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+{"text":"\\section{Introduction}\n\nThe study of electronic transport in quantum confined model systems like \nquantum rings, quantum dots, arrays of quantum dots, quantum dots embedded \nin a quantum ring, etc., has become one of the most fascinating branch of\nnanoscience and technology. With the aid of present nanotechnological\nprogress~\\cite{nanofab1,nanofab2}, these simple looking quantum confined \nsystems can be used to design nanodevices especially in electronic as \nwell as spintronic engineering. The key idea of \nmanufacturing nanodevices is based on the concept of quantum \ninterference effect~\\cite{imry1,imry2,gefen1}, and it is generally \npreserved throughout the sample having dimension smaller or comparable \nto the phase coherence length. Therefore, ring type conductors or two \npath devices are ideal candidates to exploit the effect of quantum \ninterference~\\cite{bohm}. In a ring shaped geometry, quantum \ninterference effect can be controlled by several ways, and most \nprobably, the effect can be regulated significantly by tuning the \nmagnetic flux, the so-called Aharonov-Bohm (AB) flux~\\cite{chang,\nhan,he,wel}, that threads the ring. This key feature motivates us to \nwidely use quantum interference devices in mesoscopic solid-state \nelectronic circuits~\\cite{roh}. Using a mesoscopic ring we can \nconstruct a quantum interferometer, and here we will show that the \ninterferometer \nexhibits several exotic features of electron transport which can be \nutilized in designing nanoelectronic circuits. To reveal the phenomena, \nwe make a bridge system, by inserting the interferometer between two \nelectrodes (source and drain), the so-called source-interferometer-drain \nbridge. Following the pioneering work of Aviram and Ratner~\\cite{aviram},\ntheoretical description of electron transport in a bridge system has got \nmuch progress. Later, many excellent experiments~\\cite{holl,\nkob,kob1,ji,yac,reed1,reed2} have been done in several bridge systems \nto understand the basic mechanisms underlying the electron transport. \nThough extensive studies on electron transport have already been done \nboth theoretically~\\cite{chang,han,he,wel,orella1,orella2,nitzan1,\nnitzan2,bai,muj1,muj2,cui,baer2,baer3,tagami,walc1,baer1,maiti1,\nmaiti2,maiti3,walc2,gefen,kubo,naka,aharony,konig,ern2} as well as \nexperimentally~\\cite{holl,kob,kob1,ji,yac,reed1,reed2}, yet lot of \ncontroversies are still present between the theory and experiment, and \nthe complete description of the conduction mechanism in this scale is \nnot very well defined even today. Several controlling \nfactors are there which can significantly regulate electron transport in \na conducting bridge, and all these factors have to be taken into account \nproperly to understand the transport mechanism. For our illustrative \npurposes, here we mention some of these issues.\n\\begin{enumerate}\n\\item The quantum interference effect~\\cite{baer2,baer3,tagami,walc1,\nbaer1,maiti1,maiti2} of electronic waves passing through different \narms of the bridging material becomes the most significant issue. \n\\item The coupling of the bridging material to the electrodes significantly\ncontrols the current amplitude across any bridge system~\\cite{baer1}. \n\\item Dynamical fluctuation in small-scale devices is another important \nfactor which plays an active role and can be manifested through the \nmeasurement of {\\em shot noise}~\\cite{walc2}, a direct consequence of \nthe quantization of charge.\n\\item Geometry of the conducting material between the two electrodes\nitself is an important issue to control electron transmission which\nhas been described quite elaborately by Ernzerhof \n{\\em et al.}~\\cite{ern2} through some model calculations. \n\\end{enumerate}\n\nAddition to these, several other factors of the tight-binding \nHamiltonian that describe a system also provide important effects in \nthe determination of current across a bridge system.\n\nIn this presentation we explore electron transport properties of a \nquantum interferometer based on the single particle Green's function \nformalism. The interferometer is sandwiched between two semi-infinite\none-dimensional ($1$D) metallic electrodes, viz, source and drain, and, \ntwo sub-rings of the interferometer are subject to the Aharonov-Bohm (AB) \nfluxes $\\phi_1$ and $\\phi_2$, respectively. The schematic view of the\nbridge system is depicted in Fig.~\\ref{ring}. A simple tight-binding model \nis used to describe the system and all the calculations are done \nnumerically, which illustrate conductance-energy and current-voltage \ncharacteristics as functions of the interferometer-to-electrode coupling \nstrength, magnetic fluxes and the difference of these two fluxes. \nSeveral exotic features are observed from this study. These are: (i) \nsemiconducting or metallic nature depending on the coupling strength of \nthe interferometer to the side attached electrodes, (ii) appearance of \nanti-resonant states~\\cite{chang,han,he} and (iii) unconventional \nperiodic behavior of typical conductance\/current as a function of \nthe difference of two AB fluxes.\n\nThe scheme of the paper is as follows. Following the introduction (Section \n$1$), in Section $2$, we describe the model and theoretical formulation \nfor the calculation. Section $3$ explores the results, and finally, we \nconclude our study in Section $4$.\n\n\\section{Model and synopsis of the theoretical background}\n\nLet us begin with the model presented in Fig.~\\ref{ring}. A quantum\ninterferometer with four atomic sites ($N=4$, where $N$ gives the total\nnumber of atomic sites in the interferometer) is attached symmetrically \nto two semi-infinite one-dimensional ($1$D) metallic electrodes, namely,\nsource and drain. The atomic sites $2$ and $3$ of the interferometer are \ndirectly coupled to each other, and accordingly, two sub-rings, left and \nright, are formed. These two sub-rings are subject to the AB fluxes \n$\\phi_1$ and $\\phi_2$, respectively.\n\nConsidering linear transport regime, conductance $g$ of the interferometer \ncan be obtained using the Landauer conductance formula~\\cite{land1,land2,\nland3,datta,marc},\n\\begin{equation}\ng=\\frac{2e^2}{h} T\n\\label{equ1}\n\\end{equation}\nwhere, $T$ becomes the transmission probability of an electron across \nthe interferometer. It $(T)$ can be expressed in terms of the Green's \nfunction of the interferometer and its coupling to the side-attached \nelectrodes by the relation~\\cite{datta,marc},\n\\begin{equation}\nT={\\mbox{Tr}}\\left[\\Gamma_S G_{I}^r \\Gamma_D G_{I}^a\\right]\n\\label{equ2}\n\\end{equation}\nwhere, $G_{I}^r$ and $G_{I}^a$ are respectively the retarded and advanced \nGreen's functions of the interferometer including the effects of the \nelectrodes. Here $\\Gamma_S$ and $\\Gamma_D$ describe the coupling of the \n\\begin{figure}[ht]\n{\\centering \\resizebox*{6.5cm}{3.5cm}{\\includegraphics{ring.eps}}\\par}\n\\caption{(Color online). Schematic view of a quantum interferometer\nwith four atomic sites ($N=4$) attached to two semi-infinite \none-dimensional metallic electrodes.}\n\\label{ring}\n\\end{figure}\ninterferometer to the source and drain, respectively. For the complete \nsystem i.e., the interferometer, source and drain, Green's function \nis defined as,\n\\begin{equation}\nG=\\left(E-H\\right)^{-1}\n\\label{equ3}\n\\end{equation}\nwhere, $E$ is the injecting energy of the source electron. To Evaluate\nthis Green's function, inversion of an infinite matrix is needed since\nthe complete system consists of the finite size interferometer and \ntwo semi-infinite electrodes. However, the entire system can be \npartitioned into sub-matrices corresponding to the individual sub-systems \nand the Green's function for the interferometer can be effectively \nwritten as,\n\\begin{equation}\nG_{I}=\\left(E-H_{I}-\\Sigma_S-\\Sigma_D\\right)^{-1}\n\\label{equ4}\n\\end{equation}\nwhere, $H_{I}$ is the Hamiltonian of the interferometer that can be \nexpressed within the non-interacting picture like,\n\\begin{equation}\nH_{I} = \\sum_i \\epsilon_i c_i^{\\dagger} c_i + \\sum_{<ij>} t_{ij} \n\\left(c_i^{\\dagger} c_j e^{i\\theta_{ij}}+ c_j^{\\dagger} c_i \ne^{-i\\theta_{ij}}\\right)\n\\label{equ5}\n\\end{equation}\nIn this Hamiltonian $\\epsilon_i$ gives the on-site energy for the atomic \nsite $i$, where $i$ runs from $1$ to $4$, $c_i^{\\dagger}$ ($c_i$) is the \ncreation (annihilation) operator of an electron at the site $i$ and \n$t_{ij}$ is the hopping integral between the nearest-neighbor sites \n$i$ and $j$. For the sake of simplicity, we assume that the magnitudes of\nall hopping integrals ($t_{ij}$) are identical to $t$. The phase factor \n$\\theta_{ij}$, associated with the hopping integral $t_{ij}$, comes due to \nthe fluxes $\\phi_1$ and $\\phi_2$ in the two sub-rings. The phase factors \n($\\theta_{ij}$) are chosen as, $\\theta_{12}=\\theta_{23} =\\theta_{34}=\n\\theta_{41}=2\\pi\\phi\/4\\phi_0$, $\\theta_{24}=2\\pi \\Delta \\phi\/2 \\phi_0$, \nwhere $\\phi=\\phi_1+\\phi_2$, $\\Delta \\phi=\\phi_1-\\phi_2$ and $\\phi_0=ch\/e$ \nis the elementary flux-quantum. Accordingly, a minus sign is used for \nthe phases when the electron hops in the reverse direction. For the two \n$1$D electrodes, a similar kind of tight-binding Hamiltonian is also used, \nexcept any phase factor, where the Hamiltonian is parametrized by constant \non-site potential $\\epsilon_0$ and nearest-neighbor hopping integral $t_0$. \nThe hopping integral between the source and interferometer is $\\tau_S$, \nwhile it is $\\tau_D$ between the interferometer and drain. The parameters \n$\\Sigma_S$ and $\\Sigma_D$ in Eq.~(\\ref{equ4}) represent the self-energies \ndue to the coupling of the interferometer to the source and drain, \nrespectively, where all the information of this coupling are included into \nthese self-energies~\\cite{datta}.\n\nThe current passing through the interferometer is depicted as a \nsingle-electron scattering process between the two reservoirs of charge \ncarriers. The current $I$ can be computed as a function of the applied \nbias voltage $V$ by the expression~\\cite{datta},\n\\begin{equation}\nI(V)=\\frac{e}{\\pi \\hbar}\\int \\limits_{E_F-eV\/2}^{E_F+eV\/2} T(E)~ dE\n\\label{equ8}\n\\end{equation}\nwhere $E_F$ is the equilibrium Fermi energy. Here we assume that the \nentire voltage is dropped across the interferometer-electrode interfaces, \nand it is examined that under such an assumption the $I$-$V$ characteristics \ndo not change their qualitative features. \n\nAll the results in this communication are determined at absolute zero \ntemperature, but they should valid even for some finite (low) temperatures, \nsince the broadening of the energy levels of the interferometer due to \nits coupling to the electrodes becomes much larger than that of the thermal \nbroadening~\\cite{datta}. On the other hand, at high temperature limit, all \nthese phenomena completely disappear. This is due to the fact that the \nphase coherence length decreases significantly with the rise of temperature \nwhere the contribution comes mainly from the scattering on phonons, and \naccordingly, the quantum interference effect vanishes. Our unit system\nis simplified by choosing $c=e=h=1$. \n\n\\section{Numerical results and discussion}\n\nBefore going into the discussion, let us first assign the values of\ndifferent parameters those are used for our numerical calculation. The \non-site energy $\\epsilon_i$ of the interferometer is taken as $0$ for \nall the four sites $i$, and the nearest-neighbor hopping strength $t$ \nis set to $3$. On the other hand, for two side attached $1$D electrodes \nthe on-site energy ($\\epsilon_0$) and nearest-neighbor hopping strength \n($t_0$) are fixed to $0$ and $4$, respectively. The equilibrium Fermi \nenergy $E_F$ is set to $0$. \n\nThroughout the analysis we present the basic features of electron\ntransport for two distinct regimes of electrode-to-interferometer \ncoupling.\n\n\\noindent\n\\underline{Case 1:} Weak-coupling limit\n\\vskip 0.1cm\n\\noindent\nThis limit is set by the criterion $\\tau_{S(D)} << t$. In this case,\nwe choose the values as $\\tau_S=\\tau_D=0.5$.\n\n\\noindent\n\\underline{Case 2:} Strong-coupling limit\n\\vskip 0.1cm\n\\noindent\nThis limit is described by the condition $\\tau_{S(D)} \\sim t$. In this\nregime we choose the values of hopping strengths as $\\tau_S=\\tau_D=2.5$.\n\n\\subsection{Interferometric geometry with $4$ atomic ($N=4$) sites}\n\n\\subsubsection{Conductance-energy characteristics}\n\nIn Fig.~\\ref{cond}, we plot conductance $g$ as a function of the \ninjecting electron energy $E$ for the interferometer considering $\\phi=1$, \nwhere (a), (b), (c) and (d) correspond to $\\Delta \\phi=0.2$, $0.4$, $0.6$ \n\\begin{figure}[ht]\n{\\centering \\resizebox*{8cm}{12cm}{\\includegraphics{cond.eps}}\\par}\n\\caption{(Color online). $g$-$E$ curves in the weak- (black) and\nstrong-coupling (red) limits for the interferometer with four\natomic sites ($N=4$) considering $\\phi=1$. (a) $\\Delta \\phi=0.2$, \n(b) $\\Delta \\phi=0.4$, (c) $\\Delta \\phi=0.6$ and (d) $\\Delta \\phi=0.8$.}\n\\label{cond}\n\\end{figure}\nand $0.8$, respectively. The black curves represent the results for the \nweak-coupling limit, while the results for the strong-coupling limit are \nshown by the red curves. In the limit of weak-coupling, conductance shows \nfine resonant peaks for some particular energies, while it ($g$) drops to\nzero almost for all other energies. At these resonances, conductance \nreaches the value $2$, and therefore, the transmission probability $T$ \nbecomes unity, since the relation $g=2T$ is satisfied from the Landauer \nconductance formula (see Eq.~(\\ref{equ1}) with $e=h=1$). The transmission \nprobability of getting an electron across the interferometer significantly \ndepends on the quantum interference of electronic waves passing \nthrough the different arms of the interferometer, and accordingly, the\nprobability amplitude becomes strengthened or weakened. Now all the\nresonant peaks in the conductance spectra are associated with the \nenergy eigenvalues of the interferometer, and thus it is emphasized \nthat the conductance spectrum reveals itself the electronic structure of \nthe interferometer. The situation becomes quite interesting as long as \nthe coupling strength of the interferometer to the electrodes is\nincreased from the weak regime to the strong one. In the strong-coupling\nlimit, all the resonances get substantial widths compared to the \nweak-coupling limit. The contribution for the broadening of the resonant \npeaks in this strong-coupling limit appears from the imaginary parts of \nthe self-energies $\\Sigma_S$ and $\\Sigma_D$, respectively~\\cite{datta}. \nHence, by tuning the coupling strength from the weak to strong regime,\nelectronic transmission across the interferometer can be obtained for\nthe wider range of energies, while a fine scan in the energy scale is\nneeded to get the electron conduction across the bridge in the limit\nof weak-coupling. These results provide an important signature in the \nstudy of current-voltage ($I$-$V$) characteristics. Another interesting\nfeature observed from the conductance spectra is the existence of the\nanti-resonant states. The positions of the anti-resonance states can be\nclearly noticed from the red curves, compared to the black curves since\nthe widths of these curves are too small, where they sharply drop to zero \nfor the respective energy values associated with the different values of \n$\\Delta \\phi$ (see Figs.~\\ref{cond}(a)-(d)). Such anti-resonant states \nare specific to the interferometric nature of the scattering and do not \noccur in conventional one-dimensional scattering problems of potential \nbarriers~\\cite{chang,han,he}. A clear investigation shows that the \npositions of the anti-resonances on the energy scale are independent \nof the interferometer-to-electrode coupling strength. Since the width \nof these anti-resonance states are too small, they do not provide any \nsignificant contribution in the current-voltage ($I$-$V$) characteristics. \n\n\\subsubsection{Typical conductance $g_{typ}$ as a function of \n$\\Delta \\phi$}\n\nThe effect of $\\Delta \\phi$, the difference between two AB fluxes $\\phi_1$\n\\begin{figure}[ht]\n{\\centering \\resizebox*{8cm}{11cm}{\\includegraphics{cond1.eps}}\\par}\n\\caption{(Color online). $g_{typ}$-$\\Delta \\phi$ curves in the\nstrong-coupling limit for the interferometer with four atomic sites\n($N=4$), where (a) $\\phi=0.2$, (b) $\\phi=0.4$, (c) $\\phi=0.6$ and \n(d) $\\phi=0.8$. The typical conductances are calculated at the energy \n$E=5$.