^2)$.
+
+    If the edges have a 'weight' attribute they will be used as
+    weights in this algorithm.  Unspecified weights are set to 1.
+
+    References
+    ----------
+    .. [1] Centrality Measures Based on Current Flow.
+       Ulrik Brandes and Daniel Fleischer,
+       Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
+       LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
+       https://doi.org/10.1007/978-3-540-31856-9_44
+
+    .. [2] A measure of betweenness centrality based on random walks,
+       M. E. J. Newman, Social Networks 27, 39-54 (2005).
+    """
+    if not nx.is_connected(G):
+        raise nx.NetworkXError("Graph not connected.")
+    N = G.number_of_nodes()
+    ordering = list(reverse_cuthill_mckee_ordering(G))
+    # make a copy with integer labels according to rcm ordering
+    # this could be done without a copy if we really wanted to
+    H = nx.relabel_nodes(G, dict(zip(ordering, range(N))))
+    edges = (tuple(sorted((u, v))) for u, v in H.edges())
+    betweenness = dict.fromkeys(edges, 0.0)
+    if normalized:
+        nb = (N - 1.0) * (N - 2.0)  # normalization factor
+    else:
+        nb = 2.0
+    for row, (e) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver):
+        pos = dict(zip(row.argsort()[::-1], range(1, N + 1)))
+        for i in range(N):
+            betweenness[e] += (i + 1 - pos[i]) * row.item(i)
+            betweenness[e] += (N - i - pos[i]) * row.item(i)
+        betweenness[e] /= nb
+    return {(ordering[s], ordering[t]): b for (s, t), b in betweenness.items()}

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/current_flow_closeness.py

@@ -0,0 +1,95 @@
+"""Current-flow closeness centrality measures."""
+import networkx as nx
+from networkx.algorithms.centrality.flow_matrix import (
+    CGInverseLaplacian,
+    FullInverseLaplacian,
+    SuperLUInverseLaplacian,
+)
+from networkx.utils import not_implemented_for, reverse_cuthill_mckee_ordering
+
+__all__ = ["current_flow_closeness_centrality", "information_centrality"]
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(edge_attrs="weight")
+def current_flow_closeness_centrality(G, weight=None, dtype=float, solver="lu"):
+    """Compute current-flow closeness centrality for nodes.
+
+    Current-flow closeness centrality is variant of closeness
+    centrality based on effective resistance between nodes in
+    a network. This metric is also known as information centrality.
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph.
+
+    weight : None or string, optional (default=None)
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+      The weight reflects the capacity or the strength of the
+      edge.
+
+    dtype: data type (default=float)
+      Default data type for internal matrices.
+      Set to np.float32 for lower memory consumption.
+
+    solver: string (default='lu')
+       Type of linear solver to use for computing the flow matrix.
+       Options are "full" (uses most memory), "lu" (recommended), and
+       "cg" (uses least memory).
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with current flow closeness centrality as the value.
+
+    See Also
+    --------
+    closeness_centrality
+
+    Notes
+    -----
+    The algorithm is from Brandes [1]_.
+
+    See also [2]_ for the original definition of information centrality.
+
+    References
+    ----------
+    .. [1] Ulrik Brandes and Daniel Fleischer,
+       Centrality Measures Based on Current Flow.
+       Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
+       LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
+       https://doi.org/10.1007/978-3-540-31856-9_44
+
+    .. [2] Karen Stephenson and Marvin Zelen:
+       Rethinking centrality: Methods and examples.
+       Social Networks 11(1):1-37, 1989.
+       https://doi.org/10.1016/0378-8733(89)90016-6
+    """
+    if not nx.is_connected(G):
+        raise nx.NetworkXError("Graph not connected.")
+    solvername = {
+        "full": FullInverseLaplacian,
+        "lu": SuperLUInverseLaplacian,
+        "cg": CGInverseLaplacian,
+    }
+    N = G.number_of_nodes()
+    ordering = list(reverse_cuthill_mckee_ordering(G))
+    # make a copy with integer labels according to rcm ordering
+    # this could be done without a copy if we really wanted to
+    H = nx.relabel_nodes(G, dict(zip(ordering, range(N))))
+    betweenness = dict.fromkeys(H, 0.0)  # b[n]=0 for n in H
+    N = H.number_of_nodes()
+    L = nx.laplacian_matrix(H, nodelist=range(N), weight=weight).asformat("csc")
+    L = L.astype(dtype)
+    C2 = solvername[solver](L, width=1, dtype=dtype)  # initialize solver
+    for v in H:
+        col = C2.get_row(v)
+        for w in H:
+            betweenness[v] += col.item(v) - 2 * col.item(w)
+            betweenness[w] += col.item(v)
+    return {ordering[node]: 1 / value for node, value in betweenness.items()}
+
+
+information_centrality = current_flow_closeness_centrality

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/degree_alg.py

@@ -0,0 +1,149 @@
+"""Degree centrality measures."""
+import networkx as nx
+from networkx.utils.decorators import not_implemented_for
+
+__all__ = ["degree_centrality", "in_degree_centrality", "out_degree_centrality"]
+
+
+@nx._dispatchable
+def degree_centrality(G):
+    """Compute the degree centrality for nodes.
+
+    The degree centrality for a node v is the fraction of nodes it
+    is connected to.
+
+    Parameters
+    ----------
+    G : graph
+      A networkx graph
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with degree centrality as the value.
+
+    Examples
+    --------
+    >>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)])
+    >>> nx.degree_centrality(G)
+    {0: 1.0, 1: 1.0, 2: 0.6666666666666666, 3: 0.6666666666666666}
+
+    See Also
+    --------
+    betweenness_centrality, load_centrality, eigenvector_centrality
+
+    Notes
+    -----
+    The degree centrality values are normalized by dividing by the maximum
+    possible degree in a simple graph n-1 where n is the number of nodes in G.
+
+    For multigraphs or graphs with self loops the maximum degree might
+    be higher than n-1 and values of degree centrality greater than 1
+    are possible.
+    """
+    if len(G) <= 1:
+        return {n: 1 for n in G}
+
+    s = 1.0 / (len(G) - 1.0)
+    centrality = {n: d * s for n, d in G.degree()}
+    return centrality
+
+
+@not_implemented_for("undirected")
+@nx._dispatchable
+def in_degree_centrality(G):
+    """Compute the in-degree centrality for nodes.
+
+    The in-degree centrality for a node v is the fraction of nodes its
+    incoming edges are connected to.
+
+    Parameters
+    ----------
+    G : graph
+        A NetworkX graph
+
+    Returns
+    -------
+    nodes : dictionary
+        Dictionary of nodes with in-degree centrality as values.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If G is undirected.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)])
+    >>> nx.in_degree_centrality(G)
+    {0: 0.0, 1: 0.3333333333333333, 2: 0.6666666666666666, 3: 0.6666666666666666}
+
+    See Also
+    --------
+    degree_centrality, out_degree_centrality
+
+    Notes
+    -----
+    The degree centrality values are normalized by dividing by the maximum
+    possible degree in a simple graph n-1 where n is the number of nodes in G.
+
+    For multigraphs or graphs with self loops the maximum degree might
+    be higher than n-1 and values of degree centrality greater than 1
+    are possible.
+    """
+    if len(G) <= 1:
+        return {n: 1 for n in G}
+
+    s = 1.0 / (len(G) - 1.0)
+    centrality = {n: d * s for n, d in G.in_degree()}
+    return centrality
+
+
+@not_implemented_for("undirected")
+@nx._dispatchable
+def out_degree_centrality(G):
+    """Compute the out-degree centrality for nodes.
+
+    The out-degree centrality for a node v is the fraction of nodes its
+    outgoing edges are connected to.
+
+    Parameters
+    ----------
+    G : graph
+        A NetworkX graph
+
+    Returns
+    -------
+    nodes : dictionary
+        Dictionary of nodes with out-degree centrality as values.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If G is undirected.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3)])
+    >>> nx.out_degree_centrality(G)
+    {0: 1.0, 1: 0.6666666666666666, 2: 0.0, 3: 0.0}
+
+    See Also
+    --------
+    degree_centrality, in_degree_centrality
+
+    Notes
+    -----
+    The degree centrality values are normalized by dividing by the maximum
+    possible degree in a simple graph n-1 where n is the number of nodes in G.
+
+    For multigraphs or graphs with self loops the maximum degree might
+    be higher than n-1 and values of degree centrality greater than 1
+    are possible.
+    """
+    if len(G) <= 1:
+        return {n: 1 for n in G}
+
+    s = 1.0 / (len(G) - 1.0)
+    centrality = {n: d * s for n, d in G.out_degree()}
+    return centrality

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/group.py

@@ -0,0 +1,786 @@
+"""Group centrality measures."""
+from copy import deepcopy
+
+import networkx as nx
+from networkx.algorithms.centrality.betweenness import (
+    _accumulate_endpoints,
+    _single_source_dijkstra_path_basic,
+    _single_source_shortest_path_basic,
+)
+from networkx.utils.decorators import not_implemented_for
+
+__all__ = [
+    "group_betweenness_centrality",
+    "group_closeness_centrality",
+    "group_degree_centrality",
+    "group_in_degree_centrality",
+    "group_out_degree_centrality",
+    "prominent_group",
+]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def group_betweenness_centrality(G, C, normalized=True, weight=None, endpoints=False):
+    r"""Compute the group betweenness centrality for a group of nodes.
+
+    Group betweenness centrality of a group of nodes $C$ is the sum of the
+    fraction of all-pairs shortest paths that pass through any vertex in $C$
+
+    .. math::
+
+       c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}
+
+    where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
+    shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of
+    those paths passing through some node in group $C$. Note that
+    $(s, t)$ are not members of the group ($V-C$ is the set of nodes
+    in $V$ that are not in $C$).
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph.
+
+    C : list or set or list of lists or list of sets
+      A group or a list of groups containing nodes which belong to G, for which group betweenness
+      centrality is to be calculated.
+
+    normalized : bool, optional (default=True)
+      If True, group betweenness is normalized by `1/((|V|-|C|)(|V|-|C|-1))`
+      where `|V|` is the number of nodes in G and `|C|` is the number of nodes in C.
+
+    weight : None or string, optional (default=None)
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+      The weight of an edge is treated as the length or distance between the two sides.
+
+    endpoints : bool, optional (default=False)
+      If True include the endpoints in the shortest path counts.
+
+    Raises
+    ------
+    NodeNotFound
+       If node(s) in C are not present in G.
+
+    Returns
+    -------
+    betweenness : list of floats or float
+       If C is a single group then return a float. If C is a list with
+       several groups then return a list of group betweenness centralities.
+
+    See Also
+    --------
+    betweenness_centrality
+
+    Notes
+    -----
+    Group betweenness centrality is described in [1]_ and its importance discussed in [3]_.
+    The initial implementation of the algorithm is mentioned in [2]_. This function uses
+    an improved algorithm presented in [4]_.
+
+    The number of nodes in the group must be a maximum of n - 2 where `n`
+    is the total number of nodes in the graph.
+
+    For weighted graphs the edge weights must be greater than zero.
+    Zero edge weights can produce an infinite number of equal length
+    paths between pairs of nodes.
+
+    The total number of paths between source and target is counted
+    differently for directed and undirected graphs. Directed paths
+    between "u" and "v" are counted as two possible paths (one each
+    direction) while undirected paths between "u" and "v" are counted
+    as one path. Said another way, the sum in the expression above is
+    over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs.
+
+
+    References
+    ----------
+    .. [1] M G Everett and S P Borgatti:
+       The Centrality of Groups and Classes.
+       Journal of Mathematical Sociology. 23(3): 181-201. 1999.
+       http://www.analytictech.com/borgatti/group_centrality.htm
+    .. [2] Ulrik Brandes:
+       On Variants of Shortest-Path Betweenness
+       Centrality and their Generic Computation.
+       Social Networks 30(2):136-145, 2008.
+       http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf
+    .. [3] Sourav Medya et. al.:
+       Group Centrality Maximization via Network Design.
+       SIAM International Conference on Data Mining, SDM 2018, 126–134.
+       https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf
+    .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev.
+       "Fast algorithm for successive computation of group betweenness centrality."
+       https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709
+
+    """
+    GBC = []  # initialize betweenness
+    list_of_groups = True
+    #  check weather C contains one or many groups
+    if any(el in G for el in C):
+        C = [C]
+        list_of_groups = False
+    set_v = {node for group in C for node in group}
+    if set_v - G.nodes:  # element(s) of C not in G
+        raise nx.NodeNotFound(f"The node(s) {set_v - G.nodes} are in C but not in G.")
+
+    # pre-processing
+    PB, sigma, D = _group_preprocessing(G, set_v, weight)
+
+    # the algorithm for each group
+    for group in C:
+        group = set(group)  # set of nodes in group
+        # initialize the matrices of the sigma and the PB
+        GBC_group = 0
+        sigma_m = deepcopy(sigma)
+        PB_m = deepcopy(PB)
+        sigma_m_v = deepcopy(sigma_m)
+        PB_m_v = deepcopy(PB_m)
+        for v in group:
+            GBC_group += PB_m[v][v]
+            for x in group:
+                for y in group:
+                    dxvy = 0
+                    dxyv = 0
+                    dvxy = 0
+                    if not (
+                        sigma_m[x][y] == 0 or sigma_m[x][v] == 0 or sigma_m[v][y] == 0
+                    ):
+                        if D[x][v] == D[x][y] + D[y][v]:
+                            dxyv = sigma_m[x][y] * sigma_m[y][v] / sigma_m[x][v]
+                        if D[x][y] == D[x][v] + D[v][y]:
+                            dxvy = sigma_m[x][v] * sigma_m[v][y] / sigma_m[x][y]
+                        if D[v][y] == D[v][x] + D[x][y]:
+                            dvxy = sigma_m[v][x] * sigma[x][y] / sigma[v][y]
+                    sigma_m_v[x][y] = sigma_m[x][y] * (1 - dxvy)
+                    PB_m_v[x][y] = PB_m[x][y] - PB_m[x][y] * dxvy
+                    if y != v:
+                        PB_m_v[x][y] -= PB_m[x][v] * dxyv
+                    if x != v:
+                        PB_m_v[x][y] -= PB_m[v][y] * dvxy
+            sigma_m, sigma_m_v = sigma_m_v, sigma_m
+            PB_m, PB_m_v = PB_m_v, PB_m
+
+        # endpoints
+        v, c = len(G), len(group)
+        if not endpoints:
+            scale = 0
+            # if the graph is connected then subtract the endpoints from
+            # the count for all the nodes in the graph. else count how many
+            # nodes are connected to the group's nodes and subtract that.
+            if nx.is_directed(G):
+                if nx.is_strongly_connected(G):
+                    scale = c * (2 * v - c - 1)
+            elif nx.is_connected(G):
+                scale = c * (2 * v - c - 1)
+            if scale == 0:
+                for group_node1 in group:
+                    for node in D[group_node1]:
+                        if node != group_node1:
+                            if node in group:
+                                scale += 1
+                            else:
+                                scale += 2
+            GBC_group -= scale
+
+        # normalized
+        if normalized:
+            scale = 1 / ((v - c) * (v - c - 1))
+            GBC_group *= scale
+
+        # If undirected than count only the undirected edges
+        elif not G.is_directed():
+            GBC_group /= 2
+
+        GBC.append(GBC_group)
+    if list_of_groups:
+        return GBC
+    return GBC[0]
+
+
+def _group_preprocessing(G, set_v, weight):
+    sigma = {}
+    delta = {}
+    D = {}
+    betweenness = dict.fromkeys(G, 0)
+    for s in G:
+        if weight is None:  # use BFS
+            S, P, sigma[s], D[s] = _single_source_shortest_path_basic(G, s)
+        else:  # use Dijkstra's algorithm
+            S, P, sigma[s], D[s] = _single_source_dijkstra_path_basic(G, s, weight)
+        betweenness, delta[s] = _accumulate_endpoints(betweenness, S, P, sigma[s], s)
+        for i in delta[s]:  # add the paths from s to i and rescale sigma
+            if s != i:
+                delta[s][i] += 1
+            if weight is not None:
+                sigma[s][i] = sigma[s][i] / 2
+    # building the path betweenness matrix only for nodes that appear in the group
+    PB = dict.fromkeys(G)
+    for group_node1 in set_v:
+        PB[group_node1] = dict.fromkeys(G, 0.0)
+        for group_node2 in set_v:
+            if group_node2 not in D[group_node1]:
+                continue
+            for node in G:
+                # if node is connected to the two group nodes than continue
+                if group_node2 in D[node] and group_node1 in D[node]:
+                    if (
+                        D[node][group_node2]
+                        == D[node][group_node1] + D[group_node1][group_node2]
+                    ):
+                        PB[group_node1][group_node2] += (
+                            delta[node][group_node2]
+                            * sigma[node][group_node1]
+                            * sigma[group_node1][group_node2]
+                            / sigma[node][group_node2]
+                        )
+    return PB, sigma, D
+
+
+@nx._dispatchable(edge_attrs="weight")
+def prominent_group(
+    G, k, weight=None, C=None, endpoints=False, normalized=True, greedy=False
+):
+    r"""Find the prominent group of size $k$ in graph $G$. The prominence of the
+    group is evaluated by the group betweenness centrality.
+
+    Group betweenness centrality of a group of nodes $C$ is the sum of the
+    fraction of all-pairs shortest paths that pass through any vertex in $C$
+
+    .. math::
+
+       c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)}
+
+    where $V$ is the set of nodes, $\sigma(s, t)$ is the number of
+    shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of
+    those paths passing through some node in group $C$. Note that
+    $(s, t)$ are not members of the group ($V-C$ is the set of nodes
+    in $V$ that are not in $C$).
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph.
+
+    k : int
+       The number of nodes in the group.
+
+    normalized : bool, optional (default=True)
+       If True, group betweenness is normalized by ``1/((|V|-|C|)(|V|-|C|-1))``
+       where ``|V|`` is the number of nodes in G and ``|C|`` is the number of
+       nodes in C.
+
+    weight : None or string, optional (default=None)
+       If None, all edge weights are considered equal.
+       Otherwise holds the name of the edge attribute used as weight.
+       The weight of an edge is treated as the length or distance between the two sides.
+
+    endpoints : bool, optional (default=False)
+       If True include the endpoints in the shortest path counts.
+
+    C : list or set, optional (default=None)
+       list of nodes which won't be candidates of the prominent group.
+
+    greedy : bool, optional (default=False)
+       Using a naive greedy algorithm in order to find non-optimal prominent
+       group. For scale free networks the results are negligibly below the optimal
+       results.
+
+    Raises
+    ------
+    NodeNotFound
+       If node(s) in C are not present in G.
+
+    Returns
+    -------
+    max_GBC : float
+       The group betweenness centrality of the prominent group.
+
+    max_group : list
+        The list of nodes in the prominent group.
+
+    See Also
+    --------
+    betweenness_centrality, group_betweenness_centrality
+
+    Notes
+    -----
+    Group betweenness centrality is described in [1]_ and its importance discussed in [3]_.
+    The algorithm is described in [2]_ and is based on techniques mentioned in [4]_.
+
+    The number of nodes in the group must be a maximum of ``n - 2`` where ``n``
+    is the total number of nodes in the graph.
+
+    For weighted graphs the edge weights must be greater than zero.
+    Zero edge weights can produce an infinite number of equal length
+    paths between pairs of nodes.
+
+    The total number of paths between source and target is counted
+    differently for directed and undirected graphs. Directed paths
+    between "u" and "v" are counted as two possible paths (one each
+    direction) while undirected paths between "u" and "v" are counted
+    as one path. Said another way, the sum in the expression above is
+    over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs.
+
+    References
+    ----------
+    .. [1] M G Everett and S P Borgatti:
+       The Centrality of Groups and Classes.
+       Journal of Mathematical Sociology. 23(3): 181-201. 1999.
+       http://www.analytictech.com/borgatti/group_centrality.htm
+    .. [2] Rami Puzis, Yuval Elovici, and Shlomi Dolev:
+       "Finding the Most Prominent Group in Complex Networks"
+       AI communications 20(4): 287-296, 2007.
+       https://www.researchgate.net/profile/Rami_Puzis2/publication/220308855
+    .. [3] Sourav Medya et. al.:
+       Group Centrality Maximization via Network Design.
+       SIAM International Conference on Data Mining, SDM 2018, 126–134.
+       https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf
+    .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev.
+       "Fast algorithm for successive computation of group betweenness centrality."
+       https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709
+    """
+    import numpy as np
+    import pandas as pd
+
+    if C is not None:
+        C = set(C)
+        if C - G.nodes:  # element(s) of C not in G
+            raise nx.NodeNotFound(f"The node(s) {C - G.nodes} are in C but not in G.")
+        nodes = list(G.nodes - C)
+    else:
+        nodes = list(G.nodes)
+    DF_tree = nx.Graph()
+    DF_tree.__networkx_cache__ = None  # Disable caching
+    PB, sigma, D = _group_preprocessing(G, nodes, weight)
+    betweenness = pd.DataFrame.from_dict(PB)
+    if C is not None:
+        for node in C:
+            # remove from the betweenness all the nodes not part of the group
+            betweenness.drop(index=node, inplace=True)
+            betweenness.drop(columns=node, inplace=True)