}\n\\label{cond1}\n\\end{figure}\nand $\\phi_2$, on the electron transport through the interferometer is\nalso an important issue in the present context. To visualize it, in\nFig.~\\ref{cond1}, we display the variation of the typical conductance\n($g_{typ}$) as a function of $\\Delta \\phi$ for the interferometer in\nthe limit of strong-coupling. Figures~\\ref{cond1}(a), (b), (c) and (d) \ncorrespond to the results for $\\phi=0.2$, $0.4$, $0.6$ and $0.8$, \nrespectively. The typical conductances are calculated for the fixed\nenergy $E=5$. Very interestingly we observe that, for a fixed value of\n$\\phi$, typical conductance varies periodically with $\\Delta \\phi$\nshowing $2\\phi_0$ ($=2$, since $\\phi_0=1$ in our chosen unit) \nflux-quantum periodicity, associated with the number of atomic sites ($2$) \nin the vertical line connecting two sub-rings of the quantum interferometer. \nThis period doubling behavior is completely different from the traditional \nperiodic nature, since in conventional geometries we get simple $\\phi_0$ \nflux-quantum periodicity. In the limit of weak-coupling we will also get \nthe similar behavior of periodicity ($2\\phi_0$) for the typical \nconductance with $\\Delta \\phi$, and due to the obvious reason we do not \nplot the results for this coupling limit once gain. \n\n\\subsubsection{Current-voltage characteristics}\n\nAll these features of electron transfer become much more clearly visible\nby studying the current-voltage ($I$-$V$) characteristics. The current \n$I$ passing through the interferometer is computed from the integration \nprocedure of the transmission function $T$ as prescribed in \nEq.~(\\ref{equ8}) which is not restricted in the linear response regime,\nbut it is of great significance in determining the shape of the full\ncurrent-voltage characteristics. As illustrative examples, in \nFig.~\\ref{current}, we plot the current-voltage characteristics of \nthe interferometer for the three different values of $\\phi_2$, keeping \nthe flux $\\phi_1$ in the left sub-ring to a fixed value $0.2$. The red, \nblue and black\ncurves correspond to $\\phi_2=0$, $0.1$ and $0.4$, respectively. In the\nlimit of weak-coupling (see Fig.~\\ref{current}(a)), it is observed that\nthe current exhibits staircase-like structure with fine steps as a\nfunction of the applied bias voltage $V$. This is due to the existence of\nthe sharp resonant peaks in the conductance spectrum in this coupling\nlimit, since the current is computed by the integration method of the\ntransmission function $T$. With the increase of the bias voltage $V$,\nthe electrochemical potentials on the electrodes are shifted gradually,\nand finally cross one of the quantized energy levels of the interferometer.\nAccordingly, a current channel is opened up which provides a jump in the \n\\begin{figure}[ht]\n{\\centering \\resizebox*{8cm}{10cm}{\\includegraphics{current.eps}}\\par}\n\\caption{(Color online). $I$-$V$ characteristics of the interferometer\nwith four atomic sites ($N=4$) for a fixed value of $\\phi_1=0.2$, where \nthe red, blue and black curves correspond to $\\phi_2=0$, $0.1$ and $0.4$,\nrespectively. (a) Weak-coupling limit and (b) strong-coupling limit.}\n\\label{current}\n\\end{figure}\n$I$-$V$ characteristic curve. The most important feature observed from \nthe $I$-$V$ curves for this weak-coupling limit is that, the non-zero\nvalue of the current appears beyond a finite bias voltage, the so-called\nthreshold voltage $V_{th}$. This is quite analogous to the \nsemiconducting nature of a material. Most interestingly, the results \npredict that the threshold bias voltage of electron conduction can be \ncontrolled very nicely by tuning the AB flux $\\phi_2$. The situation \nbecomes much different for the strong-coupling case. The results are \ngiven in Fig.~\\ref{current}(b). In this limit, the current varies almost \ncontinuously with the applied bias voltage and achieves much larger \namplitude than the weak-coupling case. The reason is that, in the limit \nof strong-coupling all the energy levels get broadened which provide \nlarger current in the integration procedure of the transmission function \n$T$. Thus by tuning the strength of the interferometer-to-electrode \ncoupling, we can achieve very large, even an order of magnitude, current \namplitude from the very low one for the same bias voltage $V$, which\nprovides an important signature in designing nanoelectronic devices.\nIn contrary to the weak-coupling limit, here the electron starts to\nconduct as long as the bias voltage is given i.e., $V_{th}\\rightarrow 0$,\nwhich reveals the metallic nature. Thus it can be emphasized that the\ninterferometer-to-electrode coupling is a key parameter which controls \nthe electron transport in a meaningful way. Additionally, the existence \nof the semiconducting or the metallic behavior of the interferometer \nalso significantly depends on the AB fluxes $\\phi_1$ and $\\phi_2$. The \nnature of all these $I$-$V$ curves, presented in Fig.~\\ref{current}, will \nbe exactly similar if we plot the results for the different values of \n$\\phi_1$, keeping $\\phi_2$ as a constant. \n\n\\subsubsection{Typical current amplitude $I_{typ}$ as a function of\n$\\phi_2$}\n\nNow, we draw our attention on the variation of the typical current\namplitude with anyone of these two fluxes, when the other one is fixed. \nTo explore it, in Fig.~\\ref{typcurr}, we show the variation of the typical \ncurrent amplitude ($I_{typ}$) with $\\phi_2$, considering $\\phi_1$ as a \nconstant, where (a) and (b) correspond to $\\phi_1=0$ and $0.3$, respectively. \nThe black and red lines represent the results for the weak- and \nstrong-coupling limits, respectively. The typical current amplitudes are\ncalculated for the fixed bias voltage $V=1.02$. Both for these two limiting \ncases, the typical current amplitude varies periodically with $\\phi_2$, \nexhibiting $\\phi_0$ flux-quantum periodicity, as expected. Similar feature \nis also observed for the $I_{typ}$ vs $\\phi_1$ curves, when $\\phi_2$ becomes \nconstant. Here it is also important to note that the variation of $I_{typ}$\nwith $\\Delta \\phi$ is quite similar to that as presented in Fig.~\\ref{cond1}.\nThe typical current amplitude varies periodically with $\\Delta \\phi$\n\\begin{figure}[ht]\n{\\centering \\resizebox*{8cm}{10cm}{\\includegraphics{typcurr.eps}}\\par}\n\\caption{(Color online). $I_{typ}$-$\\phi_2$ curves in the weak- (black)\nand strong-coupling (red) limits for the interferometer with four\natomic sites ($N=4$), where (a) $\\phi_1=0$ and (b) $\\phi_1=0.3$. The \ntypical current amplitudes are calculated at the bias voltage $V=1.02$.}\n\\label{typcurr}\n\\end{figure}\nshowing $2\\phi_0$ flux-quantum periodicity, following the \n$g_{typ}$-$\\Delta \\phi$ characteristics.\n\nWith the above description of electron transport for a $4$-site ($N=4$)\nquantum interferometer, now we can extend our discussion for an\ninterferometer with higher number of atomic sites i.e., $N>4$.\n\n\\subsection{Interferometric geometry with $N$ atomic ($N>4$) sites}\n\nTo get an experimentally realizable system, here we \nconsider a quantum interferometer with large number of atomic sites \ncompared to our presented mathematical model with $4$ atomic sites.\nThe schematic view of such a quantum interferometer is given in \nFig.~\\ref{ring1}, where we set $N=15$.\nThe vertical line connecting left and right sub-rings contains $5$\n\\begin{figure}[ht]\n{\\centering \\resizebox*{8cm}{3.75cm}{\\includegraphics{ring1.eps}}\\par}\n\\caption{(Color online). Schematic view of a quantum interferometer \nwith $15$ atomic sites ($N=15$) attached to two semi-infinite \none-dimensional metallic electrodes.}\n\\label{ring1}\n\\end{figure}\natomic sites, where the individual sub-rings are penetrated by AB\nfluxes $\\phi_1$ and $\\phi_2$, respectively. In this interferometric \ngeometry, the phase factors ($\\theta_{ij}$'s) are chosen according\n\\begin{figure}[ht]\n{\\centering \\resizebox*{8cm}{10cm}{\\includegraphics{cond2.eps}}\\par}\n\\caption{(Color online). $g_{typ}$-$\\Delta \\phi$ curves in the \nstrong-coupling limit for the interferometer with $15$ atomic sites\n($N=15$), where (a) $\\phi=0.4$ and (b) $\\phi=0.8$. The typical \nconductances are calculated at the energy $E=1.5$.}\n\\label{cond2}\n\\end{figure}\nto our earlier prescription. Along the circumference of the ring\n$\\theta_{ij}=2\\pi \\phi\/12 \\phi_0$ and along the vertical line\n$\\theta_{ij}=2\\pi\\Delta \\phi\/5\\phi_0$, where $\\phi$ and $\\Delta \\phi$\ncorrespond to the identical meaning as before.\n\nFor this bigger quantum interferometer ($N=15$), exactly \nsimilar features of conductance-energy and current-voltage characteristics \nare observed as we see in the case of a $4$-site interferometer. Also, \ntypical current amplitude $I_{typ}$ shows identical variation with \n$\\phi_2$ to our previous study. Only the typical conductance $g_{typ}$ \nvaries in a different way as a function of $\\Delta \\phi$. As illustrative \nexamples in Fig.~\\ref{cond2} we plot $g_{typ}$-$\\Delta \\phi$ \ncharacteristics for the quantum interferometer with $N=15$ in the limit \nof strong-coupling, where (a) and (b) correspond to $\\phi=0.4$ and $0.8$, \nrespectively. The typical conductances are determined at the energy \n$E=1.5$. From the spectra we notice that for a fixed value of $\\phi$,\ntypical conductance oscillates as a function of $\\Delta \\phi$ exhibiting\n$5 \\phi_0$ flux-quantum periodicity. This phenomenon is completely \ndifferent from the traditional periodic nature. Comparing the results\npresented in Figs.~\\ref{cond1} and \\ref{cond2} it is manifested that\nthe periodicity of $g_{typ}$-$\\Delta \\phi$ curves depends on the total\nnumber of atomic sites in the vertical line connecting left and right \nsub-rings of a quantum interferometer. Therefore, changing the length\nof the vertical line, periodicity can be changed accordingly.\n\n\\section{Closing remarks}\n\nTo summarize, we have explored electron transport properties through a \nquantum interferometer using the single particle Green's function \nformalism. We have adopted a simple tight-binding framework to illustrate \nthe bridge system, where the interferometer is sandwiched between two\nelectrodes, viz, source and drain. We have done exact numerical calculation \nto study conductance-energy and current-voltage characteristics as functions \nof the interferometer-to-electrode coupling strength, magnetic fluxes \n$\\phi_1$ and $\\phi_2$ penetrated by left and right sub-rings of the \ninterferometer and the difference of these two fluxes. Several key features \nof electron transport have been observed those may be useful in \nmanufacturing nanoelectronic devices. The most exotic features are: (i) \nexistence of semiconducting or metallic behavior, depending on the \ninterferometer-to-electrode coupling strength, (ii) appearance of the \nanti-resonant states and (iii) unconventional periodic behavior of the \ntypical conductance\/current as a function of the difference of two AB \nfluxes.\n\nThroughout our work, we have addressed the essential \nfeatures of electron transport through a quantum interferometer with \ntotal number of atomic sites $N=4$. Next, we have extended our discussion\nfor an interferometer with higher number of atomic sites where we set \n$N=15$ to achieve an experimentally realizable system. In our model \ncalculations, these typical numbers ($N=4$ and $15$) are chosen only \nfor the sake of simplicity. Though the results presented here change \nnumerically with ring size ($N$), but all the basic features remain \nexactly invariant. To be more specific, it is important to note that, \nin real situation experimentally achievable rings have typical diameters \nwithin the range $0.4$-$0.6$ $\\mu$m. In such a small ring, very high \nmagnetic fields are required to produce a quantum flux. To overcome \nthis situation, Hod {\\em et al.} have studied extensively and proposed \nhow to construct nanometer scale devices, based on Aharonov-Bohm \ninterferometry, those can be operated in moderate magnetic \nfields~\\cite{hod1,hod2,hod3,hod4,hod5}.\n\nThis is our first step to describe the electron transport in a quantum\ninterferometer. Here we have made several realistic approximations by\nignoring the effects of electron-electron correlation, electron-phonon \ninteraction, disorder, temperature, etc. Over the last few many years \npeople have studied a lot to incorporate the effect of electron-electron \ncorrelation in the study of electron transport, yet no such proper theory \nhas been well established. Thus the inclusion of electron-electron \ncorrelation in the present model is a major challenge to us. The presence \nof electron-phonon interaction in\nAharonov-Bohm interferometers provides phase shifts of the conducting\nelectrons and due to this dephasing process electron transport through an \nAB interferometer becomes highly sensitive to the AB flux $\\phi$ with the \nincrease of electron-phonon coupling strength~\\cite{hod6}. In the present \nwork, we have addressed our results considering the site energies of all \nthe atomic sites of the interferometer are identical i.e., we have treated \nthe ordered system. But in real case, the presence of impurities will \naffect the electronic structure and hence the transport properties.\nThe effect of the temperature has already been pointed out earlier, and, \nit has been examined that the presented results will not change \nsignificantly even at finite temperature, since the broadening of the \nenergy levels of the interferometer due to its coupling to the electrodes \nwill be much larger than that of the thermal broadening~\\cite{datta}.\nAt the end, we would like to mention that we need further study in such \nsystems by incorporating all these effects.\n\nThe importance of this article is mainly concerned with (i) the simplicity \nof the geometry and (ii) the smallness of the size, and our exact analysis\nmay be utilized to study electron transport in Aharonov-Bohm geometries.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\chapter{Higher-order thinking in network analysis}\n\\label{ch:intro}\n\n\\newcommand*{introduction}{introduction}\n\nA network is a model for a system of connections between things.  The core of a\nnetwork model is a graph, a mathematical object consisting of nodes (modeling\nthe things) and edges (modeling the connections between things).  This\nthesis provides new graph theoretic methods to analyze datasets associated with\nnetwork models.  Such network datasets show up in a wide variety of scientific\ndisciplines, and this thesis alone analyzes networks from the following domains.\n\n\\begin{itemize}\n\\item \\emph{Ecological systems --} Nodes are species and edges are who-eats-whom\n  relationships in food web models.  (Here, the edges are \\emph{directed} to\n  represent asymmetry---sharks regularly consume sardines, but sharks are not a\n  part of a sardine's diet!)  We examine the graph structure behind aquatic\n  layers in a food web of Florida Bay.\n  \n\\item \\emph{Human communication systems --} Nodes are people and edges are\n  messages between people.  Again, there is a natural direction to the edges.\n  Our network datasets also have temporal information associated with the graph\n  to denote when messages are sent.  We analyze the differences between communication\n  behaviors in e-mail, SMS texting, and phone call networks.\n  \n\\item \\emph{Neural systems --} Nodes are neurons and edges are synapses.  We\n  examine the patterns of connections in the complete neural network of the\n  nematode worm \\emph{C.~elegans}.\n  \n\\item \\emph{Payment systems --} Nodes are people, businesses, or accounts and\n  edges are payments between them.  We analyze transactions between addresses on\n  Bitcoin.  We again have temporal information corresponding to when payments\n  are made.\n  \n\\item \\emph{Transcription regulation networks --} Nodes are groups of genes and\n  edges represent which groups of genes regulate genetic transcription in which\n  other groups of genes.  The edges in the network data also carry information\n  about whether the regulation type is activation or suppression (in this case,\n  we say that the edges are \\emph{signed}).  We analyze the transcription regulation network\n  of the yeast \\emph{S.