+    CL = [node for _, node in sorted(zip(np.diag(betweenness), nodes), reverse=True)]
+    max_GBC = 0
+    max_group = []
+    DF_tree.add_node(
+        1,
+        CL=CL,
+        betweenness=betweenness,
+        GBC=0,
+        GM=[],
+        sigma=sigma,
+        cont=dict(zip(nodes, np.diag(betweenness))),
+    )
+
+    # the algorithm
+    DF_tree.nodes[1]["heu"] = 0
+    for i in range(k):
+        DF_tree.nodes[1]["heu"] += DF_tree.nodes[1]["cont"][DF_tree.nodes[1]["CL"][i]]
+    max_GBC, DF_tree, max_group = _dfbnb(
+        G, k, DF_tree, max_GBC, 1, D, max_group, nodes, greedy
+    )
+
+    v = len(G)
+    if not endpoints:
+        scale = 0
+        # if the graph is connected then subtract the endpoints from
+        # the count for all the nodes in the graph. else count how many
+        # nodes are connected to the group's nodes and subtract that.
+        if nx.is_directed(G):
+            if nx.is_strongly_connected(G):
+                scale = k * (2 * v - k - 1)
+        elif nx.is_connected(G):
+            scale = k * (2 * v - k - 1)
+        if scale == 0:
+            for group_node1 in max_group:
+                for node in D[group_node1]:
+                    if node != group_node1:
+                        if node in max_group:
+                            scale += 1
+                        else:
+                            scale += 2
+        max_GBC -= scale
+
+    # normalized
+    if normalized:
+        scale = 1 / ((v - k) * (v - k - 1))
+        max_GBC *= scale
+
+    # If undirected then count only the undirected edges
+    elif not G.is_directed():
+        max_GBC /= 2
+    max_GBC = float("%.2f" % max_GBC)
+    return max_GBC, max_group
+
+
+def _dfbnb(G, k, DF_tree, max_GBC, root, D, max_group, nodes, greedy):
+    # stopping condition - if we found a group of size k and with higher GBC then prune
+    if len(DF_tree.nodes[root]["GM"]) == k and DF_tree.nodes[root]["GBC"] > max_GBC:
+        return DF_tree.nodes[root]["GBC"], DF_tree, DF_tree.nodes[root]["GM"]
+    # stopping condition - if the size of group members equal to k or there are less than
+    # k - |GM| in the candidate list or the heuristic function plus the GBC is below the
+    # maximal GBC found then prune
+    if (
+        len(DF_tree.nodes[root]["GM"]) == k
+        or len(DF_tree.nodes[root]["CL"]) <= k - len(DF_tree.nodes[root]["GM"])
+        or DF_tree.nodes[root]["GBC"] + DF_tree.nodes[root]["heu"] <= max_GBC
+    ):
+        return max_GBC, DF_tree, max_group
+
+    # finding the heuristic of both children
+    node_p, node_m, DF_tree = _heuristic(k, root, DF_tree, D, nodes, greedy)
+
+    # finding the child with the bigger heuristic + GBC and expand
+    # that node first if greedy then only expand the plus node
+    if greedy:
+        max_GBC, DF_tree, max_group = _dfbnb(
+            G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy
+        )
+
+    elif (
+        DF_tree.nodes[node_p]["GBC"] + DF_tree.nodes[node_p]["heu"]
+        > DF_tree.nodes[node_m]["GBC"] + DF_tree.nodes[node_m]["heu"]
+    ):
+        max_GBC, DF_tree, max_group = _dfbnb(
+            G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy
+        )
+        max_GBC, DF_tree, max_group = _dfbnb(
+            G, k, DF_tree, max_GBC, node_m, D, max_group, nodes, greedy
+        )
+    else:
+        max_GBC, DF_tree, max_group = _dfbnb(
+            G, k, DF_tree, max_GBC, node_m, D, max_group, nodes, greedy
+        )
+        max_GBC, DF_tree, max_group = _dfbnb(
+            G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy
+        )
+    return max_GBC, DF_tree, max_group
+
+
+def _heuristic(k, root, DF_tree, D, nodes, greedy):
+    import numpy as np
+
+    # This helper function add two nodes to DF_tree - one left son and the
+    # other right son, finds their heuristic, CL, GBC, and GM
+    node_p = DF_tree.number_of_nodes() + 1
+    node_m = DF_tree.number_of_nodes() + 2
+    added_node = DF_tree.nodes[root]["CL"][0]
+
+    # adding the plus node
+    DF_tree.add_nodes_from([(node_p, deepcopy(DF_tree.nodes[root]))])
+    DF_tree.nodes[node_p]["GM"].append(added_node)
+    DF_tree.nodes[node_p]["GBC"] += DF_tree.nodes[node_p]["cont"][added_node]
+    root_node = DF_tree.nodes[root]
+    for x in nodes:
+        for y in nodes:
+            dxvy = 0
+            dxyv = 0
+            dvxy = 0
+            if not (
+                root_node["sigma"][x][y] == 0
+                or root_node["sigma"][x][added_node] == 0
+                or root_node["sigma"][added_node][y] == 0
+            ):
+                if D[x][added_node] == D[x][y] + D[y][added_node]:
+                    dxyv = (
+                        root_node["sigma"][x][y]
+                        * root_node["sigma"][y][added_node]
+                        / root_node["sigma"][x][added_node]
+                    )
+                if D[x][y] == D[x][added_node] + D[added_node][y]:
+                    dxvy = (
+                        root_node["sigma"][x][added_node]
+                        * root_node["sigma"][added_node][y]
+                        / root_node["sigma"][x][y]
+                    )
+                if D[added_node][y] == D[added_node][x] + D[x][y]:
+                    dvxy = (
+                        root_node["sigma"][added_node][x]
+                        * root_node["sigma"][x][y]
+                        / root_node["sigma"][added_node][y]
+                    )
+            DF_tree.nodes[node_p]["sigma"][x][y] = root_node["sigma"][x][y] * (1 - dxvy)
+            DF_tree.nodes[node_p]["betweenness"].loc[y, x] = (
+                root_node["betweenness"][x][y] - root_node["betweenness"][x][y] * dxvy
+            )
+            if y != added_node:
+                DF_tree.nodes[node_p]["betweenness"].loc[y, x] -= (
+                    root_node["betweenness"][x][added_node] * dxyv
+                )
+            if x != added_node:
+                DF_tree.nodes[node_p]["betweenness"].loc[y, x] -= (
+                    root_node["betweenness"][added_node][y] * dvxy
+                )
+
+    DF_tree.nodes[node_p]["CL"] = [
+        node
+        for _, node in sorted(
+            zip(np.diag(DF_tree.nodes[node_p]["betweenness"]), nodes), reverse=True
+        )
+        if node not in DF_tree.nodes[node_p]["GM"]
+    ]
+    DF_tree.nodes[node_p]["cont"] = dict(
+        zip(nodes, np.diag(DF_tree.nodes[node_p]["betweenness"]))
+    )
+    DF_tree.nodes[node_p]["heu"] = 0
+    for i in range(k - len(DF_tree.nodes[node_p]["GM"])):
+        DF_tree.nodes[node_p]["heu"] += DF_tree.nodes[node_p]["cont"][
+            DF_tree.nodes[node_p]["CL"][i]
+        ]
+
+    # adding the minus node - don't insert the first node in the CL to GM
+    # Insert minus node only if isn't greedy type algorithm
+    if not greedy:
+        DF_tree.add_nodes_from([(node_m, deepcopy(DF_tree.nodes[root]))])
+        DF_tree.nodes[node_m]["CL"].pop(0)
+        DF_tree.nodes[node_m]["cont"].pop(added_node)
+        DF_tree.nodes[node_m]["heu"] = 0
+        for i in range(k - len(DF_tree.nodes[node_m]["GM"])):
+            DF_tree.nodes[node_m]["heu"] += DF_tree.nodes[node_m]["cont"][
+                DF_tree.nodes[node_m]["CL"][i]
+            ]
+    else:
+        node_m = None
+
+    return node_p, node_m, DF_tree
+
+
+@nx._dispatchable(edge_attrs="weight")
+def group_closeness_centrality(G, S, weight=None):
+    r"""Compute the group closeness centrality for a group of nodes.
+
+    Group closeness centrality of a group of nodes $S$ is a measure
+    of how close the group is to the other nodes in the graph.
+
+    .. math::
+
+       c_{close}(S) = \frac{|V-S|}{\sum_{v \in V-S} d_{S, v}}
+
+       d_{S, v} = min_{u \in S} (d_{u, v})
+
+    where $V$ is the set of nodes, $d_{S, v}$ is the distance of
+    the group $S$ from $v$ defined as above. ($V-S$ is the set of nodes
+    in $V$ that are not in $S$).
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph.
+
+    S : list or set
+       S is a group of nodes which belong to G, for which group closeness
+       centrality is to be calculated.
+
+    weight : None or string, optional (default=None)
+       If None, all edge weights are considered equal.
+       Otherwise holds the name of the edge attribute used as weight.
+       The weight of an edge is treated as the length or distance between the two sides.
+
+    Raises
+    ------
+    NodeNotFound
+       If node(s) in S are not present in G.
+
+    Returns
+    -------
+    closeness : float
+       Group closeness centrality of the group S.
+
+    See Also
+    --------
+    closeness_centrality
+
+    Notes
+    -----
+    The measure was introduced in [1]_.
+    The formula implemented here is described in [2]_.
+
+    Higher values of closeness indicate greater centrality.
+
+    It is assumed that 1 / 0 is 0 (required in the case of directed graphs,
+    or when a shortest path length is 0).
+
+    The number of nodes in the group must be a maximum of n - 1 where `n`
+    is the total number of nodes in the graph.
+
+    For directed graphs, the incoming distance is utilized here. To use the
+    outward distance, act on `G.reverse()`.
+
+    For weighted graphs the edge weights must be greater than zero.
+    Zero edge weights can produce an infinite number of equal length
+    paths between pairs of nodes.
+
+    References
+    ----------
+    .. [1] M G Everett and S P Borgatti:
+       The Centrality of Groups and Classes.
+       Journal of Mathematical Sociology. 23(3): 181-201. 1999.
+       http://www.analytictech.com/borgatti/group_centrality.htm
+    .. [2] J. Zhao et. al.:
+       Measuring and Maximizing Group Closeness Centrality over
+       Disk Resident Graphs.
+       WWWConference Proceedings, 2014. 689-694.
+       https://doi.org/10.1145/2567948.2579356
+    """
+    if G.is_directed():
+        G = G.reverse()  # reverse view
+    closeness = 0  # initialize to 0
+    V = set(G)  # set of nodes in G
+    S = set(S)  # set of nodes in group S
+    V_S = V - S  # set of nodes in V but not S
+    shortest_path_lengths = nx.multi_source_dijkstra_path_length(G, S, weight=weight)
+    # accumulation
+    for v in V_S:
+        try:
+            closeness += shortest_path_lengths[v]
+        except KeyError:  # no path exists
+            closeness += 0
+    try:
+        closeness = len(V_S) / closeness
+    except ZeroDivisionError:  # 1 / 0 assumed as 0
+        closeness = 0
+    return closeness
+
+
+@nx._dispatchable
+def group_degree_centrality(G, S):
+    """Compute the group degree centrality for a group of nodes.
+
+    Group degree centrality of a group of nodes $S$ is the fraction
+    of non-group members connected to group members.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph.
+
+    S : list or set
+       S is a group of nodes which belong to G, for which group degree
+       centrality is to be calculated.
+
+    Raises
+    ------
+    NetworkXError
+       If node(s) in S are not in G.
+
+    Returns
+    -------
+    centrality : float
+       Group degree centrality of the group S.
+
+    See Also
+    --------
+    degree_centrality
+    group_in_degree_centrality
+    group_out_degree_centrality
+
+    Notes
+    -----
+    The measure was introduced in [1]_.
+
+    The number of nodes in the group must be a maximum of n - 1 where `n`
+    is the total number of nodes in the graph.
+
+    References
+    ----------
+    .. [1] M G Everett and S P Borgatti:
+       The Centrality of Groups and Classes.
+       Journal of Mathematical Sociology. 23(3): 181-201. 1999.
+       http://www.analytictech.com/borgatti/group_centrality.htm
+    """
+    centrality = len(set().union(*[set(G.neighbors(i)) for i in S]) - set(S))
+    centrality /= len(G.nodes()) - len(S)
+    return centrality
+
+
+@not_implemented_for("undirected")
+@nx._dispatchable
+def group_in_degree_centrality(G, S):
+    """Compute the group in-degree centrality for a group of nodes.
+
+    Group in-degree centrality of a group of nodes $S$ is the fraction
+    of non-group members connected to group members by incoming edges.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph.
+
+    S : list or set
+       S is a group of nodes which belong to G, for which group in-degree
+       centrality is to be calculated.
+
+    Returns
+    -------
+    centrality : float
+       Group in-degree centrality of the group S.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+       If G is undirected.
+
+    NodeNotFound
+       If node(s) in S are not in G.
+
+    See Also
+    --------
+    degree_centrality
+    group_degree_centrality
+    group_out_degree_centrality
+
+    Notes
+    -----
+    The number of nodes in the group must be a maximum of n - 1 where `n`
+    is the total number of nodes in the graph.
+
+    `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,
+    so for group in-degree centrality, the reverse graph is used.
+    """
+    return group_degree_centrality(G.reverse(), S)
+
+
+@not_implemented_for("undirected")
+@nx._dispatchable
+def group_out_degree_centrality(G, S):
+    """Compute the group out-degree centrality for a group of nodes.
+
+    Group out-degree centrality of a group of nodes $S$ is the fraction
+    of non-group members connected to group members by outgoing edges.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph.
+
+    S : list or set
+       S is a group of nodes which belong to G, for which group in-degree
+       centrality is to be calculated.
+
+    Returns
+    -------
+    centrality : float
+       Group out-degree centrality of the group S.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+       If G is undirected.
+
+    NodeNotFound
+       If node(s) in S are not in G.
+
+    See Also
+    --------
+    degree_centrality
+    group_degree_centrality
+    group_in_degree_centrality
+
+    Notes
+    -----
+    The number of nodes in the group must be a maximum of n - 1 where `n`
+    is the total number of nodes in the graph.
+
+    `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph,
+    so for group out-degree centrality, the graph itself is used.
+    """
+    return group_degree_centrality(G, S)