~cerevisiae}.\n  \n\\item \\emph{Transportation systems --} Nodes are locations and edges represent\n  connectivity.  We analyze an air travel reachability network, where nodes are\n  cities and edges represent how long it takes to travel from one city to\n  another using commercial airline flights.  Our analysis also incorporates\n  additional geographical data available for the nodes.\n    \n\\item \\emph{Scientific collaborations --} Nodes are scientists and there is an\n  edge between two scientists if they have co-authored a paper.  We look at\n  collaboration patterns between physicists.\n\n\\item \\emph{Social networks --} Nodes are people and edges are social\n  relationships.  We examine Facebook friendships, Twitter followers, and Stack\n  Overflow question-and-answer interactions.\n\n\\item \\emph{The World Wide Web --} Nodes are web pages and edges are hyperlinks.\n  We analyze the structure of the Stanford web graph and Wikipedia.\n\\end{itemize}\n\nThe above descriptions illustrate how graphs, which consist of nodes and edges,\nare a natural mathematical structure for network models.  The graph is the\nbackbone of the network datasets, even though we may have additional information\nsuch as timestamps, signed edges, geographical data, etc.  Consequently, we often frame our\nanalysis of and questions about networks in terms of nodes and edges.  For\nexample, we might be interested in the number of edges that a node is in (how\nmany different species does the shark consume in the food web?), the number of\nedges between subsets of the nodes (how many Facebook friendships are there\nbetween Stanford students?), or whether or not there exists a sequence of edges\nto traverse to go from one node to another (can I get from Palo Alto to Berkeley\non public transit?).\n\n\\input{CH1-FIG-intro-ffl}\n\nWhile analysis in terms of nodes and edges is natural, there is substantial\nevidence that \\emph{higher-order structures}, or small subgraph patterns between\na few nodes (\\cref{fig:intro_ffl}), are essential to the behavior of many\ncomplex systems modeled by\nnetworks~\\cite{milo2002network,yaverouglu2014revealing}.  This idea has a long\nhistory in sociology.  In the early 1900s, Simmel theorized that triangles (a\ncomplete graph on three nodes) form in social networks because two friends of\nthe same individual get opportunities to meet and become friends\nthemselves~\\cite{simmel1908sociology}, an idea later popularized by\n\\citet{granovetter1973strength} as \\emph{triadic closure}.\n\\Citet{bavelas1950communication} surveyed several studies that examined how\nsmall communication patterns used between individuals in a group affects their\nability to jointly perform tasks, and \\citet{davis1971structure} found that\ncertain directed triads were less common than predicted by a null model,\nproviding numerical evidence for the ranked groups social network theory of\n\\citet{homans1950human}.  In more recent work, \\citet{ugander2013subgraph} use\nsmall subgraph frequencies within induced Facebook friendship subgraphs to\nidentify social processes.\n\nIn network analysis more broadly, higher-order structure is often described\nthrough the idea of a \\emph{network motif}.  \\Citet{shen2002network}\nintroduced the term when they analyzed the frequency of higher-order\ninteractions through subgraph patterns, which they called network motifs, in\nthe transcription regulation network of \\emph{E. coli}.\\footnote{\\Citet{shen2002network} borrowed the phrase motif from gene\n  sequence analysis, where motifs are ``short, recurring patterns in DNA that\n  are presumed to have a biological function''~\\cite{d2006dna}.}  This style of\nanalysis was popularized in a landmark paper of \\citet{milo2002network}\n(published in \\emph{Science} the same year), where they dubbed network motifs the\n``building blocks of complex networks'' and identified motifs common in gene\nregulation networks, food webs, and electronic circuits.  In follow-up work,\n\\citet{milo2004superfamilies} showed that the profile of z-scores (with\nrespect to samples of a configuration model with the same degree distribution)\nfor the frequencies of the 13 connected, 3-node directed subgraphs were\nsufficient to distinguish biological, neurological, social, and linguistic\nnetworks.  In subsequent research, important motifs have been identified in a\nvariety of domains, including brain science~\\cite{sporns2004motifs},\necology~\\cite{camacho2007quantitative}, biology~\\cite{wuchty2003evolutionary},\nand social network analysis~\\cite{leskovec2010signed}.\n\n\\Citet{milo2002network} defined network motifs as ``patterns of interconnections\noccurring in complex networks at numbers that are significantly higher than\nthose in randomized networks.''  The ``significant'' qualifier in this\ndefinition has caused some confusion around the terminology.  Some take the\nqualifier to heart and use the term \\emph{graphlet} to specify any small network\npattern and reserve the motif designation for graphlets that occur significantly\nmore frequently with respect to some null model and some measure of statistical\nsignificance~\\cite{prvzulj2004modeling}.\\footnote{Presumably, the term graphlets\n  is a play on wavelets, which are used in signal processing to localize time\n  and frequency, but I have not found this explicitly mentioned in the\n  literature.}  However, one could just as well call a graphlet a small\nsubgraph, and indeed, many\ndo~\\cite{ugander2013subgraph,demeyer2013index,lahiri2007structure,pinar2017escape}.\nUsually, there is an implicit assumption that motifs and graphlets are\n\\emph{connected} patterns, whereas general subgraphs do not carry this\nassumption.  In this thesis, we just use ``motifs'' to specify patterns in\nnetworks and sometimes use the term subgraph when discussing the more\nmathematical points.  In \\cref{ch:honc}, we also add to the vernacular by\ndefining an ``anchored motif'', which provides a formalism for specifying a\nsubset of the nodes in a motif relevant to certain computations.  We also use\n``higher-order structures'' as a catch-all phrase.\n\nOperating under the assumption that higher-order structures are important to\nnetworks, the algorithmic data mining community has developed a litany of\nmethods for efficiently finding and counting them.  These include, to name a\nfew, algorithms for enumerating triangles that perform especially well given the\npower-law degree distributions typical of many real-world\nnetworks~\\cite{schank2005finding,latapy2008main,berry2014why}, fast algorithms\nfor enumerating general motifs via principled\nheuristics~\\cite{wernicke2006fanmod,demeyer2013index,houbraken2014index,ahmed2015efficient},\nalgorithms that approximate the total number of\ntriangles~\\cite{tsourakakis2009doulion,seshadhri2014wedge,lim2015mascot,de2016triest},\nand algorithms that approximate the total number of motifs composed of 4 or more\nnodes~\\cite{jha2015path,bressan2017counting,wang2014efficiently,slota2013fast}.\nFor the computations in this thesis, we only need the method of \\citet[Algorithm 1]{schank2005finding}\nfor enumerating triangles and the method of\n\\citet{chiba1985arboricity} for enumerating cliques.  We use these algorithms\nbecause they are straightforward to implement and are fast for the datasets\nthat we analyze.\n\nWe summarize the setup---networks consisting of nodes and edges model a broad\nrange of systems, higher-order interactions (patterns between a small number of\nnodes) are important to many networks, and we have algorithms adept at finding\nand counting these higher-order structures.  However, higher-order structures\nhave not been well integrated into the analyses, models, and algorithms that we\nactually use to study the structure of complex networks.  This thesis develops\nnew methods, or \\emph{tools}, for analyzing network data based on higher-order\nstructures, i.e., \\emph{for higher-order network analysis}.  Consequently, the\nnetwork analyst is able to examine data in terms of\nthe higher-order interactions that are important to him or her.\n\nWe demonstrate how our tools use higher-order interactions as a primitive in order\nto gain new insights into complex systems.  For example, our analysis of the neural\nnetwork for the nematode worm \\emph{C. elegans} in \\cref{sec:honc_celegans}\nfinds a cluster of 20 neurons in the frontal section of the worm that is a\nplausible control of nictation, a type of worm movement.  We discover this group\nthrough a new algorithm that looks for subsets of the graph in which a certain\n4-node ``bi-fan'' motif is contained.  The higher-order structure is key to the\ndiscovery---we need to optimize an objective function for sets of nodes that\nmodels the bi-fan motif---and \\emph{not} simply edges, as is typically done---in\norder to find this group.  Later, in \\cref{sec:ccfs_empirical}, we find that\nunder a generalized ``closure'' model for the presence of edges in the network,\n3-cliques (triangles) in \\emph{C. elegans} are more common than expected, while\n4-cliques are less common than expected, which challenges a longstanding view of\nhow edges tend to cluster together in network models of complex\nsystems~\\cite{watts1998collective}.  In this case, the consideration of\nhigher-order structure at the level of 4-cliques was necessary for the finding.\n\nHigher-order analysis is a small part of network science, but\nhigher-order thinking has still shown up in several contexts.  For example, motifs\nhave been used to generate features for both prediction\nproblems~\\cite{milenkovic2008uncovering,bonato2014dimensionality,ugander2013subgraph,chakraborty2014automatic}\nand unsupervised learning\ntasks~\\cite{yaverouglu2014revealing,henderson2012rolx}, motif frequencies are\nused to fit parameters of random graph\nmodels~\\cite{gleich2012moment,benson2014learning,bickel2011method,wasserman1996logit},\nand motif modeling is used to improve network alignment\nalgorithms~\\cite{milenkovic2010optimal,mohammadi2016triangular}.  In addition,\nthere are several higher-order approaches that do not necessarily use motif\nstructure but are similar in spirit.  Examples include ``meta paths'' to\nrepresent multi-relational networked data~\\cite{zhang2014meta,sun2011pathsim},\nhigher-order and variable-order Markov chain\nmodels~\\cite{rosvall2014memory,chierichetti2012web,benson2017spacey,wu2016general}, and\n``higher-order network'' models for aggregating temporal paths in networks where\nedges have\ntimestamps~\\cite{scholtes2014causality,scholtes2017network,scholtes2016higher}.\n\nThe new ideas in \\cref{ch:honc,ch:hoccfs} also align with recent higher-order\nnetwork analyses that directly generalize classical ideas to account for\nhigher-order structures in the network.  For example, \\citet{tsourakakis2015k}\ngeneralized the (edge) densest subgraph problem to the $k$-clique densest\nsubgraph problem, \\citet{zhang2012extracting} generalized (edge-based) $k$-cores\nto triangle-based $k$-cores, and \\citet{sariyuce2015finding} generalized\n$k$-core and $k$-truss decompositions to account for clique containment with the\nnucleus decomposition.  We describe our contributions in more detail in the\nfollowing section.\n\n\\section{Contributions}\n\nThe core contributions of this thesis are three new tools for higher-order\nnetwork analysis.  We contextualize and summarize them in\n\\cref{tab:contributions}.\n\\input{CH1-TAB-contributions}\n\nIn addition to developing the tools, we make a concerted effort to use the tools\nto gain new insights into a variety of network datasets.  Thus, there is a\nconsiderable data mining component to the chapters ahead, which are also\nmeaningful contributions.  The next few subsections provide additional details\non the tools and the insights that they provide.\n\n\\subsection{\\cref{ch:honc} -- Higher-order network clustering}\n\nThe first tool is a framework for graph clustering based on motifs.  Graph\nclustering is a broad research problem that assigns nodes in a graph to\nclusters, where the clusters are meant to represent some module in the network.\nTypically, the objective function modeling what it means to have good clusters\ninvolves a combination of the number of \\emph{edges} contained within the\nclusters and the number of \\emph{edges} that go between clusters.  We re-define\nwhat it means to be a good cluster with an objective function that considers the\nnumber of \\emph{motifs} contained within the clusters and the number of\n\\emph{motifs} that go between clusters.  Thus, if a particular motif is\nimportant for some domain, we can run an algorithm to optimize an objective\nfunction that accounts for that motif.\n\nOur main theoretical contributions include (i) defining the new objective\nfunction, \\emph{motif conductance}, which is a generalization of the classical\nconductance measure of cluster quality; and (ii) a spectral algorithm for\nfinding sets with small motif conductance.  The algorithm is accompanied by an\napproximation guarantee on the quality of the output clusters in terms of motif\nconductance.  The algorithm and quality guarantee are generalizations of the\nFiedler partition or sweep cut procedure and the Cheeger inequality.\n\nUsing this framework we show that\n\\begin{itemize}\n\\item a particular directed triangular motif reveals aquatic layers in a food\n  web;\n\\item in the \\emph{C. elegans} neural system, a cluster we find based\n  on a previously studied 4-node motif (the ``bi-fan'')\n  is a plausible control mechanism for nictation, a type of movement in the worm;\n\\item clustering with length-2 paths automatically reveals the hub structure and\n  geography of an air travel transportation reachability network;\n\\item clustering with the feedforward loop (\\cref{fig:intro_fflC}) reveals known\n  modules in transcription regulation networks with higher accuracy than\n  edge-based methods; and\n\\item clustering with particular directed triangular motifs reveal anomalous clusters in\n  the English Wikipedia web graph and the Twitter follower network.\n\\end{itemize}\n\nWe then extend the algorithm to handle the problem of \\emph{localized\n  clustering} or \\emph{targeted clustering}, where the goal is to find a (small) cluster\nof nodes containing a given seed node.  Again, existing approaches optimize the\nclassical conductance, and we instead optimize motif conductance.  Our\napproach is a generalization of the approximate Personalized PageRank\nmethod~\\cite{andersen2006local}.\n\nThe algorithmic framework and applications appeared as a publication in\n\\emph{Science}~\\cite{benson2016higher},\\footnote{See the accompanying\n  perspective piece by \\citet{prvzulj2016network} for a broader context for this\n  work.} which was joint work with David Gleich and Jure Leskovec.  The\nextension to localized clustering will appear in a paper in the proceedings of\nthe 2017 KDD conference~\\cite{yin2017local}, which was joint work with Hao Yin,\nDavid, and Jure.  Hao implemented the localized algorithm, ran the experiments for the\nresults in \\cref{sec:local}, and came up with the idea to look at recovery in the\nplanted partition model (\\cref{sec:local_synth}).\n\n\\subsection{\\cref{ch:hoccfs} -- Higher-order clustering coefficients}\n\nThe second tool is a measurement for the extent to which nodes in a network\ncluster together.  This is a generalization of the classical clustering\ncoefficient that measures the fraction of length-2 paths that induce a triangle.\nWe reinterpret the clustering coefficient as a clique expansion and closure\nprocess---a 2-clique (an edge) \\emph{expands} with an adjacent edge and we check\nif this structure \\emph{closes} by forming a 3-clique (a triangle).  We\ngeneralize this by considering an $\\ell$-clique that expands with an adjacent\nedge and checking if this structure closes by forming an $(\\ell + 1)$-clique.\nWe call the fraction of ($\\ell$-clique, adjacent edge) pairs that induce an\n$(\\ell + 1)$-clique the $\\ell$th-order clustering coefficient.\n\nWe theoretically analyze the higher-order clustering coefficient in the\nErd\\H{o}s-R\\'enyi and small-world random graph models and empirically analyze\nthe higher-order clustering coefficient on three real-world networks.  We find that\nalthough the \\emph{C. elegans} neural network has high clustering in the\ntraditional sense, it does not have high third-order clustering (3-cliques tend\nnot to expand into 4-cliques).  This could arise from the fact that 4-cliques represent\nredundant processing units and their absence leads to a more efficient neural\narchitecture.\n\nWe also make a connection between higher-order clustering coefficients and motif\nconductance.  We show that if a network has a large $\\ell$th-order clustering\ncoefficient, then there is a node whose 1-hop neighborhood subgraph has small motif\nconductance for the $\\ell$-clique motif.  This is a generalization of the $\\ell = 2$\ncase studied by \\citet{gleich2012vertex}.  In fact, I came up with the\ndefinition for the higher-order clustering coefficient when I was trying to\ngeneralize a lemma from \\citet{gleich2012vertex} to the $\\ell = 3$ case (for\ntriangle conductance).\n\nThe definitions and analysis of the higher-order clustering coefficient is based\non a paper with Hao Yin and Jure Leskovec that has been submitted for\npublication and is currently on arXiv~\\cite{yin2017higher}.  The connection to\nmotif conductance is based on part of a paper with Hao, Jure, and\nDavid Gleich that will appear in the proceedings of the 2017 KDD\nconference~\\cite{yin2017local}.  Hao helped a substantial amount on the proofs\nof \\cref{prop:ccf_er,prop:ccf_sw,thm:nbrhd_main}.\n\n\\subsection{\\cref{ch:tm} -- Motifs in temporal networks}\n\nThe third tool is a definition of motifs in temporal networks along with\nalgorithms for efficiently counting them.  The goal of the research in\n\\cref{ch:tm} is to provide the foundations of higher-order network analysis---a\ndefinition for higher-order structures and efficient algorithms for finding\nthem---for a broader class of network datasets, i.e., those that contain temporal\ninformation.  We consider a temporal network to be a collection of $(u, v, t)$\ntuples (temporal edges), where $u$ and $v$ are elements of a node set $V$ and $t\n\\in \\mathbb{R}_{+}$ is a timestamp.  We define a temporal motif by a multigraph,\nan ordering on the edges in the multigraph, and a time window $\\delta$, and we\ndefine an instance of the temporal motif in a temporal network to be a subset of\nthe temporal edges that match the edge pattern of the multigraph, appear in the\nspecified order, and all occur within $\\delta$ time units of each other.  We\nprovide a general algorithm for efficiently counting the number of instances of\ntemporal motifs in a given temporal network along with specialized fast\nalgorithms for certain classes of motifs.  We also show some basic higher-order\nanalyses on several temporal network datasets.\n\n\\Cref{ch:tm} is based on a paper with Ashwin Paranjape and Jure Leskovec that\nappeared in the proceedings of the 2017 WSDM\nconference~\\cite{paranjape2017motifs}.  Ashwin helped with the design,\nimplementation, and analysis of the algorithms and executed the experiments in\n\\cref{sec:tm_empirical}.\n\n\\subsection{Additional artifacts and impact}\n\nIn addition to the content of this thesis and the associated publications, other\nartifacts of this research include the following.\n\n\\begin{itemize}\n\\item Implementations of the motif-based spectral clustering algorithm\n \\begin{itemize}\n \\item for the SNAP software library, which is available at\\\\\n   \\url{https:\/\/github.com\/snap-stanford\/snap};\n \\item in Julia, which is available at\\\\\n   \\url{https:\/\/github.com\/arbenson\/higher-order-organization-julia}; and\n \\item in MATLAB, which is available at\\\\\n   \\url{https:\/\/github.com\/arbenson\/higher-order-organization-matlab}.\n \\end{itemize}\n\\item Implementations of the temporal motif counting algorithms for the\n  SNAP software library, which is available at\\\\\n  \\url{https:\/\/github.com\/snap-stanford\/snap}. \n \\item Julia notebooks to reproduce the results in the main text of\n   \\citet{benson2016higher}, which are available at\\\\\n   \\url{https:\/\/github.com\/arbenson\/higher-order-organization-julia}.\n\\item Metadata for the nodes in the transportation reachability network of\n  \\citet{frey2007clustering} (city latitudes, longitudes, and metropolitan\n  populations), which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/reachability.html}.\n\\item Metadata for the nodes in the Florida Bay food web\n  \\citet{frey2007clustering} (group classification), which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/Florida-bay.html}.  \n\\item The $\\dataset{email-Eu-core}$ network dataset with department\nlabels for all of the nodes, which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/email-Eu-core.html}.  \n\\item The $\\dataset{wiki-cats}$ network dataset with article names and category\n  classifications for all of the nodes, which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/wiki-topcats.html}.  \n\\item The $\\dataset{StackOverflow}$ temporal network dataset, which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/sx-stackoverflow.html}.\n\\item The $\\dataset{MathOverflow}$ temporal network dataset, which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/sx-mathoverflow.html}.\n\\item The $\\dataset{SuperUser}$ temporal network dataset, which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/sx-superuser.html}.\n\\item The $\\dataset{AskUbuntu}$ temporal network dataset, which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/sx-askubuntu.html}.\n\\item The $\\dataset{email-Eu}$ temporal network dataset and 4 department-level\n  subnetworks, which are available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/email-Eu-core-temporal.html}.\n\\item The $\\dataset{WikiTalk}$ temporal network dataset, which is available at\\\\\n  \\url{http:\/\/snap.stanford.edu\/data\/wiki-talk-temporal.html}.\n\\end{itemize}\n\nFurthermore, the ideas of this thesis have already had impact on the broader\nresearch community.  As an example, \\citet{meier2016motif} used the motif-based\nclustering algorithm in functional connectivity networks of brains and found\nsymmetry patterns between the hemispheres of the brain.\nIn their article ``Network analytics in the age of big data''\npublished in \\emph{Science}, \\citet{prvzulj2016network} contextualize the\nframework of \\cref{ch:honc} as an important step towards understanding\nthe large-scale datasets coming from a ``complex world of interconnected\nentities.''  As discussed at the beginning of this introduction, network models\nand analysis are fundamental because of their wide applicability, and\n\\Citet{prvzulj2016network} emphasize this point when (generously) discussing our work:\n``The importance of this result lies in its applicability to a broad range of\nnetworked data that we must understand to answer fundamental questions facing\nhumanity today, from climate change and impacts of genetically modified\norganisms, to the environment, to food security, human migrations, economic and\nsocietal crises, understanding diseases, aging, and personalizing medical\ntreatments.''\nIndeed, I hope that the ideas that follow are a step towards tackling these problems.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Alternative clustering algorithms for evaluation}\n\\input{CH2-TXT-alternatives}\n\n\\subsection{Comparing motif conductance and edge conductance}\n\\label{sec:comparing_cond}\n\\input{CH2-TXT-comparing-conductance}\n\n\\subsection{Motif $M_{6}$ in the Florida Bay food web}\n\\label{sec:honc_foodweb}\n\\input{CH2-TXT-foodweb}\n\n\\subsection{Coherent feedforward loops in the \\emph{S.~cerevisiae} transcriptional regulation network}\n\\label{sec:honc_yeast}\n\\input{CH2-TXT-yeast}\n\n\\subsection{Bi-directed length-2 paths in a transportation reachability network}\n\\label{sec:honc_airports}\n\\input{CH2-TXT-airports}\n\n\\subsection{The bi-fan motif in the \\emph{C.~elegans} neuronal network}\n\\label{sec:honc_celegans}\n\\input{CH2-TXT-celegans}\n\n\\subsection{Motif $M_{6}$ in the English Wikipedia article network}\n\\label{sec:honc_enwiki}\n\\input{CH2-TXT-enwiki}\n\n\\subsection{Motif $M_{6}$ in the Twitter follower network}\n\\label{sec:honc_twitter}\n\\input{CH2-TXT-twitter}\n\n\\subsection{Motif $M_{7}$ in the Stanford web graph}\n\\label{sec:honc_stanford}\n\\input{CH2-TXT-stanford}\n\n\\subsection{Semi-cliques in collaboration networks}\n\\label{sec:honc_collaboration}\n\\input{CH2-TXT-collaboration}\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Higher-order network clustering}\n\\label{ch:honc}\n\n\\section{Organization via higher-order structures}\n\\input{CH2-TXT-intro}\n\n\\section{The motif-based spectral clustering algorithm}\n\\label{sec:honc_derivation}\n\\input{CH2-TXT-theory}\n\n\\section{Case studies}\n\\input{CH2-TXT-case-studies}\n\n\\section{Scalability experiments}\n\\label{sec:honc_scalability}\n\\input{CH2-TXT-scalability}\n\n\\section{An extension to local higher-order clustering}\n\\label{sec:local}\n\\input{CH2-TXT-local}\n\n\\section{Related work and discussion}\n\\label{sec:honc_discussion}\n\\input{CH2-TXT-discussion}\n\n\n\n\n\n\\subsection{Overview}\n\\input{CH2-TXT-local-overview}\n\n\\subsection{Motif-based personalized PageRank}\n\\input{CH2-TXT-motifppr}\n\n\\subsection{Experiments on synthetic networks}\n\\label{sec:local_synth}\n\\input{CH2-TXT-local-synthetic-exps}\n\n\\subsection{Experiments on real-world networks}\n\\label{sec:local_real}\n\\input{CH2-TXT-local-real-exps}\n\n\n\n\n\n\n\n\n\\subsection{Triangular motifs}\n\\label{sec:scalability}\n\nIn this section, we demonstrate that our method scales to real-world networks\nwith billions of edges.  We tested the scalability of our method on 16 large\ndirected graphs from a variety of real-world applications.  These networks range\nfrom a couple hundred thousand to two billion edges and from 10 thousand to over\n50 million nodes.  We briefly describe the networks here and provide some\nsummary statistics in \\Cref{tab:networks_description}.\n\\begin{itemize}\n\\item \\dataset{wiki-RfA} represents which users voted for which other users for\n  adminship rights on Wikipedia~\\cite{west2014exploiting}.\n\\item \\dataset{email-EuAll} represents who emailed whom at a European research\n  institution~\\cite{leskovec2005graphs}.\n\\item \\dataset{cit-HepPh} represents citations between papers in the ``High\n  Energy Physics -- Phenomenology'' category on\n  arXiv~\\cite{gehrke2003overview}.\n\\item \\dataset{web-NotreDame} is the hyperlink structure of web pages in the\n  \\texttt{nd.edu} domain~\\cite{albert1999internet}.\n\\item \\dataset{amazon0601} consists of connections between frequently\n  co-purchased products on Amazon~\\cite{leskovec2007dynamics}.\n\\item \\dataset{wiki-Talk} represents which users wrote on which other users' talk page\n  on Wikipedia~\\cite{leskovec2010governance}.\n\\item \\dataset{ego-Gplus} is a collection of egonetworks (1-hop neighborhoods)\n  on the online social network Google+~\\cite{leskovec2012learning}.\n\\item \\dataset{uk-2014-tpd} is the hyperlink structure of top private domain\n  links in the \\texttt{.uk}\n  domain~\\cite{boldi2004webgraph,boldi2011layered,boldi2014bubing}.\n\\item \\dataset{soc-Pokec} consists of the friendship relationships on the online\n  social network Pokec~\\cite{takac2012data}.\n\\item \\dataset{uk-2014-host} is the hyperlink structure of host links on the\n  \\texttt{.uk} domain~\\cite{boldi2004webgraph,boldi2011layered,boldi2014bubing}.\n\\item \\dataset{soc-LiveJournal1} consists of the friendships on the online\n  social network LiveJournal~\\cite{backstrom2006group}.\n\\item \\dataset{enwiki-2013} is the hyperlink structure of articles on English\n  Wikipedia~\\cite{boldi2004webgraph,boldi2004ubicrawler,boldi2011layered}\n\\item \\dataset{uk-2002} is the hyperlink structure of web pages in the\n  \\texttt{.uk}\n  domain~\\cite{boldi2004webgraph,boldi2004ubicrawler,boldi2011layered}.\n\\item \\dataset{arabic-2005} is the hyperlink structure of arabic-language web\n  pages~\\cite{boldi2004webgraph,boldi2004ubicrawler,boldi2011layered}.\n\\item \\dataset{twitter-2010} represents the followers of users on the online\n  social network\n  Twitter in 2010~\\cite{boldi2004webgraph,boldi2011layered,kwak2010twitter}.\n\\item \\dataset{sk-2005} is the hyperlink structure of web pages in the\n  \\texttt{.sk}\n  domain~\\cite{boldi2004webgraph,boldi2004ubicrawler,boldi2011layered}.\n\\end{itemize}\n\n\\input{CH2-TAB-networks-description}\n\\clearpage\n\nRecall that \\Cref{alg:motif_fiedler} consists of two major computational components:\n\\begin{enumerate}\n\\item Form the weighted graph $W_M$.\n\\item Compute the eigenvector $z$ of second smallest eigenvalue of $N_M$.\n\\end{enumerate}\nAfter computing the eigenvector, we sort the vertices and loop over prefix sets\nto find the lowest motif conductance set.  We consider these final steps as part\nof the eigenvector computation for our performance experiments.\n\nFor each network in \\cref{tab:networks_description}, we ran the method for all\ndirected triangular motifs ($M_1$--$M_7$).  To compute $W_M$, we used a standard\nalgorithm that meets the $O(m^{3\/2})$\nbound~\\cite{schank2005finding,latapy2008main} with some additional\npre-processing based on the motif.  Conceptually, the algorithm is as follows:\n\n\\begin{enumerate}\n\\item Take motif type $M$ and graph $G$ as input.\n\n\\item (Pre-processing)  If $M$ is $M_1$ or $M_5$, ignore all bidirectional\n  edges in $G$ as these motifs only contain unidirectional edges.  If $M$ is\n  $M_4$, ignore all unidirectional edges in $G$ as this motif only contains\n  bidirectional edges.\n  \n\\item Form the undirected graph $G_{\\textnormal{undir}}$ by removing the\n  direction of all edges in $G$.\n  \n\\item Let $d_u$ be the degree of node $u$ in $G_{\\textnormal{undir}}$.  Order\n  the nodes in $G_{\\textnormal{undir}}$ by increasing degree, breaking ties\n  arbitrarily.  Denote this ordering by $\\psi$.\n  \n\\item For every edge undirected edge $\\{u, v\\}$ in $G_{\\textnormal{undir}}$, if\n  $\\psi_u < \\psi_v$, add directed edge $(u, v)$ to $G_{\\textnormal{dir}}$;\n  otherwise, add directed edge $(v, u)$ to $G_{\\textnormal{dir}}$.\n  \n\\item For every node in $u$ in $G_{\\textnormal{dir}}$ and every pair of directed\n  edges $(u, v)$ and $(u, w)$, check to see if edge $(v, w)$ or $(w, v)$ is in\n  $G_{\\textnormal{dir}}$.  If so, check if these three nodes form motif $M$ in\n  $G$.  If they do, increment the weights of edges $(W_M)_{uv}$,\n  $(W_M)_{uw}$, and $(W_M)_{vw}$ by $1$.\n  \n\\item Return $W_M + W_M^T$ as the motif weighted adjacency matrix.  \n\\end{enumerate}\n\nThe algorithm runs in time $O(m^{3\/2})$ time and is also\nknown as an effective heuristic for real-world\nnetworks~\\cite{berry2014why,latapy2008main}.  After, we find the largest\nconnected component of the graph corresponding to the motif adjacency matrix\n$W_M$, form the motif normalized Laplacian $N_M$ of the largest\ncomponent, and compute the eigenvector of second smallest eigenvalue of\n$N_M$.  To compute the eigenvector, we use MATLAB's \\texttt{eigs}\nroutine with tolerance 1e-4 and the ``smallest algebraic'' option for the\neigenvalue type.\n\n\\Cref{tab:scalability_results} lists the time to compute $W_M$ and the time to\ncompute the eigenvector for each network.  We omit the time to read the graph\nfrom disk because this time strongly depends on how the graph is compressed.\nAll experiments ran on a 40-core server with four 2.4 GHz Intel Xeon E7-4870\nprocessors.  All computations of $W_M$ were in serial and the computations of\nthe eigenvectors were done in parallel (as defaulted to by MATLAB).\n\nOver all networks and all motifs, the longest computation of $W_M$ (including\npre-processing time) was for $M_2$ on \\dataset{sk-2005} and took roughly 52.8\nhours.  The longest eigenvector computation was for $M_6$ on \\dataset{sk-2005}\nand took about 1.62 hours. We note that $W_M$ only needs to be computed once per\nnetwork, regardless of the eventual number of clusters that are extracted.\nAlso, the computation of $W_M$ can easily be accelerated by parallel computing\n(the enumeration of motifs can be done in parallel over nodes, for example) or\nby more sophisticated algorithms~\\cite{berry2014why}.  In this work, we perform\nthe computation of $W_M$ in serial in order to better understand the\nscalability.  Our results serve only as a rough baseline.\n\n\\input{CH2-TAB-scalability-results}\n\\clearpage\n\n\nIn theory, the worst-case time for triangle enumeration scales as $m^{1.5}$.\nWe fit a linear regression of the log of the computation time of the last step of the\nenumeration algorithm to the regressor $\\log(m)$ and a constant term:\n\\begin{equation}\\label{eqn:regression}\n\\log(\\text{time}) \\sim a\\log(m) + b\n\\end{equation}\nIf the computations truly took $cm^{1.5}$ for some constant $c$, then the\nregression coefficient for $\\log(m)$ would be $1.5$.  