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/katz.py

@@ -0,0 +1,330 @@
+"""Katz centrality."""
+import math
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+__all__ = ["katz_centrality", "katz_centrality_numpy"]
+
+
+@not_implemented_for("multigraph")
+@nx._dispatchable(edge_attrs="weight")
+def katz_centrality(
+    G,
+    alpha=0.1,
+    beta=1.0,
+    max_iter=1000,
+    tol=1.0e-6,
+    nstart=None,
+    normalized=True,
+    weight=None,
+):
+    r"""Compute the Katz centrality for the nodes of the graph G.
+
+    Katz centrality computes the centrality for a node based on the centrality
+    of its neighbors. It is a generalization of the eigenvector centrality. The
+    Katz centrality for node $i$ is
+
+    .. math::
+
+        x_i = \alpha \sum_{j} A_{ij} x_j + \beta,
+
+    where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$.
+
+    The parameter $\beta$ controls the initial centrality and
+
+    .. math::
+
+        \alpha < \frac{1}{\lambda_{\max}}.
+
+    Katz centrality computes the relative influence of a node within a
+    network by measuring the number of the immediate neighbors (first
+    degree nodes) and also all other nodes in the network that connect
+    to the node under consideration through these immediate neighbors.
+
+    Extra weight can be provided to immediate neighbors through the
+    parameter $\beta$.  Connections made with distant neighbors
+    are, however, penalized by an attenuation factor $\alpha$ which
+    should be strictly less than the inverse largest eigenvalue of the
+    adjacency matrix in order for the Katz centrality to be computed
+    correctly. More information is provided in [1]_.
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph.
+
+    alpha : float, optional (default=0.1)
+      Attenuation factor
+
+    beta : scalar or dictionary, optional (default=1.0)
+      Weight attributed to the immediate neighborhood. If not a scalar, the
+      dictionary must have a value for every node.
+
+    max_iter : integer, optional (default=1000)
+      Maximum number of iterations in power method.
+
+    tol : float, optional (default=1.0e-6)
+      Error tolerance used to check convergence in power method iteration.
+
+    nstart : dictionary, optional
+      Starting value of Katz iteration for each node.
+
+    normalized : bool, optional (default=True)
+      If True normalize the resulting values.
+
+    weight : None or string, optional (default=None)
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+      In this measure the weight is interpreted as the connection strength.
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with Katz centrality as the value.
+
+    Raises
+    ------
+    NetworkXError
+       If the parameter `beta` is not a scalar but lacks a value for at least
+       one node
+
+    PowerIterationFailedConvergence
+        If the algorithm fails to converge to the specified tolerance
+        within the specified number of iterations of the power iteration
+        method.
+
+    Examples
+    --------
+    >>> import math
+    >>> G = nx.path_graph(4)
+    >>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
+    >>> centrality = nx.katz_centrality(G, 1 / phi - 0.01)
+    >>> for n, c in sorted(centrality.items()):
+    ...     print(f"{n} {c:.2f}")
+    0 0.37
+    1 0.60
+    2 0.60
+    3 0.37
+
+    See Also
+    --------
+    katz_centrality_numpy
+    eigenvector_centrality
+    eigenvector_centrality_numpy
+    :func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank`
+    :func:`~networkx.algorithms.link_analysis.hits_alg.hits`
+
+    Notes
+    -----
+    Katz centrality was introduced by [2]_.
+
+    This algorithm it uses the power method to find the eigenvector
+    corresponding to the largest eigenvalue of the adjacency matrix of ``G``.
+    The parameter ``alpha`` should be strictly less than the inverse of largest
+    eigenvalue of the adjacency matrix for the algorithm to converge.
+    You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
+    eigenvalue of the adjacency matrix.
+    The iteration will stop after ``max_iter`` iterations or an error tolerance of
+    ``number_of_nodes(G) * tol`` has been reached.
+
+    For strongly connected graphs, as $\alpha \to 1/\lambda_{\max}$, and $\beta > 0$,
+    Katz centrality approaches the results for eigenvector centrality.
+
+    For directed graphs this finds "left" eigenvectors which corresponds
+    to the in-edges in the graph. For out-edges Katz centrality,
+    first reverse the graph with ``G.reverse()``.
+
+    References
+    ----------
+    .. [1] Mark E. J. Newman:
+       Networks: An Introduction.
+       Oxford University Press, USA, 2010, p. 720.
+    .. [2] Leo Katz:
+       A New Status Index Derived from Sociometric Index.
+       Psychometrika 18(1):39–43, 1953
+       https://link.springer.com/content/pdf/10.1007/BF02289026.pdf
+    """
+    if len(G) == 0:
+        return {}
+
+    nnodes = G.number_of_nodes()
+
+    if nstart is None:
+        # choose starting vector with entries of 0
+        x = {n: 0 for n in G}
+    else:
+        x = nstart
+
+    try:
+        b = dict.fromkeys(G, float(beta))
+    except (TypeError, ValueError, AttributeError) as err:
+        b = beta
+        if set(beta) != set(G):
+            raise nx.NetworkXError(
+                "beta dictionary must have a value for every node"
+            ) from err
+
+    # make up to max_iter iterations
+    for _ in range(max_iter):
+        xlast = x
+        x = dict.fromkeys(xlast, 0)
+        # do the multiplication y^T = Alpha * x^T A + Beta
+        for n in x:
+            for nbr in G[n]:
+                x[nbr] += xlast[n] * G[n][nbr].get(weight, 1)
+        for n in x:
+            x[n] = alpha * x[n] + b[n]
+
+        # check convergence
+        error = sum(abs(x[n] - xlast[n]) for n in x)
+        if error < nnodes * tol:
+            if normalized:
+                # normalize vector
+                try:
+                    s = 1.0 / math.hypot(*x.values())
+                except ZeroDivisionError:
+                    s = 1.0
+            else:
+                s = 1
+            for n in x:
+                x[n] *= s
+            return x
+    raise nx.PowerIterationFailedConvergence(max_iter)
+
+
+@not_implemented_for("multigraph")
+@nx._dispatchable(edge_attrs="weight")
+def katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight=None):
+    r"""Compute the Katz centrality for the graph G.
+
+    Katz centrality computes the centrality for a node based on the centrality
+    of its neighbors. It is a generalization of the eigenvector centrality. The
+    Katz centrality for node $i$ is
+
+    .. math::
+
+        x_i = \alpha \sum_{j} A_{ij} x_j + \beta,
+
+    where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$.
+
+    The parameter $\beta$ controls the initial centrality and
+
+    .. math::
+
+        \alpha < \frac{1}{\lambda_{\max}}.
+
+    Katz centrality computes the relative influence of a node within a
+    network by measuring the number of the immediate neighbors (first
+    degree nodes) and also all other nodes in the network that connect
+    to the node under consideration through these immediate neighbors.
+
+    Extra weight can be provided to immediate neighbors through the
+    parameter $\beta$.  Connections made with distant neighbors
+    are, however, penalized by an attenuation factor $\alpha$ which
+    should be strictly less than the inverse largest eigenvalue of the
+    adjacency matrix in order for the Katz centrality to be computed
+    correctly. More information is provided in [1]_.
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX graph
+
+    alpha : float
+      Attenuation factor
+
+    beta : scalar or dictionary, optional (default=1.0)
+      Weight attributed to the immediate neighborhood. If not a scalar the
+      dictionary must have an value for every node.
+
+    normalized : bool
+      If True normalize the resulting values.
+
+    weight : None or string, optional
+      If None, all edge weights are considered equal.
+      Otherwise holds the name of the edge attribute used as weight.
+      In this measure the weight is interpreted as the connection strength.
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with Katz centrality as the value.
+
+    Raises
+    ------
+    NetworkXError
+       If the parameter `beta` is not a scalar but lacks a value for at least
+       one node
+
+    Examples
+    --------
+    >>> import math
+    >>> G = nx.path_graph(4)
+    >>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
+    >>> centrality = nx.katz_centrality_numpy(G, 1 / phi)
+    >>> for n, c in sorted(centrality.items()):
+    ...     print(f"{n} {c:.2f}")
+    0 0.37
+    1 0.60
+    2 0.60
+    3 0.37
+
+    See Also
+    --------
+    katz_centrality
+    eigenvector_centrality_numpy
+    eigenvector_centrality
+    :func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank`
+    :func:`~networkx.algorithms.link_analysis.hits_alg.hits`
+
+    Notes
+    -----
+    Katz centrality was introduced by [2]_.
+
+    This algorithm uses a direct linear solver to solve the above equation.
+    The parameter ``alpha`` should be strictly less than the inverse of largest
+    eigenvalue of the adjacency matrix for there to be a solution.
+    You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest
+    eigenvalue of the adjacency matrix.
+
+    For strongly connected graphs, as $\alpha \to 1/\lambda_{\max}$, and $\beta > 0$,
+    Katz centrality approaches the results for eigenvector centrality.
+
+    For directed graphs this finds "left" eigenvectors which corresponds
+    to the in-edges in the graph. For out-edges Katz centrality,
+    first reverse the graph with ``G.reverse()``.
+
+    References
+    ----------
+    .. [1] Mark E. J. Newman:
+       Networks: An Introduction.
+       Oxford University Press, USA, 2010, p. 173.
+    .. [2] Leo Katz:
+       A New Status Index Derived from Sociometric Index.
+       Psychometrika 18(1):39–43, 1953
+       https://link.springer.com/content/pdf/10.1007/BF02289026.pdf
+    """
+    import numpy as np
+
+    if len(G) == 0:
+        return {}
+    try:
+        nodelist = beta.keys()
+        if set(nodelist) != set(G):
+            raise nx.NetworkXError("beta dictionary must have a value for every node")
+        b = np.array(list(beta.values()), dtype=float)
+    except AttributeError:
+        nodelist = list(G)
+        try:
+            b = np.ones((len(nodelist), 1)) * beta
+        except (TypeError, ValueError, AttributeError) as err:
+            raise nx.NetworkXError("beta must be a number") from err
+
+    A = nx.adjacency_matrix(G, nodelist=nodelist, weight=weight).todense().T
+    n = A.shape[0]
+    centrality = np.linalg.solve(np.eye(n, n) - (alpha * A), b).squeeze()
+
+    # Normalize: rely on truediv to cast to float, then tolist to make Python numbers
+    norm = np.sign(sum(centrality)) * np.linalg.norm(centrality) if normalized else 1
+    return dict(zip(nodelist, (centrality / norm).tolist()))

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/second_order.py

@@ -0,0 +1,141 @@
+"""Copyright (c) 2015 – Thomson Licensing, SAS
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted (subject to the limitations in the
+disclaimer below) provided that the following conditions are met:
+
+* Redistributions of source code must retain the above copyright
+notice, this list of conditions and the following disclaimer.
+
+* Redistributions in binary form must reproduce the above copyright
+notice, this list of conditions and the following disclaimer in the
+documentation and/or other materials provided with the distribution.
+
+* Neither the name of Thomson Licensing, or Technicolor, nor the names
+of its contributors may be used to endorse or promote products derived
+from this software without specific prior written permission.
+
+NO EXPRESS OR IMPLIED LICENSES TO ANY PARTY'S PATENT RIGHTS ARE
+GRANTED BY THIS LICENSE.  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT
+HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED
+WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
+MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
+DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
+BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE
+OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN
+IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+"""
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+# Authors: Erwan Le Merrer ([email protected])
+
+__all__ = ["second_order_centrality"]
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(edge_attrs="weight")
+def second_order_centrality(G, weight="weight"):
+    """Compute the second order centrality for nodes of G.
+
+    The second order centrality of a given node is the standard deviation of
+    the return times to that node of a perpetual random walk on G:
+
+    Parameters
+    ----------
+    G : graph
+      A NetworkX connected and undirected graph.
+
+    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
+    -------
+    nodes : dictionary
+       Dictionary keyed by node with second order centrality as the value.
+
+    Examples
+    --------
+    >>> G = nx.star_graph(10)
+    >>> soc = nx.second_order_centrality(G)
+    >>> print(sorted(soc.items(), key=lambda x: x[1])[0][0])  # pick first id
+    0
+
+    Raises
+    ------
+    NetworkXException
+        If the graph G is empty, non connected or has negative weights.
+
+    See Also
+    --------
+    betweenness_centrality
+
+    Notes
+    -----
+    Lower values of second order centrality indicate higher centrality.
+
+    The algorithm is from Kermarrec, Le Merrer, Sericola and Trédan [1]_.
+
+    This code implements the analytical version of the algorithm, i.e.,
+    there is no simulation of a random walk process involved. The random walk
+    is here unbiased (corresponding to eq 6 of the paper [1]_), thus the
+    centrality values are the standard deviations for random walk return times
+    on the transformed input graph G (equal in-degree at each nodes by adding
+    self-loops).
+
+    Complexity of this implementation, made to run locally on a single machine,
+    is O(n^3), with n the size of G, which makes it viable only for small
+    graphs.
+
+    References
+    ----------
+    .. [1] Anne-Marie Kermarrec, Erwan Le Merrer, Bruno Sericola, Gilles Trédan
+       "Second order centrality: Distributed assessment of nodes criticity in
+       complex networks", Elsevier Computer Communications 34(5):619-628, 2011.
+    """
+    import numpy as np
+
+    n = len(G)
+
+    if n == 0:
+        raise nx.NetworkXException("Empty graph.")
+    if not nx.is_connected(G):
+        raise nx.NetworkXException("Non connected graph.")
+    if any(d.get(weight, 0) < 0 for u, v, d in G.edges(data=True)):
+        raise nx.NetworkXException("Graph has negative edge weights.")
+
+    # balancing G for Metropolis-Hastings random walks
+    G = nx.DiGraph(G)
+    in_deg = dict(G.in_degree(weight=weight))
+    d_max = max(in_deg.values())
+    for i, deg in in_deg.items():
+        if deg < d_max:
+            G.add_edge(i, i, weight=d_max - deg)
+
+    P = nx.to_numpy_array(G)
+    P /= P.sum(axis=1)[:, np.newaxis]  # to transition probability matrix
+
+    def _Qj(P, j):
+        P = P.copy()
+        P[:, j] = 0
+        return P
+
+    M = np.empty([n, n])
+
+    for i in range(n):
+        M[:, i] = np.linalg.solve(
+            np.identity(n) - _Qj(P, i), np.ones([n, 1])[:, 0]
+        )  # eq 3
+
+    return dict(
+        zip(
+            G.nodes,
+            (float(np.sqrt(2 * np.sum(M[:, i]) - n * (n + 1))) for i in range(n)),
+        )
+    )  # eq 6