Because of the\npre-processing of the algorithm, the number of edges $m$ depends on the motif.\nFor example, with motifs $M_1$ and $M_5$, we only count the number of\nunidirectional edges.  The pre-processing time, which is linear in the total\nnumber of edges, is not included in the time.  The regression coefficient for\n$\\log(m)$ (the variable $a$ in \\cref{eqn:regression}) is smaller than $1.5$ for\neach motif (\\cref{tab:scale_conf}).  The largest regression coefficient is\n$1.31$ for $M_3$ (with 95\\% confidence interval $1.31 \\pm 0.19$).  The\nregression coefficient over the aggregation of data points (the ``combined''\ncolumn in \\cref{tab:scale_conf}) is $1.17$ (with 95\\% confidence interval $1.17\n\\pm 0.09$).  We conclude that on real-world datasets, the algorithm for\ncomputing $W_M$ performs much better than the worst-case guarantees.\n\n\\input{CH2-TAB-scalability-regression}\n\\clearpage\n\n\\subsection{Larger $k$-clique motifs}\n\nOn smaller graphs, we can compute larger motifs.  To demonstrate this point, we\nform the motif adjacency matrix $W_M$ based on the $k$-cliques motif for $k = 4,\n\\dots, 9$ using the algorithm of \\citet{chiba1985arboricity} with the additional\npre-processing of computing the ($k-1$)-core of the graph.  (This pre-processing\nimproves the running time in practice but does not affect the asymptotic\ncomplexity.)  The motif adjacency matrices for $k$-cliques are sparser than the\nadjacency matrix of the original graph, so we do not worry about spatial\ncomplexity for these motifs.\n\nWe evaluate this procedure on nine real-world networks, ranging from roughly\nfour thousand nodes and 88 thousand edges to over two million nodes and around\nfive million edges.  We briefly describe the networks here and list summary\nstatistics in \\cref{tab:cliques_perf}.\n\\begin{itemize}\n\\item \\dataset{ego-Facebook} is the union of ego networks from the user\n  friendship graph of the online social network\n  Facebook~\\cite{leskovec2012learning}.\n\n\\item \\dataset{ca-AstroPh} represents scientists who have co-authored a paper\n  listed on the AstroPhysics category on\n  arXiv~\\cite{leskovec2005graphs}.\n\n\\item \\dataset{soc-Slashdot0811} represents who tagged whom during an event on\n  the online social network Slashdot~\\cite{leskovec2009community}.  The\n  original network data is directed and signed, and we ignore both of these\n  properties here.\n\n\\item \\dataset{com-DBLP} represents scientists who have co-authored a paper\n  listed on DBLP~\\cite{yang2012defining}.\n\n\\item \\dataset{com-Youtube} consists of friendships on the online social network\n  YouTube~\\cite{mislove2007measurement}.\n\\end{itemize}\nWe also use the \\dataset{wiki-RfA}, \\dataset{email-EuAll}, \\dataset{cit-HepPh},\nand \\dataset{wiki-Talk} datasets described earlier, although we now consider the\ngraphs to be undirected.\n\nEach network contains at least one $9$-clique and hence at least one $k$-clique\nfor $k < 9$.  All computations ran on the same server as for the triangular\nmotifs and again there was no parallelism.  We terminated computations after two\nhours.  For five of the nine networks, it takes less than two hours to compute\n$W_M$ for any $k$-clique motif, $k = 4, \\ldots, 9$ (\\cref{tab:cliques_perf}).\nFurthermore, on all of these networks, the computation takes less than two hours\nfor $k = 4, 5, 6$.  The smallest network (in terms of number of nodes and number\nof edges) is \\dataset{ego-Facebook}, where it took just under two hours to\ncomptue $W_M$ for the $6$-clique motif and over two hours for the $7$-clique\nmotif.  This network has around 80,000 edges.  On the other hand, for\n$\\dataset{com-Youtube}$, which contains nearly 3 million edges, we can compute $W_M$ for the\n$9$-clique motif in under a minute.  We conclude that it is possible to use our\nframework with motifs much larger than the three-node motifs on which we\nperformed many of our experiments.  However, the number of edges is not a\ngood predictor of the running time to compute $W_M$.  This makes sense because the\ncomplexity of the algorithm of \\citet{chiba1985arboricity} is $O(a^{k-2}m)$,\nwhere $a$ is the arboricity of the graph.  Hence, the dependence on the number\nof edges is always linear, and the arboricity drives the running time.\n\n\\input{CH2-TAB-clique-performance}\n\\clearpage\n\n\n\\subsection{Review of cuts, volumes, conductance and the graph Laplacian for weighted, undirected graphs}\n\\input{CH2-TXT-review}\n\n\\subsection{Definition of network motifs}\n\\label{sec:motif_def}\n\\input{CH2-TXT-motif-definitions}\n\n\\subsection{Motif conductance}\n\\label{sec:motif_cond}\n\\input{CH2-TXT-motif-conductance}\n\n\\subsection{The motif adjacency matrix and the motif Laplacian}\n\\label{sec:motif_adjacency}\n\\input{CH2-TXT-motif-adjacency}\n\n\\subsection{The spectral algorithm for finding a single cluster}\n\\label{sec:honc_alg}\n\\input{CH2-TXT-algorithm}\n\n\\subsection{Interlude for matrix computations}\n\\label{sec:honc_matrix}\n\\input{CH2-TXT-matrix}\n\n\\subsection{Motif Cheeger inequality for network motifs with three nodes}\n\\label{sec:motif_cheeger}\n\\input{CH2-TXT-cheeger-3}\n\n\\subsection{Motif Cheeger inequality for network motifs with four or more nodes}\n\\label{sec:fournode}\n\\input{CH2-TXT-cheeger-4}\n\n\\subsection{Analysis of computational complexity}\n\\label{sec:honc_basic_complexity}\n\\input{CH2-TXT-complexity}\n\n\\subsection{Methods for simultaneously finding multiple clusters}\n\\label{sec:honc_multiple_clusters}\n\\input{CH2-TXT-multiple-clusters}\n\n\\subsection{Extensions for multiple motifs, weighted motifs, and weighted, signed, and colored networks}\n\\label{sec:honc_extensions}\n\\input{CH2-TXT-extensions}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Higher-order clustering coefficients}\n\\label{ch:hoccfs}\n\n\\section{The clustering coefficient and closure probabilities}\n\\input{CH3-TXT-intro}\n\n\\section{Definitions of higher-order clustering coefficients}\n\\input{CH3-TXT-definitions}\n\n\\section{Theoretical Analysis}\n\\input{CH3-TXT-theory}\n\n\\section{Empircal Analysis}\n\\label{sec:ccfs_empirical}\n\\input{CH3-TXT-empirical}\n\n\\section{Relating higher-order clustering coefficients to motif conductance through 1-hop neighborhoods}\n\\label{sec:nbrhood_cond}\n\\input{CH3-TXT-neighborhoods}\n\n\\section{Related work and discussion}\n\\input{CH3-TXT-discussion}\n\n\n\n\n\n\n\n\n\\subsection{A formal relationship between 1-hop neighborhoods and motif conductance}\n\\sectionmark{1-hop neighborhoods and motif conductance}\n\\label{sec:nbrhd_thy}\n\\input{CH3-TXT-neighborhoods_theory}\n\n\\subsection{Experiments}\n\\label{sec:nbrhd_exps}\n\\input{CH3-TXT-neighborhoods_experiments}\n\n\n\n\n\\subsection{A method to compute higher-order clustering coefficients}\n\\input{CH3-TXT-complexity}\n\n\\subsection{Probabilistic interpretations}\n\\input{CH3-TXT-probabilistic}\n\n\\subsection{Bounds on higher-order clustering coefficients}\n\\input{CH3-TXT-bounds}\n\n\\subsection{Analysis for the $G_{n, p}$ model}\n\\input{CH3-TXT-er_analysis}\n\n\\subsection{Analysis for the small-world model}\n\\input{CH3-TXT-sw_analysis}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{General counting framework}\\label{sec:tm_general_framework}\n\nWe begin with a general framework for counting the number of instances of a of a\ntemporal motif $M = (K, \\sigma, \\delta)$.  To start, consider $H$ to\nbe the static directed graph induced by $K$ (i.e., $H$ consists of all\nunique edges in $K$).  A sequence of temporal edges is an instance\nof $M$ if and only if\n\\begin{enumerate}\n\\item the static subgraph induced by the edges in the sequence is isomorphic to $H$\n\\item the ordering of the edges in the sequence matches $\\sigma$,\n\\item all the edges in the sequence span a time window of at most $\\delta$ time units.\n\\end{enumerate}\nThis leads to the following general algorithm for counting instances of $M$ in a\ntemporal graph $T$:\n\\begin{enumerate}\n\\item\nIdentify all instances $H'$ of the static motif $H$ within the static graph $G$\ninduced by the temporal graph $T$.  For example, there are three instances of\n$H$ induced in \\cref{fig:intro1} ($\\{(a, b), (a, c), (c, a)\\}$, $\\{(a, d), (a, c), (c, a)\\}$, and\n$\\{(c, d), (c, a), (a, c)\\}$).\n\\item\nFor each static motif instance $H'$, gather all temporal edges between pairs of\nnodes forming an edge in $H'$ into an ordered sequence $S =$ $(u_1, v_1, t_1)$,\n$\\ldots$, $(u_L, v_L, t_L)$.\n\\item\nCount the number of (potentially non-contiguous) subsequences of edges in $S$\noccurring within $\\delta$ time units that match $(K, \\sigma)$.\n\\end{enumerate}\n\nThe first step can use known algorithms for enumerating motifs in static graphs\nsuch as those by \\citet{wernicke2006fanmod}, and the second step is a simple\nmatter of fetching the appropriate temporal edges.  To perform the third step\nefficiently, we develop a dynamic programming approach for counting the number\nof subsequences (instances of motif $M$) that match a particular pattern within\na larger sequence ($S$).  The key idea is that, as we stream through an input\nsequence of edges, the count of a given length-$l$ pattern (i.e., motif) with a\ngiven final edge is computed from the current count of the length-($l-1$) prefix\nof the pattern.  Inductively, we maintain auxiliary counters of all of the\nprefixes of the pattern (motif).  Second, we also require that all edges in the\nmotif be at most $\\delta$ time apart.  Thus, we use the notion of a moving time\nwindow such that any two edges in the time window are at most $\\delta$ time\napart.  The auxiliary counters now keep track of only the subsequences occurring\nwithin the current time window.  Last, it is important to note that the\nalgorithm only \\emph{counts} the number of instances of motifs rather\nthan \\emph{enumerating} them.  This is critical to making the algorithm fast.\n\n\\input{CH4-ALG-general}\n\nFor simplicity of presentation, \\cref{alg:general} counts \\emph{all} possible\n$l$-edge motifs that occur in a given sequence of edges.  The data structure\n$\\textnormal{counts}[\\cdot]$ maintains auxiliary counts of all (ordered) patterns of length\nat most $l$. Specifically, $\\textnormal{counts}[e_1\\cdots e_r]$ is the number of times the\nsubsequence $[e_1\\cdots e_r]$ occurs in the current time window (if $r < l$) or\nthe number of times the subsequence has occurred within all time windows of\nlength $\\delta$ (if $r = l$).  Crucially, we also assume the keys of $\\textnormal{counts}[\\cdot]$ are\naccessed in order of length.  Moving the time window forward by adding a new\nedge into the window, all edges $(e = (u, v), t)$ farther than $\\delta$ time\nfrom the new edge are removed from the window and the appropriate counts are\ndecremented (the \\emph{DecrementCounts()} method)---first, the single edge\ncounts ($[e]$) are updated. Based on these updates, length-$2$ subsequences\nformed with $e$ as its first edge are updated and so on, up through\nlength-($l-1$) subsequences. On the other hand, when an edge $e$ is added to the\nwindow, similar updates take place, but in reverse order, from longest to\nshortest subsequences, in order to increment counts in subsequences where $e$ is\nthe last edge (the \\emph{IncrementCounts()} method). Importantly, length-$l$\nsubsequence counts are incremented in this step but never decremented.  As the\ntime window moves from the beginning to the end of the sequence of edges, the\nalgorithm accumulates counts of all length-$l$ subsequences in all possible time\nwindows of length $\\delta$.\n\n\\input{CH4-TAB-general-algorithm-example}\n\nWhile \\cref{alg:general} maintains counts for all $l$-edge motifs, we could\nsimply maintain the $\\textnormal{counts}[\\cdot]$ data structure for the contiguous prefixes\nof the motif $M$.  As an example, \\cref{tab:general_alg_example} shows the\nexecution of \\cref{alg:general} for a particular sequence of edges and a particular\nmotif, where counts of only the necessary contiguous subsequences of the motif are maintained.\nIn general, there are $O(l^2)$ contiguous subsequences of an $l$-edge motif $M$,\nand there are $O(\\lvert H \\rvert^l)$ total keys in $\\textnormal{counts}[\\cdot]$, where\n$\\lvert H \\rvert$ is the number of edges in the static subgraph $H$ induced by\n$M$, in order to count all $l$-edge motifs in the sequence (i.e., not just motif\n$M$).\n\nWe now analyze the complexity of the overall 3-step algorithm.  We assume that\nthe temporal graph $T$ has edges sorted by timestamps, which is reasonable if\nedges are logged in their order of occurrence, and we also assume that we have pre-processed\n$T$ so that we can access the sorted list of all edges between $u$ and $v$ in\n$O(1)$ time.  Given an instance $H'$ of $H$, constructing the time-sorted\nsequence $S$ in step 2 of the algorithm then takes\n$O(\\log({\\lvert H \\rvert}) {\\lvert S \\rvert})$\ntime (by merging sorted lists).  Each edge inputted to\n\\cref{alg:general} is processed exactly twice: once to increment counts when\nit enters the time window and once to decrement counts when it exits the time\nwindow.  As presented in \\cref{alg:general}, each update changes $O(\\vert H \\vert^l)$\ncounters resulting in an overall complexity of $O(\\vert H \\vert^l {\\lvert S \\rvert})$.\nHowever, one could modify \\cref{alg:general} to only update counts for\ncontiguous subsequences of the sequence $M$, which would change $O(l^2)$\ncounters and have overall complexity $O(l^2 {\\lvert S \\rvert})$.  We are typically only\ninterested in small constant values of $\\vert H \\rvert$ and $l$ (for our\nexperiments in \\cref{sec:tm_experiments}, $\\lvert H \\rvert \\le 3$ and $l = 3$),\nin which case the running time of \\cref{alg:general} \nis linear in the size of the input to the algorithm, i.e., $O({\\lvert S \\rvert})$.\nWhat remains is to identify all instances $H'$ of $H$.  Of course, this depends\non the structure of $H$.  For example, if $H$ is an edge, this can be done in\nlinear time.  If $H$ is a triangle, this can be done in $O(a \\lvert G \\rvert)$ time,\nwhere $\\lvert G \\rvert$ is the number of edges in the static graph $G$ induced\nby $T$ and $a$ is the arboricity of $G$~\\cite{chiba1985arboricity}.\n\nIn the remainder of this section we analyze our 3-step algorithm with respect to\ndifferent types of motifs (2-node, stars, and triangles) and argue benefits and\ndeficiencies of the proposed framework.  We show that for $2$-node motifs, our\ngeneral counting framework takes time linear in the total number of edges $m$.\nSince all the input data needs to be examined for computing exact counts, this\nmeans the algorithm is optimal for $2$-node motifs. However, we also show that\nfor star and triangle motifs the algorithm is not optimal, which then motivates\nus to design faster algorithms in \\cref{sec:tm_faster_algs}.\n\n\\xhdr{General algorithm for 2-node motifs}\nWe first show how to map $2$-node motifs to the framework described above.  Any\ninduced graph $H$ of a $2$-node $\\delta$-temporal motif is either a single or a\nbidirectional edge.  In either case, it is straightforward to enumerate over all\ninstances of $H$ in the induced static graph.  This leads to the following procedure.\nFor each pair of nodes $u$ and $v$ for which there is at least one edge,\ngather and sort the edges in either direction between $u$ and $v$\nand call \\cref{alg:general} with these edges.  To obtain the total motif count,\nwe simply need to sum the results from each call to \\cref{alg:general}.\n\nWe only need to input each edge to \\cref{alg:general} once, and\nunder the assumption that we can\naccess the sorted directed edges from one node to another in $O(1)$ time, the\nmerging of edges into sorted order takes linear time (merging 2 sorted lists).\nTherefore, the total running time is $O(l^2m)$, where $l$ is the number\nof edges in the motif, which is linear in the number of temporal edges $m$.  \nWe are mostly interested in small patterns, i.e., cases\nwhen $l$ is a small constant.  Thus, this methodology is optimal (linear in the\ninput size, $m$) for counting $2$-node $\\delta$-temporal motif instances.\n\n\\xhdr{General algorithm for star motifs}\nNext, we consider $k$-node, $l$-edge star motifs $M$, whose induced static graph\n$H$ consists of a center node and $k-1$ neighbors, where edges may occur in\neither direction between the center node and a neighbor node.  For example, in\nthe top left corner of \\cref{fig:three_edge_motifs}, $M_{1,1}$ is a star motif\nwith all edges pointing toward the center node (the orange node).  In such\nmotifs, the induced static graph $H$ contains at most $2(k - 1) = 2k - 2$ static\nedges---one incoming and outgoing edge from the center node to each neighbor\nnode.  We have the following method for counting the number of instances of\n$k$-node, $l$-edge star motifs. For each node $u$ in the static graph and for\neach unique set of $k - 1$ neighbors, gather and sort the edges in either\ndirection between $u$ and the neighbors.  Then, count the number of instances of\n$M$ using \\cref{alg:general}.  The counts from each call to \\cref{alg:general}\nare summed over all center nodes and sets of $k - 1$ neighbors.\n\nThe major drawback of this approach is that we have to loop over each\nsize-$(k-1)$ neighbor set.  This can be prohibitively expensive even when $k =\n3$ if the center node has large degree.  In \\cref{sec:tm_faster_algs}, we will\ndesign an algorithm that avoids this issue for the case when the star motif has\n$k = 3$ nodes and $l = 3$ edges.\n\n\\input{CH4-FIG-three-edge-motifs.tex}\n\\clearpage\n\n\\xhdr{General algorithm for triangle motifs}\nWith triangle motifs, the induced graph $H$ consists of 3 nodes and at least one\ndirected edge between any pair of nodes (see \\cref{fig:three_edge_motifs}\nfor all eight of the $3$-edge triangle motifs).  The induced static graph $H$ of $M$\ncontains at least three and at most six static edges.  A straightforward\nalgorithm for counting $l$-edge triangle motifs in a temporal graph $T$ is:\n\\begin{enumerate}\n\\item Use a fast static graph triangle enumeration algorithm to find all\n  triangles in the static graph $G$ induced by $T$ (using, e.g.,\n  one of the methods analyzed by \\citet{latapy2008main}).\n\\item For each triangle, merge all temporal edges from each pair of nodes to\n  get a time-sorted list of edges.  Use \\cref{alg:general} to count the\n  number of instances of $M$.\n\\end{enumerate}\nThis approach is problematic as the edges between a pair of nodes may\nparticipate in many triangles. \\Cref{fig:worstcase} shows a worst-case example\nfor the motif $M = (w, u), (w, v), (u, v)$ with $\\delta = \\infty$.\nIn the figure, the timestamps are ordered by their index. There are\n$m-2n$ edges between $u$ and $v$, and each of these edges forms an instance of\n$M$ with every $w_i$. Thus, the overall worst-case running time of the algorithm\nis $O(\\Delta_{\\textnormal{enum}} + \\min(m\\tau, mn))$, where $\\Delta_{\\textnormal{enum}}$ is the time to enumerate\nthe number of triangles $\\tau$ in the static graph.  (The $mn$ running time comes\nfrom the case that each edge can participate in at most $n$ triangles).\nIn the following section, we devise an algorithm that significantly reduces the dependency on $\\tau$ from\nlinear to sub-linear (specifically, to $O(\\Delta_{\\textnormal{enum}} + \\min(m\\sqrt{\\tau}, mn)$)\nwhen there are $l = 3$ edges.\n\n\\input{CH4-FIG-worstcase}\n\n\n\\subsection{Faster algorithms}\\label{sec:tm_faster_algs}\n\nThe general counting algorithm from the previous section counts the number of\ninstances of any $k$-node, $l$-edge $\\delta$-temporal motif, and is also optimal\nfor $2$-node motifs.  However, the computational cost may be expensive for other\nmotifs such as stars and triangles.  We now develop specialized algorithms that\ncount certain motif classes faster.  Specifically, we design faster algorithms\nfor counting all 3-node, 3-edge star and 3-node, 3-edge triangle motifs\n(\\cref{fig:three_edge_motifs} illustrates these motifs).  Our algorithm for\nstars is linear in the input size, so it is optimal up to constant factors.\n\n\n\n\\xhdr{Fast algorithm for 3-node, 3-edge stars}\nWith $3$-node, $3$-edge star motifs, the drawback of the previous algorithmic\napproach is the need to loop over all pairs of neighbors given a center node.\nInstead, we will count all instances of star motifs for a given center node in\njust a single pass over the edges adjacent to the center node.\n\n\\input{CH4-FIG-star-classes}\n\nWe use a dynamic programming approach for counting star motifs.  First, note\nthat every temporal edge in a star with center $u$ is defined by\n\\begin{enumerate} \n\\item a neighbor node,\n\\item a direction of the edge (outward from or inward toward $u$), and\n\\item the timestamp.\n\\end{enumerate}\nWith this characterization, there are 3 classes of star motifs with 3 nodes and 3 edges\n(illustrated in \\cref{fig:star_classes}).\n\\begin{enumerate}\n\\item ``pre'' -- the first two edges contain node $u$ and neighbor $v$, and the third\nedge contains node $u$ and a different neighbor $w$.\n\\item ``post'' -- the last two edges contain node $u$ and neighbor $w$, and the first\nedge contains node $u$ and a different neighbor $v$.\n\\item ``peri'' -- the first and third edges contain node $u$ and neighbor $v$, and the second\nedge contains node $u$ and a different neighbor $w$.\n\\end{enumerate}\nEach class consists of $2^3 = 8$ motifs corresponding to the different combinations\nof the direction of the edges from $u$ to the neighbors (see \\cref{fig:three_edge_motifs}, which shows all of the $3 \\cdot 8 = 24$ 3-node, 3-edge stars).\n\nNow, suppose we process the time-ordered sequence of all edges containing the center\nnode $u$.  We maintain the following counters when processing an edge with\ntimestamp $t_j$:\n\\begin{itemize}\n\\item $\\textnormal{pre\\_sum}[\\dir1, \\dir2]$ is the number of sequentially ordered pairs of\n  edges in $[t_j - \\delta, t_j)$ where the first edge points in direction\n    $\\dir1$ and the second edge points in direction $\\dir2$.  We can think of a\n    direction as being outward from $u$ or inward toward $u$.\n    \n\\item $\\textnormal{post\\_sum}[\\dir1, \\dir2]$ is the number of sequentially ordered\npairs of edges in $(t_j, t _j + \\delta]$ where the first edge points in\ndirection $\\dir1$ and the second edge points in direction $\\dir2$.\n\n\\item $\\textnormal{peri\\_sum}[\\dir1, \\dir2]$ is the number of pairs of edges where the first\n  edge is in direction $\\dir1$ and occurred at time $t < t_j$, the second edge\n  is in direction $\\dir2$ and occurred at time $t' > t_j$, and $t' - t \\le \\delta$.\n  \n\\end{itemize}\n\nIf we are currently processing an edge, the ``pre'' class gets\n$\\textnormal{pre\\_sum}[\\dir1, \\dir2]$ new motif instances for any choice of directions $\\dir1$\nand $\\dir2$ (specifying the first two edge directions) and the current edge\nserves as the third edge in the motif (hence specifying the third edge\ndirection).  Similar updates are made with the $\\textnormal{post\\_sum}[\\cdot,\\cdot]$ and\n$\\textnormal{peri\\_sum}[\\cdot,\\cdot]$ counters, where the current edge serves as the first\nor second edge in the motif, respectively.\n\nIn order for our algorithm to be fast, we must be able to efficiently update\nthese counters when processing edges.  To aid in this, we introduce two\nadditional counters:\n\\begin{itemize}\n\\item $\\textnormal{pre\\_nodes}[\\textnormal{dir}, v_i]$ is the number of times node $v_i$ has appeared\n  in an edge with $u$ with direction $\\textnormal{dir}$ in the time window $[t_j - \\delta, t_j)$.\n  We can think of the direction as being outward from $u$ to $v_i$ or\n  inward from $v_i$ toward $u$.\n  \n\\item $\\textnormal{post\\_nodes}[\\textnormal{dir}, v_i]$ is the number of times node $v_i$ has appeared\n  in an edge with $u$ with direction $\\textnormal{dir}$ in the time window $(t_j, t_j + \\delta]$.\n\\end{itemize}\n\nFollowing the ideas of \\cref{alg:general}, it is easy to update these counters\nwhen we process a new edge.  Consequently, $\\textnormal{pre\\_sum}[\\cdot,\\cdot]$,\n$\\textnormal{post\\_sum}[\\cdot,\\cdot]$, and $\\textnormal{peri\\_sum}[\\cdot,\\cdot]$ can be maintained when\nprocessing an edge with just a few simple primitives:\n\\begin{itemize}\n\\item\n  \\emph{Push()} and \\emph{Pop()} update the counts for\n  $\\textnormal{pre\\_nodes}[\\cdot,\\cdot]$, $\\textnormal{post\\_nodes}[\\cdot,\\cdot]$,\n  $\\textnormal{pre\\_sum}[\\cdot,\\cdot]$ and $\\textnormal{post\\_sum}[\\cdot,\\cdot]$ when edges enter and leave\n  the time windows $[t_j - \\delta, t_j)$ and $(t_j, t_j + \\delta]$.\n\\item\n  \\emph{ProcessCurrent()} updates motif counts involving the current edge and\n  updates the counter $\\textnormal{peri\\_sum}[\\cdot,\\cdot]$.\n\\end{itemize}\n\nWe describe the general procedure in \\cref{alg:fast_framework}, which calls the\nsubroutines \\emph{Push()}, \\emph{Pop()}, and\n\\emph{ProcessCurrent()}, and \\cref{alg:fast_wedges} implements\nthese subroutines.  The $\\textnormal{pre\\_count}[\\cdot,\\cdot,\\cdot]$,\n$\\textnormal{post\\_count}[\\cdot,\\cdot,\\cdot]$, and $\\textnormal{peri\\_count}[\\cdot,\\cdot,\\cdot]$ counters\nin \\cref{alg:fast_wedges} maintain the counts of the three different classes of\nstars described above.  The edges input to each call of \\cref{alg:fast_wedges}\nassumes that an edge consists of the following two pieces of information:\n\\begin{enumerate}\n \\item the neighbor node $\\textnormal{nbr}$ of $u$ and\n \\item the direction $\\textnormal{dir}$ of the edge.\n\\end{enumerate}\nThe timestamps are all handled in \\cref{alg:fast_framework}.\n\nWe have intentionally separated the logic of\nthe \\emph{Push()}, \\emph{Pop()}, and \\emph{ProcessCurrent()}\nfrom \\cref{alg:fast_framework}.  The reason for this is that our fast $3$-node,\n$3$-edge triangle counting procedure follows the same logic\nof \\cref{alg:fast_framework}, only with different implementations of these\nsubroutines.  We will get to this idea shortly.\n\nFinally, we note that our counting scheme incorrectly includes instances of\n$2$-node motifs such as $M = (u, v_i)$, $(u, v_i)$, $(u, v_i)$, but we can use\nour fast $2$-node motif counting algorithm to account for this.  Putting\neverything together, we have the following procedure:\n\\begin{enumerate}\n\\item For each node $u$ in the temporal graph $T$, get a time-ordered\n  list of all edges containing $u$.\n  \n\\item Use \\cref{alg:fast_framework,alg:fast_wedges} to count\n  star motif instances.\n  \n\\item For each neighbor $v$ of a star center $u$, subtract the 2-node motif\n  counts using \\cref{alg:general}.\n\\end{enumerate}\n\nIf the $m$ temporal edges of $T$ are time-sorted, the first step takes linear\ntime.  The second and third steps also run in linear time in the input size.\nEach edge is used in steps (ii) and (iii) exactly twice: once for each end point\nas the center node.  Thus, the overall complexity of the algorithm is $O(m)$.\nThis is optimal (linear in the size of the dataset).  We also need $O(n)$\nmemory (in addition to storing the graph) to maintain the counters in \\cref{alg:fast_framework}.\n\n\\input{CH4-ALG-fast_framework}\n\n\\input{CH4-ALG-wedges}\n\\clearpage\n\n\\xhdr{Fast algorithm for 3-edge triangle motifs}\nWhile our fast 3-node, 3-edge star counting routine relied on counting motif instances for all\nedges adjacent to a given \\emph{node}, our fast triangle algorithm is based on\ncounting instances for all edges adjacent to a given \\emph{pair of nodes}.\nSpecifically, given a pair of nodes $u$ and $v$ and a list of common neighbors\n$w_1, \\ldots, w_d$, we count the number of motif instances for triangles $(w_i, u, v)$,\n$i = 1, \\ldots, d$.  Given all of the temporal edges in these $d$ static triangles,\nthe counting procedures are nearly identical to the case of stars.\nWe use the same general counting method (\\cref{alg:fast_framework}), but the behavior of the\nsubroutines \\emph{Push()}, \\emph{Pop()}, and \\emph{ProcessCurrent()} depends on\nwhether or not the edge is between $u$ and $v$.\n\n\\input{CH4-ALG-triad_subroutine}\n\nThese methods are implemented in \\cref{alg:fast_triangles}.  The input\nis a list of edges adjacent to a given pair of neighbors $u$ and $v$, where each\nedge consists of three pieces of information:\n\\begin{enumerate}\n \\item a neighbor node $\\textnormal{nbr}$,\n \\item an indicator of whether or not the node $\\textnormal{nbr}$ connects to node $u$ or node $v$, and\n \\item the direction $\\textnormal{dir}$ of the edge.\n\\end{enumerate}\n\nThe node counters ($\\textnormal{pre\\_nodes}[\\cdot,\\cdot,\\cdot]$ and\n$\\textnormal{post\\_nodes}[\\cdot,\\cdot,\\cdot]$) in \\cref{alg:fast_triangles} have an extra\ndimension compared to \\cref{alg:fast_wedges} to indicate whether the counts\ncorrespond to edges containing node $u$ or node $v$ (denoted by ``$\\textnormal{u\\_or\\_v}$'').\nFor example, $\\textnormal{pre\\_nodes}[u, \\textnormal{dir}, w_i]$ is the number of times node $w_i$ has\nappeared in an edge with $u$ with directed $\\textnormal{dir}$ (incoming to $u$ or outgoing\nfrom $u$) in the time window $[t_j - \\delta, t_j)$.  Similarly, the sum counters\n($\\textnormal{pre\\_sum}[\\cdot,\\cdot,\\cdot]$, $\\textnormal{peri\\_sum}[\\cdot,\\cdot,\\cdot]$ and\n$\\textnormal{post\\_sum}[\\cdot,\\cdot,\\cdot]$) have an extra dimension to denote if the first\nedge is incident on node $u$ or node $v$.  For example,\n$\\textnormal{pre\\_sum}[u, \\dir1, \\dir2]$ is the number of sequentially ordered pairs of edges\ncontaining some node $w_i$ in the time window $[t_j - \\delta, t_j)$, where the\nfirst edge is adjacent to $u$ in direction $\\dir1$ and the second edge is\nadjacent to $v$ in direction $\\dir2$.\n\n\\input{CH4-ALG-triad_main}\n\\clearpage\n\nRecall that the problem with counting triangle motifs by the general framework\nin \\cref{alg:general} is that a pair of nodes with many edges might have to be\ncounted for many triangles in the graph.  However, with\n\\cref{alg:fast_triangles}, we can simultaneously count all triangles\nadjacent to a given pair of nodes.  What remains is that we must assign each\ntriangle in the static graph to a pair of nodes.  Here, we propose to assign\neach triangle to the pair of nodes in that triangle containing the largest\nnumber of edges (this is sketched in \\cref{alg:triangle}).  The core\nidea of this assignment procedure is that we should simultaneously\nprocess as many triangles as possible for pairs of nodes with many\nedges. The following theorem says that this is faster than simply counting for\neach triangle (described in \\cref{sec:tm_general_framework}).  Specifically, we\nreduce $O(\\Delta_{\\textnormal{enum}} + \\min(m\\tau, mn))$ complexity to $O(\\Delta_{\\textnormal{enum}} + \\min(m\\sqrt{\\tau}, mn))$, where\n$\\tau$ is the number of triangles in the graph.\n\n\\begin{theorem}\\label{thm:fast_triangles}\nIn the worse case, \\cref{alg:triangle} runs in time\n$O(\\Delta_{\\textnormal{enum}} + \\min(m\\sqrt{\\tau}, mn))$\nwhere $\\Delta_{\\textnormal{enum}}$ is the time to enumerate all triangles in the static graph\n$G$, $m$ is the total number of temporal edges, $n$ is the number of nodes, and\n$\\tau$ is the number of static triangles in $G$.\n\\end{theorem}\n\\begin{proof}\n\nLet $\\rho_e$ be the number of temporal edges along the static edge $e$, and\nlet $z_e \\ge 1$ be the number of times that timestamps along this static edge\nare used in a call to \\cref{alg:fast_triangles} by\n\\cref{alg:triangle}.  Since \\cref{alg:fast_triangles} runs in\nlinear time in the number of edges in its input, the total running time is on\nthe order of $\\sum_{e}\\rho_ez_e$.  Each static edge appears in at\nmost $n$ triangles and has at most $m$ temporal edges, so $\\sum_{e}\\rho_ez_e \\le mn$.\nWe now consider the case when $\\sqrt{\\tau} \\le n$.\n\nThe $\\rho_e$ are fixed, and we wish to find the values of $z_e$ that maximize\nthe summation.  Without loss of generality, write $\\{\\rho_e\\}$ by\n$\\rho_1 \\ge \\rho_2 \\ge \\ldots \\ge \\rho_r$, where $r$ is the number of\nstatic edges in the static graph induced by the temporal graph.  Consequently,\n$z_i \\le i$, $i = 1, \\ldots, r$.  There are $\\tau$ static triangles,\nso \\cref{alg:fast_triangles} will look at the timestamps along $3\\tau$ static\nedges.  Thus, $\\sum_{i}z_i \\le 3\\tau$.  The summation\n$\\sum_{i=1}^{r}\\rho_iz_i$ is maximized when $z_1 = 1$, $z_2 = 2$, and so on up\nto some index $J = O(\\sqrt{\\tau})$ for which\n$\\sum_{i=1}^{J}z_i = \\sum_{i=1}^{J}i = 3\\tau$.\nNow given that the $z_i$ are fixed and the $\\rho_i$\nare ordered, the summation is maximized when\n$\\rho_1 = \\rho_2 = \\ldots = \\rho_j = m \/ j$.\nIn this case,\n\\[\n\\sum_{i}\\rho_iz_i \n= \\sum_{i=1}^{J}\\frac{m}{J} \\cdot i \n= \\frac{m}{J}\\sum_{i=1}^{J} \n= O\\left(\\frac{m}{J} \\cdot J^2\\right) \n= O(m\\sqrt{\\tau}).