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/subgraph_alg.py

@@ -0,0 +1,339 @@
+"""
+Subraph centrality and communicability betweenness.
+"""
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+__all__ = [
+    "subgraph_centrality_exp",
+    "subgraph_centrality",
+    "communicability_betweenness_centrality",
+    "estrada_index",
+]
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def subgraph_centrality_exp(G):
+    r"""Returns the subgraph centrality for each node of G.
+
+    Subgraph centrality  of a node `n` is the sum of weighted closed
+    walks of all lengths starting and ending at node `n`. The weights
+    decrease with path length. Each closed walk is associated with a
+    connected subgraph ([1]_).
+
+    Parameters
+    ----------
+    G: graph
+
+    Returns
+    -------
+    nodes:dictionary
+        Dictionary of nodes with subgraph centrality as the value.
+
+    Raises
+    ------
+    NetworkXError
+        If the graph is not undirected and simple.
+
+    See Also
+    --------
+    subgraph_centrality:
+        Alternative algorithm of the subgraph centrality for each node of G.
+
+    Notes
+    -----
+    This version of the algorithm exponentiates the adjacency matrix.
+
+    The subgraph centrality of a node `u` in G can be found using
+    the matrix exponential of the adjacency matrix of G [1]_,
+
+    .. math::
+
+        SC(u)=(e^A)_{uu} .
+
+    References
+    ----------
+    .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez,
+       "Subgraph centrality in complex networks",
+       Physical Review E 71, 056103 (2005).
+       https://arxiv.org/abs/cond-mat/0504730
+
+    Examples
+    --------
+    (Example from [1]_)
+    >>> G = nx.Graph(
+    ...     [
+    ...         (1, 2),
+    ...         (1, 5),
+    ...         (1, 8),
+    ...         (2, 3),
+    ...         (2, 8),
+    ...         (3, 4),
+    ...         (3, 6),
+    ...         (4, 5),
+    ...         (4, 7),
+    ...         (5, 6),
+    ...         (6, 7),
+    ...         (7, 8),
+    ...     ]
+    ... )
+    >>> sc = nx.subgraph_centrality_exp(G)
+    >>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)])
+    ['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90']
+    """
+    # alternative implementation that calculates the matrix exponential
+    import scipy as sp
+
+    nodelist = list(G)  # ordering of nodes in matrix
+    A = nx.to_numpy_array(G, nodelist)
+    # convert to 0-1 matrix
+    A[A != 0.0] = 1
+    expA = sp.linalg.expm(A)
+    # convert diagonal to dictionary keyed by node
+    sc = dict(zip(nodelist, map(float, expA.diagonal())))
+    return sc
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def subgraph_centrality(G):
+    r"""Returns subgraph centrality for each node in G.
+
+    Subgraph centrality  of a node `n` is the sum of weighted closed
+    walks of all lengths starting and ending at node `n`. The weights
+    decrease with path length. Each closed walk is associated with a
+    connected subgraph ([1]_).
+
+    Parameters
+    ----------
+    G: graph
+
+    Returns
+    -------
+    nodes : dictionary
+       Dictionary of nodes with subgraph centrality as the value.
+
+    Raises
+    ------
+    NetworkXError
+       If the graph is not undirected and simple.
+
+    See Also
+    --------
+    subgraph_centrality_exp:
+        Alternative algorithm of the subgraph centrality for each node of G.
+
+    Notes
+    -----
+    This version of the algorithm computes eigenvalues and eigenvectors
+    of the adjacency matrix.
+
+    Subgraph centrality of a node `u` in G can be found using
+    a spectral decomposition of the adjacency matrix [1]_,
+
+    .. math::
+
+       SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}},
+
+    where `v_j` is an eigenvector of the adjacency matrix `A` of G
+    corresponding to the eigenvalue `\lambda_j`.
+
+    Examples
+    --------
+    (Example from [1]_)
+    >>> G = nx.Graph(
+    ...     [
+    ...         (1, 2),
+    ...         (1, 5),
+    ...         (1, 8),
+    ...         (2, 3),
+    ...         (2, 8),
+    ...         (3, 4),
+    ...         (3, 6),
+    ...         (4, 5),
+    ...         (4, 7),
+    ...         (5, 6),
+    ...         (6, 7),
+    ...         (7, 8),
+    ...     ]
+    ... )
+    >>> sc = nx.subgraph_centrality(G)
+    >>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)])
+    ['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90']
+
+    References
+    ----------
+    .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez,
+       "Subgraph centrality in complex networks",
+       Physical Review E 71, 056103 (2005).
+       https://arxiv.org/abs/cond-mat/0504730
+
+    """
+    import numpy as np
+
+    nodelist = list(G)  # ordering of nodes in matrix
+    A = nx.to_numpy_array(G, nodelist)
+    # convert to 0-1 matrix
+    A[np.nonzero(A)] = 1
+    w, v = np.linalg.eigh(A)
+    vsquare = np.array(v) ** 2
+    expw = np.exp(w)
+    xg = vsquare @ expw
+    # convert vector dictionary keyed by node
+    sc = dict(zip(nodelist, map(float, xg)))
+    return sc
+
+
+@not_implemented_for("directed")
+@not_implemented_for("multigraph")
+@nx._dispatchable
+def communicability_betweenness_centrality(G):
+    r"""Returns subgraph communicability for all pairs of nodes in G.
+
+    Communicability betweenness measure makes use of the number of walks
+    connecting every pair of nodes as the basis of a betweenness centrality
+    measure.
+
+    Parameters
+    ----------
+    G: graph
+
+    Returns
+    -------
+    nodes : dictionary
+        Dictionary of nodes with communicability betweenness as the value.
+
+    Raises
+    ------
+    NetworkXError
+        If the graph is not undirected and simple.
+
+    Notes
+    -----
+    Let `G=(V,E)` be a simple undirected graph with `n` nodes and `m` edges,
+    and `A` denote the adjacency matrix of `G`.
+
+    Let `G(r)=(V,E(r))` be the graph resulting from
+    removing all edges connected to node `r` but not the node itself.
+
+    The adjacency matrix for `G(r)` is `A+E(r)`,  where `E(r)` has nonzeros
+    only in row and column `r`.
+
+    The subraph betweenness of a node `r`  is [1]_
+
+    .. math::
+
+         \omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}},
+         p\neq q, q\neq r,
+
+    where
+    `G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq}`  is the number of walks
+    involving node r,
+    `G_{pq}=(e^{A})_{pq}` is the number of closed walks starting
+    at node `p` and ending at node `q`,
+    and `C=(n-1)^{2}-(n-1)` is a normalization factor equal to the
+    number of terms in the sum.
+
+    The resulting `\omega_{r}` takes values between zero and one.
+    The lower bound cannot be attained for a connected
+    graph, and the upper bound is attained in the star graph.
+
+    References
+    ----------
+    .. [1] Ernesto Estrada, Desmond J. Higham, Naomichi Hatano,
+       "Communicability Betweenness in Complex Networks"
+       Physica A 388 (2009) 764-774.
+       https://arxiv.org/abs/0905.4102
+
+    Examples
+    --------
+    >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)])
+    >>> cbc = nx.communicability_betweenness_centrality(G)
+    >>> print([f"{node} {cbc[node]:0.2f}" for node in sorted(cbc)])
+    ['0 0.03', '1 0.45', '2 0.51', '3 0.45', '4 0.40', '5 0.19', '6 0.03']
+    """
+    import numpy as np
+    import scipy as sp
+
+    nodelist = list(G)  # ordering of nodes in matrix
+    n = len(nodelist)
+    A = nx.to_numpy_array(G, nodelist)
+    # convert to 0-1 matrix
+    A[np.nonzero(A)] = 1
+    expA = sp.linalg.expm(A)
+    mapping = dict(zip(nodelist, range(n)))
+    cbc = {}
+    for v in G:
+        # remove row and col of node v
+        i = mapping[v]
+        row = A[i, :].copy()
+        col = A[:, i].copy()
+        A[i, :] = 0
+        A[:, i] = 0
+        B = (expA - sp.linalg.expm(A)) / expA
+        # sum with row/col of node v and diag set to zero
+        B[i, :] = 0
+        B[:, i] = 0
+        B -= np.diag(np.diag(B))
+        cbc[v] = float(B.sum())
+        # put row and col back
+        A[i, :] = row
+        A[:, i] = col
+    # rescale when more than two nodes
+    order = len(cbc)
+    if order > 2:
+        scale = 1.0 / ((order - 1.0) ** 2 - (order - 1.0))
+        cbc = {node: value * scale for node, value in cbc.items()}
+    return cbc
+
+
+@nx._dispatchable
+def estrada_index(G):
+    r"""Returns the Estrada index of a the graph G.
+
+    The Estrada Index is a topological index of folding or 3D "compactness" ([1]_).
+
+    Parameters
+    ----------
+    G: graph
+
+    Returns
+    -------
+    estrada index: float
+
+    Raises
+    ------
+    NetworkXError
+        If the graph is not undirected and simple.
+
+    Notes
+    -----
+    Let `G=(V,E)` be a simple undirected graph with `n` nodes  and let
+    `\lambda_{1}\leq\lambda_{2}\leq\cdots\lambda_{n}`
+    be a non-increasing ordering of the eigenvalues of its adjacency
+    matrix `A`. The Estrada index is ([1]_, [2]_)
+
+    .. math::
+        EE(G)=\sum_{j=1}^n e^{\lambda _j}.
+
+    References
+    ----------
+    .. [1] E. Estrada, "Characterization of 3D molecular structure",
+       Chem. Phys. Lett. 319, 713 (2000).
+       https://doi.org/10.1016/S0009-2614(00)00158-5
+    .. [2] José Antonio de la Peñaa, Ivan Gutman, Juan Rada,
+       "Estimating the Estrada index",
+       Linear Algebra and its Applications. 427, 1 (2007).
+       https://doi.org/10.1016/j.laa.2007.06.020
+
+    Examples
+    --------
+    >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)])
+    >>> ei = nx.estrada_index(G)
+    >>> print(f"{ei:0.5}")
+    20.55
+    """
+    return sum(subgraph_centrality(G).values())

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/tests/test_current_flow_betweenness_centrality.py