\n\\]\n\\end{proof}\n\n\\input{CH4-FIG-fast-worstcase}\n\nWe now show a worst-case example for the algorithm, which\nis illustrated in \\cref{fig:fast_worstcase}.\nThe graph consists of $k + 1$ nodes $u$, $v_1$, $\\ldots$, $v_k$, and the\nstatic edges are $(v_i, v_j)$, $1 \\le i < j \\le k$, and $(v_r, u)$, $1 \\le r \\le k$.\nEach edge $(v_i, v_j)$ has a single temporal timestamp, and each edge $(v_i, u)$\nhas an equal number of timestamps.  Without loss of generality, the first\n$k$ edges in the ordering of the static edges is $\\rho_1 = (v_1, u)$, $\\rho_2 = (v_2, u)$,\n$\\ldots$, $\\rho_k = (v_k, u)$ (the remainder of the ordering is arbitrary).\nFinally, suppose that we are counting the motif $M = (a, b), (a, c), (b, c)$.  \n\nIn this case, the call to the \\cref{alg:fast_triangles} subroutine for edge $(v_r, u)$ ($1 \\le r \\le k$)\nin \\cref{alg:triangle} processes temporal edges along $(v_r, v_2)$, $\\ldots$, $(v_r, v_k)$\nand $(v_r, u)$, $\\ldots$, $(v_k, u)$.  Each temporal edge along $(v_i, v_j)$ is processed\nonly once, and the temporal edges along $(v_r, u)$ are processed $r$ times.  Thus, the\ntotal number of temporal edges processed is\n\\begin{align*}\n{k \\choose 2} + \\sum_{r = 1}^{k}r\\frac{m - {k \\choose 2}}{k}\n= O(k^2) + O(mk)\n= O(\\tau + m\\sqrt{\\tau}).\n\\end{align*}\n\nIn this case, \\cref{alg:general} has the same asymptotic running time.  Each of\nthe $O(k^2)$ static triangles contains $O(m \/ k)$ temporal edges.\n\nFinally, we emphasize that, if we ignore the pre-processing time in \\cref{alg:fast_triangles},\nthe amount of computation in \\cref{alg:fast_triangles}\nis always less than the computation done by \\cref{alg:general}\nfor counting triangles.  Thus, regardless of the asymptotics,\n\\cref{alg:fast_triangles} is still a strict improvement over the general algorithm.\n\n\n\\subsection{Data}\\label{sec:tm_data}\n\n\\input{CH4-TAB-network-descriptions}\n\nWe gathered a variety of datasets in order to study the patterns of\n$\\delta$-temporal motifs in several domains.  The datasets are described below\nand summary statistics are in \\cref{tab:datasets}.  The time resolution of the\nedges in all datasets is one second.\n\n\\begin{itemize}\n\\item $\\dataset{email-Eu}$\\footnote{This is the same network as $\\dataset{email-Eu-core}$\nin \\cref{sec:local_real}, but we are now using the temporal information.}\nis a collection of emails between members of a European\nresearch institution~\\cite{leskovec2007graph}.  An edge $(u, v, t)$ signifies\nthat person $u$ sent person $v$ an email at time $t$.\n\n\\item $\\dataset{Phonecall-Eu}$ was constructed from telephone call records for a major European\nservice provider.  An edge $(u, v, t)$ signifies that person $u$ called person\n$v$ starting at time $t$.\n\n\\item $\\dataset{SMS-A}$ is a collection of short messaging service (SMS) texting\nrecords from a company charging account~\\cite{wu2010evidence}.\nIn this dataset, an edge $(u, v, t)$ means that person $u$ sent\nan SMS message to person $v$ at time $t$.\n\n\\item $\\dataset{CollegeMsg}$ is comprised of private messages sent on an online social network\nat the University of California, Irvine~\\cite{panzarasa2009patterns}.  Users\ncould search the network for others and then initiate conversation based on\nprofile information.  An edge $(u, v, t)$ means that user $u$ sent a private\nmessage to user $v$ at time $t$.\n\n\\item $\\dataset{StackOverflow}$ is constructed from communication on Stack Overflow.\nOn stack exchange web sites, users post questions and receive answers from other\nusers, and users may comment on both questions and answers.  We derive a\ntemporal network by creating an edge $(u, v, t)$ if, at time $t$, user $u$: (1)\nposts an answer to user $v$'s question, (2) comments on user $v$'s question, or\n(3) comments on user $v$'s answer.  We formed the temporal network from the\nentirety of Stack Overflow's history up to March 6, 2016.\n\n\\item $\\dataset{Bitcoin}$ consists of all payments made with the decentralized\ndigital currency and payment system Bitcoin up to October 19, 2014~\\cite{kondor2014rich}.\nNodes in the network correspond to Bitcoin addresses, and an individual may have\nseveral addresses.  An edge $(u, v, t)$ signifies that bitcoin was transferred\nfrom address $u$ to address $v$ at time $t$.  \n\n\\item $\\dataset{FBWall}$ consists of wall posts between users on the social network\nFacebook located in the New Orleans region~\\cite{viswanath2009evolution}.  Any friend of\na given user can see all posts on that user's wall, so communication is public\namong friends.  An edge $(u, v, t)$ means that user $u$ posted on user $v$'s\nwall at time $t$.\n\n\\item $\\dataset{WikiTalk}$ represents edits on user talk pages on\nWikipedia~\\cite{leskovec2010governance}.  An edge $(u, v, t)$ means that\nuser $u$ edited user $v$'s talk page at time $t$.\n\n\\item $\\dataset{Phonecall-ME}$ is constructed from phone call records of a large\ntelecommunications service provider in the Middle East.  An edge $(u, v, t)$ \nmeans that user $u$ initiated a call to user $v$ at time $t$.\n\n\\item $\\dataset{SMS-ME}$ is constructed from SMS texting records from the same\ntelecommunications service provider in the Middle East.\nAn edge $(u, v, t)$ means that user $u$ sent an\nSMS message to user $v$ at time $t$.\n\\end{itemize}\n\n\\subsection{Empirical observations of motif counts}\\label{sec:tm_empirical}\n\nWe first examine the distribution of 2- and 3-node, 3-edge motif instance counts\nfrom 8 of the datasets described in \\cref{sec:tm_data} with $\\delta = 1$ hour\n(\\cref{fig:raw_counts}).  We choose 1 hour for the time window as this is close\nto the median time for a node to take part in three edges in most of our\ndatasets. We make a few empirical observations uniquely available due to\ntemporal motifs and provide possible explanations for these observations.\n\n\\input{CH4-FIG-raw-counts}\n\\clearpage\n\n\n\\input{CH4-FIG-blocking}\n\\xhdr{Blocking communication}\nIf an individual typically waits for a reply from one individual before\nproceeding to communicate with another individual, we consider it a\n\\emph{blocking} form of communication. A typical conversation between two\nindividuals characterized by fast exchanges happening back and forth is blocking\nas it requires complete attention of both individuals. We capture this\nbehavior in the ``blocking motifs'' $M_{5,1}$, $M_{5,2}$ and $M_{6,2}$, which\ncontain 3 edges between two nodes with at least one edge in either direction\n(\\cref{fig:blocking}, left).  However, if the reply doesn't arrive soon,\nwe might expect the individual to communicate with others without waiting for a\nreply from the first individual. This is a non-blocking form of communication\nand is captured by the ``non-blocking motifs'' $M_{4,1}$, $M_{4,3}$ and $M_{6,3}$\nhaving edges originating from the same source but directed to different\ndestinations (\\cref{fig:blocking}, right)\n\nThe fractions of counts corresponding to the blocking and non-blocking motifs\nout of the counts for all 36 motifs in \\cref{fig:three_edge_motifs}\nuncover several interesting characteristics in communication networks ($\\delta =\n1$ hour; see \\cref{fig:blocking}). In $\\dataset{FBWall}$ and $\\dataset{SMS-A}$, blocking\ncommunication is vastly more common, while in $\\dataset{email-Eu}$ non-blocking\ncommunication is prevalent.  Email is not a dynamic method of\ncommunication and replies within an hour are rare.  Thus, we would expect\nnon-blocking behavior.  Interestingly, the $\\dataset{CollegeMsg}$ dataset shows both\nbehaviors as we might expect individuals to engage in multiple conversations\nsimultaneously.  In complete contrast, the $\\dataset{Phonecall-Eu}$ dataset shows neither\nbehavior.  A simple explanation is that that a single edge (a phone call)\ncaptures an entire conversation and hence blocking behavior does not emerge.\n\n\\input{CH4-FIG-switching}\n\\xhdr{Cost of switching}\nAmongst the non-blocking motifs discussed above, $M_{4,1}$ captures two\nconsecutive ``switches'' between pairs of nodes whereas $M_{4,3}$ and $M_{6,3}$ each\nhave a single switch (\\cref{fig:switching}, right). In communication networks, a switch\ncorresponds to a change in message destination for a node $u$.  Prevalence of\n$M_{4,1}$ indicates a lower cost of switching targets, whereas prevalence of the\nother two motifs are indicative of a higher cost.  We observe in\n\\cref{fig:switching} that the ratio of 2-switch to 1-switch motif counts\nis the least in $\\dataset{StackOverflow}$, followed by $\\dataset{WikiTalk}$, $\\dataset{CollegeMsg}$ and then\n$\\dataset{email-Eu}$. On Stack Overflow and Wikipedia talk pages, there is a high\ncost to switch targets because of peer engagement and depth of discussion.  On the other\nhand, in the $\\dataset{CollegeMsg}$ dataset there is less cost to switch because it\nlacks depth of discussion within the time frame of $\\delta = $ 1 hour.\nIn $\\dataset{email-Eu}$, there is almost no peer engagement, and cost of switching is\nnegligible.\n\n\\input{CH4-FIG-cycles}\n\\xhdr{Cycles in $\\dataset{Bitcoin}$}\nOf the eight $3$-edge triangle motifs, $M_{2,4}$ and $M_{3,5}$ are cyclic, i.e.,\nthe target of each edge serves as the source of another edge.\\footnote{By ``cyclic''\nwe do not mean that the temporal edges must form a cycle.  Rather, the multigraph\n$K$ in the formal motif definition is a cycle.}  We observe in\n\\cref{fig:cycles} that the fraction of triangles that are cyclic is much higher in\n$\\dataset{Bitcoin}$ compared to any other dataset.\nThis can be attributed to the transactional nature of $\\dataset{Bitcoin}$ where\nthe total amount of bitcoin is limited. Since remittance (outgoing edges) is typically associated\nwith earnings (incoming edges), we should expect cyclic behavior.\n\n\n\n\\input{CH4-FIG-stackoverflow-time}\n\\xhdr{Motif counts at varying time scales}\nWe now explore how motif counts change at different time scales.  For the\n$\\dataset{StackOverflow}$ dataset we counted the number of instances of $2$- and\n$3$-node, $3$-edge $\\delta$-temporal motifs for $\\delta = $ 60, 300, 1800, and\n3600 seconds (\\cref{fig:so_counts_over_time}).  These counts determine the\nnumber of motifs that completed in the intervals\n[0, 60], (60, 300], (300, 1800s], and (1800, 3600] seconds\n(e.g., subtracting 60 second counts from 300 second counts gives the interval (60, 300]).\nObservations at smaller timescales reveal phenomenon which start to get eclipsed\nat larger timescales.  For instance, on short time scales, motif $M_{1,1}$\n(\\cref{fig:so_counts_over_time}, top-left corner) is quite common.  We\nsuspect this arises from multiple, quick comments on the original question, so\nthe original poster receives many incoming edges.  At larger time scales, this\nbehavior is still frequent but relatively less so.  Now let us compare counts\nfor $M_{1,5}$, $M_{1,6}$, $M_{2,5}$, $M_{2,6}$ (the four in the top right\ncorner) with counts for $M_{3,3}$, $M_{3,4}$, $M_{4,3}$, $M_{4,4}$ (the four in\nthe center). The former counts likely correspond to conversations with the\noriginal poster while the latter are constructed by the same user interacting with\nmultiple questions.  Between 300 and 1800 seconds (5 to 30 minutes),\nthe former counts are relatively more common while the latter counts only become\nmore common after 1800 seconds. A possible explanation is that the typical\nlength of discussions on a post is about 30 minutes, and later on, users answer\nother questions.\n\n\\input{CH4-FIG-collegemsg}\n\nNext, we examine messaging behavior in the $\\dataset{CollegeMsg}$ dataset at fine-grained\ntime intervals.  We counted the number of motifs consisting of a single node\nsending three outgoing messages to one or two neighbors (motifs $M_{6,1}$,\n$M_{6,3}$, $M_{4,1}$, and $M_{4,3}$) in the time bins $[10(i-1), 10i)$ seconds,\n$i = 1, \\ldots, 500$ (\\cref{fig:IM_over_time}).  We first notice that at\nsmall time scales, the motif consisting of three edges to a single neighbor\n($M_{6,1}$) occurs frequently.  This pattern could emerge from a succession of\nquick introductory messages.  Overall, motif counts increase from roughly 1\nminute to 20 minutes and then decline.  After 5 minutes, counts\nfor the three motifs with one switch in the target ($M_{6,1}$, $M_{6,3}$, and\n$M_{4,3}$) grow at a faster rate than the counts for the motif with two switches\n($M_{4,1}$).  As mentioned above, this pattern could emerge from a tendency to\nsend several messages in one conversation before switching to a conversation\nwith another friend.\n\n\\subsection{Algorithm scalability}\\label{sec:tm_scalability}\n  \nFinally, we performed scalability experiments of our algorithms.  All algorithms\nwere implemented in C++, and all experiments ran using a single thread of a\n2.4GHz Intel Xeon E7-4870 processor.  We did not measure the time to load\ndatasets into memory, but our timings include all pre-processing time needed by\nthe algorithms (e.g., the triangle counting algorithms first find triangles in\nthe static graph).  We emphasize that our implementation is single threaded, and\nthere is ample room for parallelism in our algorithms.\n\n\\input{CH4-TAB-triangle-speedups}\n\nFirst, we used both the general counting method (\\cref{alg:general}) and the\nfast counting method (\\cref{alg:triangle}) to count the number of all eight\n3-edge $\\delta$-temporal triangle motifs in our datasets ($\\delta =$ 1 hour).\n\\Cref{tab:scalability} reports the running times of the algorithms for all\ndatasets with at least one million triangles in the static graph. For all of\nthese datasets, our fast temporal triangle counting algorithm provides\nsignificant performance gains over the general counting method, ranging between\na 1.29x and a 56.5x speedup.  The gains of the fast algorithm are the largest\nfor $\\dataset{Bitcoin}$, which is due to some pairs of nodes having many edges between\nthem and also participating in many triangles.\n\n\\input{CH4-FIG-scalability}\n\nSecond, we measured the time to count various $3$-edge $\\delta$-temporal motifs\nin our largest dataset, $\\dataset{Phonecall-ME}$.  Specifically, we measured the time to\ncompute (1) $2$-node motifs, (2) $3$-node stars, and (3) triangles on the first\n$k$ million edges in the dataset for $k = 250, 500, \\ldots, 2000$\n(\\cref{fig:scalability}).  The time to compute the $2$-node, $3$-edge motifs and\nthe $3$-node, $3$-edge stars scales linearly, as expected from our complexity\nanalysis.  The time to count triangle motifs grows superlinearly and becomes the\ndominant cost when there is a large number of edges.  For practical purposes,\nthe running times are quite modest.  With two billion edges, our methods take\nless than 3.5 hours to complete (executing sequentially).\n\n\n\n\n\\chapter{Motifs in temporal networks}\n\\label{ch:tm}\n\\chaptermark{Motifs in temporal networks}\n\n\\section{Analyzing network data with timestamped edges}\n\\label{sec:tm_introduction}\n\\input{CH4-TXT-introduction}\n\n\\section{Definitions of temporal networks and temporal motifs}\n\\label{sec:tm_preliminaries}\n\\input{CH4-TXT-preliminaries}\n\n\\section{Counting algorithms}\n\\label{sec:tm_algorithms}\n\\input{CH4-TXT-algorithms}\n\n\\section{Experiments}\n\\label{sec:tm_experiments}\n\\input{CH4-TXT-experiments}\n\n\\section{Prior definitions of temporal motifs and other related work}\n\\sectionmark{Prior definitions and related work}\n\\label{sec:tm_related}\n\\input{CH4-TXT-related}\n\n\\section{Discussion}\n\\label{sec:tm_discussion}\n\\input{CH4-TXT-discussion}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf Introduction }\n\\label{sec:introduction}\nIn recent years, random graph theory has been applied to model many complex real-world phenomena. A basic random graph used to model complex networks is the \\gls{ER} graph \\cite{erdos1959random}, where edges between the nodes appear with equal probabilities.  In \\cite{gilbert1961random}, the author introduces another random graph called \\gls{RGG} where nodes have some random position in a metric space and the edges are determined by the position of these nodes.  Since then, \\gls{RGG}  properties have been widely studied \\cite{penrose2003random}.\n\n \\glspl{RGG} are very useful to model problems in which the geographical distance is a critical factor.  For example, {\\glspl{RGG}} have been applied to wireless communication network \\cite{bettstetter2002minimum}, sensor network \\cite{yick2008wireless} and to study the dynamics of a viral spreading in a specific network of interactions  \\cite{preciado2009spectral}, \\cite{ganesh2005effect}.  