@@ -0,0 +1,197 @@
+import pytest
+
+import networkx as nx
+from networkx import approximate_current_flow_betweenness_centrality as approximate_cfbc
+from networkx import edge_current_flow_betweenness_centrality as edge_current_flow
+
+np = pytest.importorskip("numpy")
+pytest.importorskip("scipy")
+
+
+class TestFlowBetweennessCentrality:
+    def test_K4_normalized(self):
+        """Betweenness centrality: K4"""
+        G = nx.complete_graph(4)
+        b = nx.current_flow_betweenness_centrality(G, normalized=True)
+        b_answer = {0: 0.25, 1: 0.25, 2: 0.25, 3: 0.25}
+        for n in sorted(G):
+            assert b[n] == pytest.approx(b_answer[n], abs=1e-7)
+        G.add_edge(0, 1, weight=0.5, other=0.3)
+        b = nx.current_flow_betweenness_centrality(G, normalized=True, weight=None)
+        for n in sorted(G):
+            assert b[n] == pytest.approx(b_answer[n], abs=1e-7)
+        wb_answer = {0: 0.2222222, 1: 0.2222222, 2: 0.30555555, 3: 0.30555555}
+        b = nx.current_flow_betweenness_centrality(G, normalized=True, weight="weight")
+        for n in sorted(G):
+            assert b[n] == pytest.approx(wb_answer[n], abs=1e-7)
+        wb_answer = {0: 0.2051282, 1: 0.2051282, 2: 0.33974358, 3: 0.33974358}
+        b = nx.current_flow_betweenness_centrality(G, normalized=True, weight="other")
+        for n in sorted(G):
+            assert b[n] == pytest.approx(wb_answer[n], abs=1e-7)
+
+    def test_K4(self):
+        """Betweenness centrality: K4"""
+        G = nx.complete_graph(4)
+        for solver in ["full", "lu", "cg"]:
+            b = nx.current_flow_betweenness_centrality(
+                G, normalized=False, solver=solver
+            )
+            b_answer = {0: 0.75, 1: 0.75, 2: 0.75, 3: 0.75}
+            for n in sorted(G):
+                assert b[n] == pytest.approx(b_answer[n], abs=1e-7)
+
+    def test_P4_normalized(self):
+        """Betweenness centrality: P4 normalized"""
+        G = nx.path_graph(4)
+        b = nx.current_flow_betweenness_centrality(G, normalized=True)
+        b_answer = {0: 0, 1: 2.0 / 3, 2: 2.0 / 3, 3: 0}
+        for n in sorted(G):
+            assert b[n] == pytest.approx(b_answer[n], abs=1e-7)
+
+    def test_P4(self):
+        """Betweenness centrality: P4"""
+        G = nx.path_graph(4)
+        b = nx.current_flow_betweenness_centrality(G, normalized=False)
+        b_answer = {0: 0, 1: 2, 2: 2, 3: 0}
+        for n in sorted(G):
+            assert b[n] == pytest.approx(b_answer[n], abs=1e-7)
+
+    def test_star(self):
+        """Betweenness centrality: star"""
+        G = nx.Graph()
+        nx.add_star(G, ["a", "b", "c", "d"])
+        b = nx.current_flow_betweenness_centrality(G, normalized=True)
+        b_answer = {"a": 1.0, "b": 0.0, "c": 0.0, "d": 0.0}
+        for n in sorted(G):
+            assert b[n] == pytest.approx(b_answer[n], abs=1e-7)
+
+    def test_solvers2(self):
+        """Betweenness centrality: alternate solvers"""
+        G = nx.complete_graph(4)
+        for solver in ["full", "lu", "cg"]:
+            b = nx.current_flow_betweenness_centrality(
+                G, normalized=False, solver=solver
+            )
+            b_answer = {0: 0.75, 1: 0.75, 2: 0.75, 3: 0.75}
+            for n in sorted(G):
+                assert b[n] == pytest.approx(b_answer[n], abs=1e-7)
+
+
+class TestApproximateFlowBetweennessCentrality:
+    def test_K4_normalized(self):
+        "Approximate current-flow betweenness centrality: K4 normalized"
+        G = nx.complete_graph(4)
+        b = nx.current_flow_betweenness_centrality(G, normalized=True)
+        epsilon = 0.1
+        ba = approximate_cfbc(G, normalized=True, epsilon=0.5 * epsilon)
+        for n in sorted(G):
+            np.testing.assert_allclose(b[n], ba[n], atol=epsilon)
+
+    def test_K4(self):
+        "Approximate current-flow betweenness centrality: K4"
+        G = nx.complete_graph(4)
+        b = nx.current_flow_betweenness_centrality(G, normalized=False)
+        epsilon = 0.1
+        ba = approximate_cfbc(G, normalized=False, epsilon=0.5 * epsilon)
+        for n in sorted(G):
+            np.testing.assert_allclose(b[n], ba[n], atol=epsilon * len(G) ** 2)
+
+    def test_star(self):
+        "Approximate current-flow betweenness centrality: star"
+        G = nx.Graph()
+        nx.add_star(G, ["a", "b", "c", "d"])
+        b = nx.current_flow_betweenness_centrality(G, normalized=True)
+        epsilon = 0.1
+        ba = approximate_cfbc(G, normalized=True, epsilon=0.5 * epsilon)
+        for n in sorted(G):
+            np.testing.assert_allclose(b[n], ba[n], atol=epsilon)
+
+    def test_grid(self):
+        "Approximate current-flow betweenness centrality: 2d grid"
+        G = nx.grid_2d_graph(4, 4)
+        b = nx.current_flow_betweenness_centrality(G, normalized=True)
+        epsilon = 0.1
+        ba = approximate_cfbc(G, normalized=True, epsilon=0.5 * epsilon)
+        for n in sorted(G):
+            np.testing.assert_allclose(b[n], ba[n], atol=epsilon)
+
+    def test_seed(self):
+        G = nx.complete_graph(4)
+        b = approximate_cfbc(G, normalized=False, epsilon=0.05, seed=1)
+        b_answer = {0: 0.75, 1: 0.75, 2: 0.75, 3: 0.75}
+        for n in sorted(G):
+            np.testing.assert_allclose(b[n], b_answer[n], atol=0.1)
+
+    def test_solvers(self):
+        "Approximate current-flow betweenness centrality: solvers"
+        G = nx.complete_graph(4)
+        epsilon = 0.1
+        for solver in ["full", "lu", "cg"]:
+            b = approximate_cfbc(
+                G, normalized=False, solver=solver, epsilon=0.5 * epsilon
+            )
+            b_answer = {0: 0.75, 1: 0.75, 2: 0.75, 3: 0.75}
+            for n in sorted(G):
+                np.testing.assert_allclose(b[n], b_answer[n], atol=epsilon)
+
+    def test_lower_kmax(self):
+        G = nx.complete_graph(4)
+        with pytest.raises(nx.NetworkXError, match="Increase kmax or epsilon"):
+            nx.approximate_current_flow_betweenness_centrality(G, kmax=4)
+
+
+class TestWeightedFlowBetweennessCentrality:
+    pass
+
+
+class TestEdgeFlowBetweennessCentrality:
+    def test_K4(self):
+        """Edge flow betweenness centrality: K4"""
+        G = nx.complete_graph(4)
+        b = edge_current_flow(G, normalized=True)
+        b_answer = dict.fromkeys(G.edges(), 0.25)
+        for (s, t), v1 in b_answer.items():
+            v2 = b.get((s, t), b.get((t, s)))
+            assert v1 == pytest.approx(v2, abs=1e-7)
+
+    def test_K4_normalized(self):
+        """Edge flow betweenness centrality: K4"""
+        G = nx.complete_graph(4)
+        b = edge_current_flow(G, normalized=False)
+        b_answer = dict.fromkeys(G.edges(), 0.75)
+        for (s, t), v1 in b_answer.items():
+            v2 = b.get((s, t), b.get((t, s)))
+            assert v1 == pytest.approx(v2, abs=1e-7)
+
+    def test_C4(self):
+        """Edge flow betweenness centrality: C4"""
+        G = nx.cycle_graph(4)
+        b = edge_current_flow(G, normalized=False)
+        b_answer = {(0, 1): 1.25, (0, 3): 1.25, (1, 2): 1.25, (2, 3): 1.25}
+        for (s, t), v1 in b_answer.items():
+            v2 = b.get((s, t), b.get((t, s)))
+            assert v1 == pytest.approx(v2, abs=1e-7)
+
+    def test_P4(self):
+        """Edge betweenness centrality: P4"""
+        G = nx.path_graph(4)
+        b = edge_current_flow(G, normalized=False)
+        b_answer = {(0, 1): 1.5, (1, 2): 2.0, (2, 3): 1.5}
+        for (s, t), v1 in b_answer.items():
+            v2 = b.get((s, t), b.get((t, s)))
+            assert v1 == pytest.approx(v2, abs=1e-7)
+
+
+@pytest.mark.parametrize(
+    "centrality_func",
+    (
+        nx.current_flow_betweenness_centrality,
+        nx.edge_current_flow_betweenness_centrality,
+        nx.approximate_current_flow_betweenness_centrality,
+    ),
+)
+def test_unconnected_graphs_betweenness_centrality(centrality_func):
+    G = nx.Graph([(1, 2), (3, 4)])
+    G.add_node(5)
+    with pytest.raises(nx.NetworkXError, match="Graph not connected"):
+        centrality_func(G)

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/tests/test_dispersion.py

@@ -0,0 +1,73 @@
+import networkx as nx
+
+
+def small_ego_G():
+    """The sample network from https://arxiv.org/pdf/1310.6753v1.pdf"""
+    edges = [
+        ("a", "b"),
+        ("a", "c"),
+        ("b", "c"),
+        ("b", "d"),
+        ("b", "e"),
+        ("b", "f"),
+        ("c", "d"),
+        ("c", "f"),
+        ("c", "h"),
+        ("d", "f"),
+        ("e", "f"),
+        ("f", "h"),
+        ("h", "j"),
+        ("h", "k"),
+        ("i", "j"),
+        ("i", "k"),
+        ("j", "k"),
+        ("u", "a"),
+        ("u", "b"),
+        ("u", "c"),
+        ("u", "d"),
+        ("u", "e"),
+        ("u", "f"),
+        ("u", "g"),
+        ("u", "h"),
+        ("u", "i"),
+        ("u", "j"),
+        ("u", "k"),
+    ]
+    G = nx.Graph()
+    G.add_edges_from(edges)
+
+    return G
+
+
+class TestDispersion:
+    def test_article(self):
+        """our algorithm matches article's"""
+        G = small_ego_G()
+        disp_uh = nx.dispersion(G, "u", "h", normalized=False)
+        disp_ub = nx.dispersion(G, "u", "b", normalized=False)
+        assert disp_uh == 4
+        assert disp_ub == 1
+
+    def test_results_length(self):
+        """there is a result for every node"""
+        G = small_ego_G()
+        disp = nx.dispersion(G)
+        disp_Gu = nx.dispersion(G, "u")
+        disp_uv = nx.dispersion(G, "u", "h")
+        assert len(disp) == len(G)
+        assert len(disp_Gu) == len(G) - 1
+        assert isinstance(disp_uv, float)
+
+    def test_dispersion_v_only(self):
+        G = small_ego_G()
+        disp_G_h = nx.dispersion(G, v="h", normalized=False)
+        disp_G_h_normalized = nx.dispersion(G, v="h", normalized=True)
+        assert disp_G_h == {"c": 0, "f": 0, "j": 0, "k": 0, "u": 4}
+        assert disp_G_h_normalized == {"c": 0.0, "f": 0.0, "j": 0.0, "k": 0.0, "u": 1.0}
+
+    def test_impossible_things(self):
+        G = nx.karate_club_graph()
+        disp = nx.dispersion(G)
+        for u in disp:
+            for v in disp[u]:
+                assert disp[u][v] >= 0

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/tests/test_group.py

@@ -0,0 +1,278 @@
+"""
+Tests for Group Centrality Measures
+"""
+
+
+import pytest
+
+import networkx as nx
+
+
+class TestGroupBetweennessCentrality:
+    def test_group_betweenness_single_node(self):
+        """
+        Group betweenness centrality for single node group
+        """
+        G = nx.path_graph(5)
+        C = [1]
+        b = nx.group_betweenness_centrality(
+            G, C, weight=None, normalized=False, endpoints=False
+        )
+        b_answer = 3.0
+        assert b == b_answer
+
+    def test_group_betweenness_with_endpoints(self):
+        """
+        Group betweenness centrality for single node group
+        """
+        G = nx.path_graph(5)
+        C = [1]
+        b = nx.group_betweenness_centrality(
+            G, C, weight=None, normalized=False, endpoints=True
+        )
+        b_answer = 7.0
+        assert b == b_answer
+
+    def test_group_betweenness_normalized(self):
+        """
+        Group betweenness centrality for group with more than
+        1 node and normalized
+        """
+        G = nx.path_graph(5)
+        C = [1, 3]
+        b = nx.group_betweenness_centrality(
+            G, C, weight=None, normalized=True, endpoints=False
+        )
+        b_answer = 1.0
+        assert b == b_answer
+
+    def test_two_group_betweenness_value_zero(self):
+        """
+        Group betweenness centrality value of 0
+        """
+        G = nx.cycle_graph(7)
+        C = [[0, 1, 6], [0, 1, 5]]
+        b = nx.group_betweenness_centrality(G, C, weight=None, normalized=False)
+        b_answer = [0.0, 3.0]
+        assert b == b_answer
+
+    def test_group_betweenness_value_zero(self):
+        """
+        Group betweenness centrality value of 0
+        """
+        G = nx.cycle_graph(6)
+        C = [0, 1, 5]
+        b = nx.group_betweenness_centrality(G, C, weight=None, normalized=False)
+        b_answer = 0.0
+        assert b == b_answer
+
+    def test_group_betweenness_disconnected_graph(self):
+        """
+        Group betweenness centrality in a disconnected graph
+        """
+        G = nx.path_graph(5)
+        G.remove_edge(0, 1)
+        C = [1]
+        b = nx.group_betweenness_centrality(G, C, weight=None, normalized=False)
+        b_answer = 0.0
+        assert b == b_answer
+
+    def test_group_betweenness_node_not_in_graph(self):
+        """
+        Node(s) in C not in graph, raises NodeNotFound exception
+        """
+        with pytest.raises(nx.NodeNotFound):
+            nx.group_betweenness_centrality(nx.path_graph(5), [4, 7, 8])
+
+    def test_group_betweenness_directed_weighted(self):
+        """
+        Group betweenness centrality in a directed and weighted graph
+        """
+        G = nx.DiGraph()
+        G.add_edge(1, 0, weight=1)
+        G.add_edge(0, 2, weight=2)
+        G.add_edge(1, 2, weight=3)
+        G.add_edge(3, 1, weight=4)
+        G.add_edge(2, 3, weight=1)
+        G.add_edge(4, 3, weight=6)
+        G.add_edge(2, 4, weight=7)
+        C = [1, 2]
+        b = nx.group_betweenness_centrality(G, C, weight="weight", normalized=False)
+        b_answer = 5.0
+        assert b == b_answer
+
+
+class TestProminentGroup:
+    np = pytest.importorskip("numpy")
+    pd = pytest.importorskip("pandas")
+
+    def test_prominent_group_single_node(self):
+        """
+        Prominent group for single node
+        """
+        G = nx.path_graph(5)
+        k = 1
+        b, g = nx.prominent_group(G, k, normalized=False, endpoints=False)
+        b_answer, g_answer = 4.0, [2]
+        assert b == b_answer and g == g_answer
+
+    def test_prominent_group_with_c(self):
+        """
+        Prominent group without some nodes
+        """
+        G = nx.path_graph(5)
+        k = 1
+        b, g = nx.prominent_group(G, k, normalized=False, C=[2])
+        b_answer, g_answer = 3.0, [1]
+        assert b == b_answer and g == g_answer
+
+    def test_prominent_group_normalized_endpoints(self):
+        """
+        Prominent group with normalized result, with endpoints
+        """
+        G = nx.cycle_graph(7)
+        k = 2
+        b, g = nx.prominent_group(G, k, normalized=True, endpoints=True)
+        b_answer, g_answer = 1.7, [2, 5]
+        assert b == b_answer and g == g_answer
+
+    def test_prominent_group_disconnected_graph(self):
+        """
+        Prominent group of disconnected graph
+        """
+        G = nx.path_graph(6)
+        G.remove_edge(0, 1)
+        k = 1
+        b, g = nx.prominent_group(G, k, weight=None, normalized=False)
+        b_answer, g_answer = 4.0, [3]
+        assert b == b_answer and g == g_answer
+
+    def test_prominent_group_node_not_in_graph(self):
+        """
+        Node(s) in C not in graph, raises NodeNotFound exception
+        """
+        with pytest.raises(nx.NodeNotFound):
+            nx.prominent_group(nx.path_graph(5), 1, C=[10])
+
+    def test_group_betweenness_directed_weighted(self):
+        """
+        Group betweenness centrality in a directed and weighted graph
+        """
+        G = nx.DiGraph()
+        G.add_edge(1, 0, weight=1)
+        G.add_edge(0, 2, weight=2)
+        G.add_edge(1, 2, weight=3)
+        G.add_edge(3, 1, weight=4)
+        G.add_edge(2, 3, weight=1)
+        G.add_edge(4, 3, weight=6)
+        G.add_edge(2, 4, weight=7)
+        k = 2
+        b, g = nx.prominent_group(G, k, weight="weight", normalized=False)
+        b_answer, g_answer = 5.0, [1, 2]
+        assert b == b_answer and g == g_answer
+
+    def test_prominent_group_greedy_algorithm(self):
+        """
+        Group betweenness centrality in a greedy algorithm
+        """
+        G = nx.cycle_graph(7)
+        k = 2
+        b, g = nx.prominent_group(G, k, normalized=True, endpoints=True, greedy=True)
+        b_answer, g_answer = 1.7, [6, 3]
+        assert b == b_answer and g == g_answer
+
+
+class TestGroupClosenessCentrality:
+    def test_group_closeness_single_node(self):
+        """
+        Group closeness centrality for a single node group
+        """
+        G = nx.path_graph(5)
+        c = nx.group_closeness_centrality(G, [1])
+        c_answer = nx.closeness_centrality(G, 1)
+        assert c == c_answer
+
+    def test_group_closeness_disconnected(self):
+        """
+        Group closeness centrality for a disconnected graph
+        """
+        G = nx.Graph()
+        G.add_nodes_from([1, 2, 3, 4])
+        c = nx.group_closeness_centrality(G, [1, 2])
+        c_answer = 0
+        assert c == c_answer
+
+    def test_group_closeness_multiple_node(self):
+        """
+        Group closeness centrality for a group with more than
+        1 node
+        """
+        G = nx.path_graph(4)
+        c = nx.group_closeness_centrality(G, [1, 2])
+        c_answer = 1
+        assert c == c_answer
+
+    def test_group_closeness_node_not_in_graph(self):
+        """
+        Node(s) in S not in graph, raises NodeNotFound exception
+        """
+        with pytest.raises(nx.NodeNotFound):
+            nx.group_closeness_centrality(nx.path_graph(5), [6, 7, 8])
+
+
+class TestGroupDegreeCentrality:
+    def test_group_degree_centrality_single_node(self):
+        """
+        Group degree centrality for a single node group
+        """
+        G = nx.path_graph(4)
+        d = nx.group_degree_centrality(G, [1])
+        d_answer = nx.degree_centrality(G)[1]
+        assert d == d_answer
+
+    def test_group_degree_centrality_multiple_node(self):
+        """
+        Group degree centrality for group with more than
+        1 node
+        """
+        G = nx.Graph()
+        G.add_nodes_from([1, 2, 3, 4, 5, 6, 7, 8])
+        G.add_edges_from(
+            [(1, 2), (1, 3), (1, 6), (1, 7), (1, 8), (2, 3), (2, 4), (2, 5)]
+        )
+        d = nx.group_degree_centrality(G, [1, 2])
+        d_answer = 1
+        assert d == d_answer
+
+    def test_group_in_degree_centrality(self):
+        """
+        Group in-degree centrality in a DiGraph
+        """
+        G = nx.DiGraph()
+        G.add_nodes_from([1, 2, 3, 4, 5, 6, 7, 8])
+        G.add_edges_from(
+            [(1, 2), (1, 3), (1, 6), (1, 7), (1, 8), (2, 3), (2, 4), (2, 5)]
+        )
+        d = nx.group_in_degree_centrality(G, [1, 2])
+        d_answer = 0
+        assert d == d_answer
+
+    def test_group_out_degree_centrality(self):
+        """
+        Group out-degree centrality in a DiGraph
+        """
+        G = nx.DiGraph()
+        G.add_nodes_from([1, 2, 3, 4, 5, 6, 7, 8])
+        G.add_edges_from(
+            [(1, 2), (1, 3), (1, 6), (1, 7), (1, 8), (2, 3), (2, 4), (2, 5)]
+        )
+        d = nx.group_out_degree_centrality(G, [1, 2])
+        d_answer = 1
+        assert d == d_answer
+
+    def test_group_degree_centrality_node_not_in_graph(self):
+        """
+        Node(s) in S not in graph, raises NetworkXError
+        """
+        with pytest.raises(nx.NetworkXError):
+            nx.group_degree_centrality(nx.path_graph(5), [6, 7, 8])

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/tests/test_laplacian_centrality.py

@@ -0,0 +1,221 @@
+import pytest
+
+import networkx as nx
+
+np = pytest.importorskip("numpy")
+sp = pytest.importorskip("scipy")
+
+
+def test_laplacian_centrality_null_graph():
+    G = nx.Graph()
+    with pytest.raises(nx.NetworkXPointlessConcept):
+        d = nx.laplacian_centrality(G, normalized=False)
+
+
+def test_laplacian_centrality_single_node():
+    """See gh-6571"""
+    G = nx.empty_graph(1)
+    assert nx.laplacian_centrality(G, normalized=False) == {0: 0}
+    with pytest.raises(ZeroDivisionError):
+        nx.laplacian_centrality(G, normalized=True)
+
+
+def test_laplacian_centrality_unconnected_nodes():
+    """laplacian_centrality on a unconnected node graph should return 0
+
+    For graphs without edges, the Laplacian energy is 0 and is unchanged with
+    node removal, so::
+
+        LC(v) = LE(G) - LE(G - v) = 0 - 0 = 0
+    """
+    G = nx.empty_graph(3)
+    assert nx.laplacian_centrality(G, normalized=False) == {0: 0, 1: 0, 2: 0}
+
+
+def test_laplacian_centrality_empty_graph():
+    G = nx.empty_graph(3)
+    with pytest.raises(ZeroDivisionError):
+        d = nx.laplacian_centrality(G, normalized=True)
+
+
+def test_laplacian_centrality_E():
+    E = nx.Graph()
+    E.add_weighted_edges_from(
+        [(0, 1, 4), (4, 5, 1), (0, 2, 2), (2, 1, 1), (1, 3, 2), (1, 4, 2)]
+    )
+    d = nx.laplacian_centrality(E)
+    exact = {
+        0: 0.700000,
+        1: 0.900000,
+        2: 0.280000,
+        3: 0.220000,
+        4: 0.260000,
+        5: 0.040000,
+    }
+
+    for n, dc in d.items():
+        assert exact[n] == pytest.approx(dc, abs=1e-7)
+
+    # Check not normalized
+    full_energy = 200
+    dnn = nx.laplacian_centrality(E, normalized=False)
+    for n, dc in dnn.items():
+        assert exact[n] * full_energy == pytest.approx(dc, abs=1e-7)
+
+    # Check unweighted not-normalized version
+    duw_nn = nx.laplacian_centrality(E, normalized=False, weight=None)
+    print(duw_nn)
+    exact_uw_nn = {
+        0: 18,
+        1: 34,
+        2: 18,
+        3: 10,
+        4: 16,
+        5: 6,
+    }
+    for n, dc in duw_nn.items():
+        assert exact_uw_nn[n] == pytest.approx(dc, abs=1e-7)
+
+    # Check unweighted version
+    duw = nx.laplacian_centrality(E, weight=None)
+    full_energy = 42
+    for n, dc in duw.items():
+        assert exact_uw_nn[n] / full_energy == pytest.approx(dc, abs=1e-7)
+
+
+def test_laplacian_centrality_KC():
+    KC = nx.karate_club_graph()
+    d = nx.laplacian_centrality(KC)
+    exact = {
+        0: 0.2543593,
+        1: 0.1724524,
+        2: 0.2166053,
+        3: 0.0964646,
+        4: 0.0350344,
+        5: 0.0571109,
+        6: 0.0540713,
+        7: 0.0788674,
+        8: 0.1222204,
+        9: 0.0217565,
+        10: 0.0308751,
+        11: 0.0215965,
+        12: 0.0174372,
+        13: 0.118861,
+        14: 0.0366341,
+        15: 0.0548712,
+        16: 0.0172772,
+        17: 0.0191969,
+        18: 0.0225564,
+        19: 0.0331147,
+        20: 0.0279955,
+        21: 0.0246361,
+        22: 0.0382339,
+        23: 0.1294193,
+        24: 0.0227164,
+        25: 0.0644697,
+        26: 0.0281555,
+        27: 0.075188,
+        28: 0.0364742,
+        29: 0.0707087,
+        30: 0.0708687,
+        31: 0.131019,
+        32: 0.2370821,
+        33: 0.3066709,
+    }
+    for n, dc in d.items():
+        assert exact[n] == pytest.approx(dc, abs=1e-7)
+
+    # Check not normalized
+    full_energy = 12502
+    dnn = nx.laplacian_centrality(KC, normalized=False)
+    for n, dc in dnn.items():
+        assert exact[n] * full_energy == pytest.approx(dc, abs=1e-3)
+
+
+def test_laplacian_centrality_K():
+    K = nx.krackhardt_kite_graph()
+    d = nx.laplacian_centrality(K)
+    exact = {
+        0: 0.3010753,
+        1: 0.3010753,
+        2: 0.2258065,
+        3: 0.483871,
+        4: 0.2258065,
+        5: 0.3870968,
+        6: 0.3870968,
+        7: 0.1935484,
+        8: 0.0752688,
+        9: 0.0322581,
+    }
+    for n, dc in d.items():
+        assert exact[n] == pytest.approx(dc, abs=1e-7)
+
+    # Check not normalized
+    full_energy = 186
+    dnn = nx.laplacian_centrality(K, normalized=False)
+    for n, dc in dnn.items():
+        assert exact[n] * full_energy == pytest.approx(dc, abs=1e-3)
+
+
+def test_laplacian_centrality_P3():
+    P3 = nx.path_graph(3)
+    d = nx.laplacian_centrality(P3)
+    exact = {0: 0.6, 1: 1.0, 2: 0.6}
+    for n, dc in d.items():
+        assert exact[n] == pytest.approx(dc, abs=1e-7)
+
+
+def test_laplacian_centrality_K5():
+    K5 = nx.complete_graph(5)
+    d = nx.laplacian_centrality(K5)
+    exact = {0: 0.52, 1: 0.52, 2: 0.52, 3: 0.52, 4: 0.52}
+    for n, dc in d.items():
+        assert exact[n] == pytest.approx(dc, abs=1e-7)
+
+
+def test_laplacian_centrality_FF():
+    FF = nx.florentine_families_graph()
+    d = nx.laplacian_centrality(FF)
+    exact = {
+        "Acciaiuoli": 0.0804598,
+        "Medici": 0.4022989,
+        "Castellani": 0.1724138,
+        "Peruzzi": 0.183908,
+        "Strozzi": 0.2528736,
+        "Barbadori": 0.137931,
+        "Ridolfi": 0.2183908,
+        "Tornabuoni": 0.2183908,
+        "Albizzi": 0.1954023,
+        "Salviati": 0.1149425,
+        "Pazzi": 0.0344828,
+        "Bischeri": 0.1954023,
+        "Guadagni": 0.2298851,
+        "Ginori": 0.045977,
+        "Lamberteschi": 0.0574713,
+    }
+    for n, dc in d.items():
+        assert exact[n] == pytest.approx(dc, abs=1e-7)
+
+
+def test_laplacian_centrality_DG():
+    DG = nx.DiGraph([(0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 6), (5, 7), (5, 8)])
+    d = nx.laplacian_centrality(DG)
+    exact = {
+        0: 0.2123352,
+        5: 0.515391,
+        1: 0.2123352,
+        2: 0.2123352,
+        3: 0.2123352,
+        4: 0.2123352,
+        6: 0.2952031,
+        7: 0.2952031,
+        8: 0.2952031,
+    }
+    for n, dc in d.items():
+        assert exact[n] == pytest.approx(dc, abs=1e-7)
+
+    # Check not normalized
+    full_energy = 9.50704
+    dnn = nx.laplacian_centrality(DG, normalized=False)
+    for n, dc in dnn.items():
+        assert exact[n] * full_energy == pytest.approx(dc, abs=1e-4)

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/tests/test_reaching.py

@@ -0,0 +1,117 @@
+"""Unit tests for the :mod:`networkx.algorithms.centrality.reaching` module."""
+import pytest
+
+import networkx as nx
+
+
+class TestGlobalReachingCentrality:
+    """Unit tests for the global reaching centrality function."""
+
+    def test_non_positive_weights(self):
+        with pytest.raises(nx.NetworkXError):
+            G = nx.DiGraph()
+            nx.global_reaching_centrality(G, weight="weight")
+
+    def test_negatively_weighted(self):
+        with pytest.raises(nx.NetworkXError):
+            G = nx.Graph()
+            G.add_weighted_edges_from([(0, 1, -2), (1, 2, +1)])
+            nx.global_reaching_centrality(G, weight="weight")
+
+    def test_directed_star(self):
+        G = nx.DiGraph()
+        G.add_weighted_edges_from([(1, 2, 0.5), (1, 3, 0.5)])
+        grc = nx.global_reaching_centrality
+        assert grc(G, normalized=False, weight="weight") == 0.5
+        assert grc(G) == 1
+
+    def test_undirected_unweighted_star(self):
+        G = nx.star_graph(2)
+        grc = nx.global_reaching_centrality
+        assert grc(G, normalized=False, weight=None) == 0.25
+
+    def test_undirected_weighted_star(self):
+        G = nx.Graph()
+        G.add_weighted_edges_from([(1, 2, 1), (1, 3, 2)])
+        grc = nx.global_reaching_centrality
+        assert grc(G, normalized=False, weight="weight") == 0.375
+
+    def test_cycle_directed_unweighted(self):
+        G = nx.DiGraph()
+        G.add_edge(1, 2)
+        G.add_edge(2, 1)
+        assert nx.global_reaching_centrality(G, weight=None) == 0
+
+    def test_cycle_undirected_unweighted(self):
+        G = nx.Graph()
+        G.add_edge(1, 2)
+        assert nx.global_reaching_centrality(G, weight=None) == 0
+
+    def test_cycle_directed_weighted(self):
+        G = nx.DiGraph()
+        G.add_weighted_edges_from([(1, 2, 1), (2, 1, 1)])
+        assert nx.global_reaching_centrality(G) == 0
+
+    def test_cycle_undirected_weighted(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=1)
+        grc = nx.global_reaching_centrality
+        assert grc(G, normalized=False) == 0
+
+    def test_directed_weighted(self):
+        G = nx.DiGraph()
+        G.add_edge("A", "B", weight=5)
+        G.add_edge("B", "C", weight=1)
+        G.add_edge("B", "D", weight=0.25)
+        G.add_edge("D", "E", weight=1)
+
+        denom = len(G) - 1
+        A_local = sum([5, 3, 2.625, 2.0833333333333]) / denom
+        B_local = sum([1, 0.25, 0.625]) / denom
+        C_local = 0
+        D_local = sum([1]) / denom
+        E_local = 0
+
+        local_reach_ctrs = [A_local, C_local, B_local, D_local, E_local]
+        max_local = max(local_reach_ctrs)
+        expected = sum(max_local - lrc for lrc in local_reach_ctrs) / denom
+        grc = nx.global_reaching_centrality
+        actual = grc(G, normalized=False, weight="weight")
+        assert expected == pytest.approx(actual, abs=1e-7)
+
+
+class TestLocalReachingCentrality:
+    """Unit tests for the local reaching centrality function."""
+
+    def test_non_positive_weights(self):
+        with pytest.raises(nx.NetworkXError):
+            G = nx.DiGraph()
+            G.add_weighted_edges_from([(0, 1, 0)])
+            nx.local_reaching_centrality(G, 0, weight="weight")
+
+    def test_negatively_weighted(self):
+        with pytest.raises(nx.NetworkXError):
+            G = nx.Graph()
+            G.add_weighted_edges_from([(0, 1, -2), (1, 2, +1)])
+            nx.local_reaching_centrality(G, 0, weight="weight")
+
+    def test_undirected_unweighted_star(self):
+        G = nx.star_graph(2)
+        grc = nx.local_reaching_centrality
+        assert grc(G, 1, weight=None, normalized=False) == 0.75
+
+    def test_undirected_weighted_star(self):
+        G = nx.Graph()
+        G.add_weighted_edges_from([(1, 2, 1), (1, 3, 2)])
+        centrality = nx.local_reaching_centrality(
+            G, 1, normalized=False, weight="weight"
+        )
+        assert centrality == 1.5
+
+    def test_undirected_weighted_normalized(self):
+        G = nx.Graph()
+        G.add_weighted_edges_from([(1, 2, 1), (1, 3, 2)])
+        centrality = nx.local_reaching_centrality(
+            G, 1, normalized=True, weight="weight"
+        )
+        assert centrality == 1.0

venv/lib/python3.10/site-packages/networkx/algorithms/centrality/tests/test_second_order_centrality.py

@@ -0,0 +1,82 @@
+"""
+Tests for second order centrality.
+"""
+
+import pytest
+
+pytest.importorskip("numpy")
+pytest.importorskip("scipy")
+
+import networkx as nx
+
+
+def test_empty():
+    with pytest.raises(nx.NetworkXException):
+        G = nx.empty_graph()
+        nx.second_order_centrality(G)
+
+
+def test_non_connected():
+    with pytest.raises(nx.NetworkXException):
+        G = nx.Graph()
+        G.add_node(0)
+        G.add_node(1)
+        nx.second_order_centrality(G)
+
+
+def test_non_negative_edge_weights():
+    with pytest.raises(nx.NetworkXException):
+        G = nx.path_graph(2)
+        G.add_edge(0, 1, weight=-1)
+        nx.second_order_centrality(G)
+
+
+def test_weight_attribute():
+    G = nx.Graph()
+    G.add_weighted_edges_from([(0, 1, 1.0), (1, 2, 3.5)], weight="w")
+    expected = {0: 3.431, 1: 3.082, 2: 5.612}
+    b = nx.second_order_centrality(G, weight="w")
+
+    for n in sorted(G):
+        assert b[n] == pytest.approx(expected[n], abs=1e-2)
+
+
+def test_one_node_graph():
+    """Second order centrality: single node"""
+    G = nx.Graph()
+    G.add_node(0)
+    G.add_edge(0, 0)
+    assert nx.second_order_centrality(G)[0] == 0
+
+
+def test_P3():
+    """Second order centrality: line graph, as defined in paper"""
+    G = nx.path_graph(3)
+    b_answer = {0: 3.741, 1: 1.414, 2: 3.741}
+
+    b = nx.second_order_centrality(G)
+
+    for n in sorted(G):
+        assert b[n] == pytest.approx(b_answer[n], abs=1e-2)
+
+
+def test_K3():
+    """Second order centrality: complete graph, as defined in paper"""
+    G = nx.complete_graph(3)
+    b_answer = {0: 1.414, 1: 1.414, 2: 1.414}
+
+    b = nx.second_order_centrality(G)
+
+    for n in sorted(G):
+        assert b[n] == pytest.approx(b_answer[n], abs=1e-2)
+
+
+def test_ring_graph():
+    """Second order centrality: ring graph, as defined in paper"""
+    G = nx.cycle_graph(5)
+    b_answer = {0: 4.472, 1: 4.472, 2: 4.472, 3: 4.472, 4: 4.472}
+
+    b = nx.second_order_centrality(G)
+
+    for n in sorted(G):
+        assert b[n] == pytest.approx(b_answer[n], abs=1e-2)