Another motivation for \\glspl{RGG} in arbitrary dimensions is multivariate statistics of high-dimensional data. In this case, the coordinates of the nodes can represent the attributes of the data. Then, the metric imposed by the \\gls{RGG} depicts the similarity between the data.\n\nIn this work,  the \\gls{RGG} is constructed by considering a finite set $\\mathcal{X}_{n}$ of $n$ nodes, $x_{1},...,x_{n},$ distributed uniformly and independently on the $d$-dimensional torus $\\mathbb{T}^d \\equiv [0, 1]^d$. We choose a torus instead of a cube in order to avoid boundary effects. Given a geographical distance, $r_{n} >0 $, we form a graph by connecting two nodes $x_{i}, x_{j} \\in \\mathcal{X}_{n}$  if their $\\ell_{p}$-distance, $p \\in [1, \\infty]$ is at most $r_{n}$, i.e., $\\|x_{i}-x_{j} \\|_{p} \\leq r_{n}$, where $\\|.\\|_{p}$ is the $\\ell_{p}$-metric defined as\n\\begin{equation*}\n\\small\n\\label{eqq1}\n\\| x_{i}- x_{j} \\|_{p} =\\left\\{\n\\begin{array}{ll}\n \\left(  \\sum_{k=1}^d \\vert x_{i}^{(k)}-x_{j}^{(k)}\\vert^p \\right)^{1\/p} &   p \\in [1, \\infty),\\\\ \n \\max\\lbrace \\vert x_{i}^{(k)}-x_{j}^{(k)}\\vert, \\ k \\in [1, d]  \\rbrace & p=\\infty.\n    \\end{array}\n\\right. \n\\end{equation*}\n\nThe \\gls{RGG} is denoted by $G(\\mathcal{X}_{n}, r_{n})$.  Note that for the case $p=2$ we obtain the Euclidean metric on $\\mathbb{R}^d$.   Typically, the function $r_{n}$ is chosen such that $r_{n}\\rightarrow 0$ when $n \\rightarrow \\infty$. \n\n\nThe degree of a vertex in $G(\\mathcal{X}_{n}, r_{n})$ is the number of edges connected to it.  The average vertex degree in $G(\\mathcal{X}_{n}, r_{n})$ is given by \\cite{penrose2003random}\n\\begin{equation*}\na_{n} = \\theta^{(d)} nr_{n}^d,\n\\end{equation*} \nwhere $\\theta^{(d)}=\\pi^{d\/2}\/\\Gamma(d\/2+1)$ denotes the volume of the $d$-dimensional unit hypersphere in $\\mathbb{T}^d$ and $\\Gamma(.)$ is the Gamma function. \n\nDifferent values of $r_{n}$, or equivalently $a_{n}$, lead to different geometric structures in \\glspl{RGG}. In \\cite{penrose2003random}, different interesting regimes are introduced: the \\textit{connectivity regime} in which $a_{n}$ scales as $\\log(n)$ or faster, i.e., $\\Omega(\\log(n))$\\footnote{The notation $f(n) =\\Omega(g(n))$ indicates that $f(n)$ is bounded below by $g(n)$ asymptotically, i.e., $\\exists K>0$ and $ n_{o} \\in \\mathbb{N}$ such that $\\forall n > n_{0}$ $f(n) \\geq K g(n)$.},  the  \\textit{thermodynamic regime} in which $a_{n}\\equiv \\gamma$, for $\\gamma >0$ and the \\textit{dense regime}, i.e., $a_{n}\\equiv \\Theta(n).$ \n\n\n\\glspl{RGG} can be described by a variety of random matrices such as adjacency matrices, transition probability matrices and normalized Laplacian. The spectral properties of those random matrices are powerful tools to predict and analyze complex networks behavior.  In this work, we  give a special attention to the \\gls{test} of the adjacency matrix of \\glspl{RGG}  in the connectivity regime.\n\n\nSome works analyzed the spectral properties of \\glspl{RGG} in different regimes. In particular, in the thermodynamic regime, the authors in \\cite{bordenave2008eigenvalues}, \\cite{blackwell2007spectra} show that the spectral measure of the adjacency matrix of \\glspl{RGG} has a limit as $n \\to \\infty$. However, due to the difficulty to compute exactly this spectral measure,  Bordenave in \\cite{bordenave2008eigenvalues} proposes an approximation for it as $\\gamma \\rightarrow \\infty$.\n\n In the connectivity regime, the work in \\cite{preciado2009spectral} provides a closed form expression for the asymptotic spectral moments of the adjacency matrix of $G(\\mathcal{X}_{n}, r_{n})$. Additionnaly,  Bordenave in \\cite{bordenave2008eigenvalues} characterizes the spectral measure of the adjacency matrix normalized by $n$ in the dense regime. However, in the connectivity regime and as $n\\rightarrow \\infty$,  the normalization factor $n$ puts to zero all the eigenvalues of the adjacency matrix that are finite and only the infinite eigenvalues in the adjacency matrix are nonzero in the normalized adjacency matrix. Motivated by this results, in this work we analyze the behavior of the eigenvalues of the adjacency matrix  without normalization in the connectivity regime and in a wider range of the connectivity regime.\n\n\nFirst, we propose an approximation for the actual \\gls{test} of the \\gls{RGG}. Then, we provide a bound on the Levy distance between this approximation and the actual distribution. More precisely, for $\\epsilon>0$ we show that the \\glspl{test} of the adjacency matrices of the \\gls{RGG} and the \\gls{DGG} with nodes in a grid converge to the same limit when $a_{n}$ scales as $\\Omega (\\log^{\\epsilon}(n)\\sqrt{n})$ for $d=1$ and as $\\Omega (\\log^{2}(n))$ for $d\\geq 2$. \nThen, under the $\\ell_{\\infty}$-metric we provide an analytical approximation for the eigenvalues of the adjacency matrix of \\glspl{RGG} by taking the $d$-dimensional discrete Fourier transform (DFT) of an $n=\\mathrm{N}^{d}$ tensor of rank $d$ obtained from the first block row of the adjacency matrix of the \\gls{DGG}.\n\nThe rest of this paper is organized as follows.  In Section \\ref{sec:systemmodel} we describe the model, then we present our main results on the concentration of the \\gls{test} of large \\glspl{RGG} in  the connectivity  regime.  Numerical results are given in Section \\ref{sec:results} to validate the theoretical results. Finally, conclusions are given in Section \\ref{sec:conclusion}.\n\\begin{comment}\n\\begin{figure*}[!t]\n\\centering\n\\subfloat[Thermodynamic regime]{\\includegraphics[width=2.1in]{thermodynamic}%\n\\label{fig_first_case}}\n\\hfil\n\\subfloat[Connectivity regime]{\\includegraphics[width=2.1in]{connected}%\n\\label{fig_second_case}}\n\\caption{An illustration of an \\gls{RGG} on a torus with $n=200$ nodes with a connection radius $r_{n}$= $0.02$ in (a) and $r_{n}$= $0.1$ in (b). }\n\\label{fig_RGG1}\n\\end{figure*}\n\n\\end{comment}\n\\section{\\bf\\bf Spectral Analysis of \\glspl{RGG}}\n\\label{sec:systemmodel} \nTo study the spectrum of $G(\\mathcal{X}_{n}, r_{n})$ we introduce an auxiliary graph called the \\gls{DGG}. The \\gls{DGG} denoted by $G(\\mathcal{D}_{n}, r_{n})$ is formed by letting $\\mathcal{D}_{n}$ be the set of $n$ grid points that are at the intersections of axes parallel hyperplanes with separation $n^{-1\/d}$, and connecting two points $x'_{i}$, $x'_{j}$ $\\in \\mathcal{D}_{n}$ if $\\|x'_{i}-x'_{j} \\|_{p} \\leq r_{n}$ with $p\\in [1, \\infty]$. Given two nodes, we assume that there is always at most one edge between them. There is no edge from a vertex to itself. Moreover,  we assume that the edges are not directed. \n \n  Let $\\mathbf{A}(\\mathcal{X}_{n})$  be the  adjacency matrix of $G(\\mathcal{X}_{n}, r_{n})$, with entries \n\\begin{equation*}\n\\mathbf{A}(\\mathcal{X}_{n})_{i j}=\\chi [x_{i} \\sim x_{j}],\n\\label{RW}\n\\end{equation*}\nwhere the term $\\chi[x_{i}\\thicksim x_{j}] $ takes the value 1 when there is a connection between nodes $x_{i}$ and $x_{j}$ in $G(\\mathcal{X}_{n}, r_{n})$ and zero otherwise, represented as\n\\begin{equation*}\n\\label{eqq1}\n\\chi[x_{i}\\thicksim x_{j}] =\\left\\{\n    \\begin{array}{ll}\n        1, &  \\| x_{i} - x_{j}\\|_{p} \\leq r_{n}, \\ \\ i \\neq j,  \\ \\ p \\in [1, \\infty]\\\\\n       0, & \\mathrm{otherwise}.\n    \\end{array}\n\\right.\n\\end{equation*}\n\nA similar definition holds for $\\mathbf{A}(\\mathcal{D}_{n})$ defined over $G(\\mathcal{D}_{n}, r_{n})$. The matrices $\\mathbf{A}({\\mathcal{X}_{n}})$  and $\\mathbf{A}({\\mathcal{D}_{n}})$ are symmetric and their spectrum consists of real eigenvalues. We denote by $\\lbrace \\lambda_{i}, i=1,..,n \\rbrace$ and $\\lbrace \\mu_{i}, i=1,..,n \\rbrace$ the sets of all real eigenvalues of the real symmetric square matrices $\\mathbf{A}({\\mathcal{D}_{n}})$  and $\\mathbf{A}({\\mathcal{X}_{n}})$ of order $n$, respectively.\nThe empirical spectral distribution functions $v_{n}(x)$ and $v'_{n}(x)$ of the adjacency matrices of an \\gls{RGG} and a \\gls{DGG}, respectively  are defined as\n\\begin{equation*}\nv_{n}(x)=\\dfrac{1}{n} \\sum\\limits_{i=1}^n \\mathrm{I}\\lbrace \\mu_{i} \\leq x\\rbrace \\ \\ \\mathrm{and }\\  \\ \\ v'_{n}(x)=\\dfrac{1}{n} \\sum\\limits_{i=1}^n \\mathrm{I}\\lbrace \\lambda_{i}\\leq  x\\rbrace,\n\\end{equation*}\nwhere $\\mathrm{I}\\lbrace \\mathrm{B} \\rbrace$ denotes the indicator of an event $\\mathrm{B}$.\n\nLet $a'_{n}$ be the degree of the nodes in $G(\\mathcal{D}_{n}, r_{n})$. In the following Lemma \\ref{bound}  we provide an upper bound for $a'_{n}$ under any $\\ell_{p}$-metric.\n\n\\begin{lemma}\n\\label{bound}\n For any chosen $\\ell_{p}$-metric with $p \\in [1,\\infty]$ and $d\\geq1$, we have\n\\begin{equation*}\na'_{n} \\leq d^{\\frac{1}{p}}2^{d}a_{n}  \\left( 1+\\frac{1}{2a_{n}^{1\/d}}\\right)^d.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{app:bound}\n\\end{proof}\n\nTo prove our result on the concentration of the \\gls{test} of \\glspl{RGG} and investigate its relationship with the \\gls{test} of \\glspl{DGG} under any $\\ell_{p}$-metric, we use the Levy distance between two distribution functions defined as follows.\n \n\\begin{definition}(\\cite{taylor2012introduction}, page 257)\nLet $v_{n}^A$ and $v_{n}^B$ be two distribution functions on $\\mathbb{R}$. The Levy distance $L(v_{n}^A, v_{n}^B)$ between them is the infimum of all positive $\\epsilon$ such that, for all $x \\in \\mathbb{R}$ \n\\begin{equation*}\nv_{n}^A(x-\\epsilon) -\\epsilon \\leq v_{n}^B(x)\\leq v_{n}^A(x+\\epsilon)+\\epsilon.\n\\end{equation*}\n\\end{definition}\n\n\\begin{lemma}(\\cite{bai2008methodologies}, page 614)\n\\label{Difference Inequality}\nLet A and B be two $n$ $\\times$ $n$ Hermitian matrices with eigenvalues $\\lambda_{1},...,\\lambda_{n}$ and $\\mu_{1},...,\\mu_{n}$, respectively. Then\n\\begin{equation*}\nL^{3}(v_{n}^A, v_{n}^B) \\leqslant \\dfrac{1}{n}tr(A-B)^2,\n\\end{equation*}\nwhere $L(v_{n}^{A},v_{n}^{B})$ denotes the Levy distance between the empirical distribution functions $v_{n}^{A}$ and $v_{n}^{B}$ of the eigenvalues of $A$ and $B$, respectively.\n\\end{lemma}\n\n\nLet $\\mathrm{M}_{n}$ be the minimum bottleneck matching distance corresponding to the minimum length such that there exists a perfect matching of the random nodes to the grid points for which the distance between every pair of matched points is at most $\\mathrm{M}_{n}$.\n\nSharp bounds for $\\mathrm{M}_{n}$ are given in \\cite{shor1991minimax}\\cite{leighton1986tight}\\cite{goel2004sharp}. We repeat them in the following lemma for convenience.\n\\begin{lemma}\nUnder any $\\ell_{p}$-norm, the bottleneck matching is \n \\begin{itemize}\n \\item $\\mathrm{M}_{n} = O \\left( \\left( \\dfrac{\\log n}{n}\\right)^{1\/d}\\right),$ \\ \\ when $d \\geq 3$ \\cite{shor1991minimax}.\n \\vspace{0.1cm}\n \n  \\item $\\mathrm{M}_{n} = O \\left(\\left(\\dfrac{\\log^{3\/2} n}{n}\\right)^{1\/2} \\right),$ \\ \\ when $d= 2$ \\cite{leighton1986tight}.\n   \\vspace{0.1cm}\n  \n    \\item $\\mathrm{M}_{n} = O \\left(\\sqrt{\\dfrac{\\log \\epsilon^{-1}}{n}}\\right),$ with prob. $\\geq 1-\\epsilon$,\\ $d= 1$  \\cite{goel2004sharp}.\n \\end{itemize}\n\\end{lemma}\n\nUnder the condition $\\mathrm{M}_{n}= o(r_{n})$, we provide an upper bound for the Levy distance between $v_{n}$ and $v'_{n}$ in the following lemma.\n\n\\begin{lemma}\n\\label{lemafirst}\nFor $d \\geq 1$, $p \\in [1, \\infty]$ and $\\mathrm{M}_{n}= o(r_{n})$, the Levy distance between $v_{n}$ and $v'_{n}$ is upper bounded as\n\\begin{equation}\n\\begin{aligned}\n\\label{equation0}\n& L^{3} \\left( v_{n}, v'_{n} \\right)  \\leq d^{\\frac{1}{p}}  2^{d+1} \\left|  \\frac{1}{n}\\sum\\limits_{\\substack{ i}}^{} \\mathbf{N}(x_{i}) - a_{n} \\right|  \\\\\n &+d^{\\frac{1}{p}} 2^{d+1}  \\left| a_{n} -\\frac{2}{n}  \\sum\\limits_{\\substack{ i}}^{} \\mathrm{L}_{i} \\right| +a'_{n},\n\\end{aligned}\n\\end{equation}\nwhere,  $\\mathbf{N}(x_{i})$ denotes the degree of $x_{i}$ in $G(\\mathcal{X}_{n}, r_{n})$ and $\\mathrm{L}_{i} \\sim \\mathrm{Bin}\\left(n, \\theta^{(d)}\\left( r_{n}-2\\mathrm{M}_{n}\\right) \\right)$. \\comment{Due to space constraints we omit the details of the proof}. \n\\end{lemma}\n\\begin{proof}\nSee Appendix  \\ref{app:lemmafirst}.\n\\end{proof}\n\nThe condition enforced on $r_{n}$, i.e., $\\mathrm{M}_{n}= o(r_{n})$ implies that for $\\epsilon >0$, (\\ref{equation0}) holds when $a_{n}$ scales as $\\Omega (\\log^{\\epsilon}(n)\\sqrt{n})$ for $d=1$, as $\\Omega (\\log^{\\frac{3}{2}+\\epsilon}(n))$ for $d= 2$ and as $\\Omega (\\log^{1+\\epsilon}(n))$ for $d\\geq 3$.\n\nIn what follows, we show that the \\gls{test} of the adjacency matrix of $G(\\mathcal{X}_{n}, r_{n})$ concentrate around the \\gls{test} of the adjacency matrix of $G(\\mathcal{D}_{n}, r_{n})$ in the connectivity regime in the sense of convergence in probability.\n\nNotice that the term $ \\sum\\limits_{\\substack{ i}}^{} \\mathbf{N}(x_{i}) \/ 2$  in Lemma \\ref{lemafirst} counts the number of edges in $G(\\mathcal{X}_{n}, r_{n})$. For convenience, we denote $ \\sum\\limits_{\\substack{ i}}^{} \\mathbf{N}(x_{i}) \/ 2$ as $\\xi_{n}$.  To show our main result we apply the Chebyshev inequality given in Lemma \\ref{Chebyschev-inequality} on the random variable $\\xi_{n}$. For that, we need to determine $\\mathrm{Var}(\\xi_{n})$\n\n\\begin{lemma}(Chebyshev Inequality)\n\\label{Chebyschev-inequality}\nLet $\\mathrm{X}$ be a random variable with an expected value $\\mathbb{E}\\mathrm{X}$ and a variance $\\mathrm{Var}\\left(\\mathrm{X}\\right)$. Then, for any $t>0$\n\\begin{equation*}\n\\mathbb{P}  \\lbrace \\vert \\mathrm{X}-\\mathbb{E}\\mathrm{\\mathrm{X}} \\vert  \\geq t \\rbrace \\leq \\dfrac{\\mathrm{Var}(\\mathrm{X})}{t^2}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{lemma}\n\\label{variance-edges}\nWhen $x_{1},...,x_{n}$ are i.i.d. uniformly distributed in the $d$-dimensional unit torus $\\mathbb{T}^{d}=[0, 1]$\n\\begin{equation*}\n\\mathrm{Var} \\left(\\xi_{n}\\right) \\leq [\\theta^{(d)}+2\\theta^{(d)} a_{n} ].\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nThe proof follows along the same lines of Proposition A.1 in \\cite{muller2008two} when extended to a unit torus and applied to i.i.d. and uniformly distributed nodes.\n\\end{proof}\n\n\n\n\\begin{comment}\n Let us consider a $d$-dimensional {\\gls{DGG}} with $n= \\mathrm{N}^d$ nodes and assume the use of the $\\ell_{\\infty}$-metric. Then, the  degree of a vertex in $G(\\mathcal{D}_{n}, r_{n})$ is given as {\\cite{nyberg2014laplacian}}\n\\begin{equation*}\na'_{n}=(2k_{n}+1)^d-1, \\ \\ \\ \\mathrm{ with} \\ \\ k_{n}= \\left\\lfloor \\mathrm{N}r_{n} \\right\\rfloor,\n\\label{eq:degree}\n\\end{equation*}\nwhere $ \\left\\lfloor x \\right\\rfloor$ is the integer part, i.e.,  the greatest integer less than or equal to $x$. Note that when $d=1$, the Chebyshev distance and the Euclidean distance are the same.  In the following lemma, we provide an upper bound on the degree of the each node $a'_{n}$ in $G(\\mathcal{D}_{n}, r_{n})$ for any $\\ell_{p}$-norm\n\\end{comment}\n\nWe can now state the main theorem on the concentration of the adjacency matrix of $G(\\mathcal{X}_{n}, r_{n})$.\n \n\\begin{theorem}\n\\label{theorem-connectivity} \nFor  $d \\geq 1$, $p\\in[1, \\infty]$, $a \\geq 1$, $\\mathrm{M}_{n}= o(r_{n})$ and  $t >0$,  we have\n\\begin{equation*}\n\\mathbb{P} \\lbrace L^{3} \\left( v_{n}, v'_{n} \\right) >t\\rbrace  \\leq   2n \\exp\\left( \\dfrac{-a_{n}\\varepsilon^2}{3} \\left(1-\\dfrac{2\\mathrm{M}_{n}}{r_{n}}\\right)   \\right)\n\\end{equation*}\n\\begin{equation*}\n + \\frac{n \\left[\\theta^{(d)}(r_{n}-2\\mathrm{M}_{n})(a-1)+1\\right]^n}{a^{\\left(\\frac{t}{d^{\\frac{1}{p}}2^{d+3}}+\\frac{a_{n}(2-c)}{4}\\right)}}\n\\end{equation*}\n\\begin{equation}\n\\ \\ \\ \\ +\\frac{d^{\\frac{2}{p}} 2^{2d+6}\\left[ \\theta^{(d)}+2\\theta^{(d)} a_{n} \\right]}{n^2t^2},\n\\end{equation}\nwhere $\\varepsilon= \\left( \\frac{t}{d^{\\frac{1}{p}}2^{d+2}a_{n}}+\\dfrac{(2-c)}{4}- \\frac{2\\mathrm{M}_{n}}{r_{n}} \\right)$ and $c=\\left(1+\\frac{1}{2a_{n}^{1\/d}} \\right)^d$.\n\\\\\n\nIn particular,  for every $t>0$, $a \\geq 2$, $\\epsilon>0$ and $a_{n}$ that scales as $\\Omega (\\log^{\\epsilon}(n)\\sqrt{n})$ when $d=1$, as $\\Omega (\\log^{2}(n))$ when $d\\geq 2$, we have\n\n\\begin{equation*}\n\\lim_{n \\to\\infty} \\mathbb{P} \\left\\lbrace L^{3} \\left( v_{n}, v'_{n} \\right) >t  \\right\\rbrace  = 0.\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\n  See Appendix \\ref{app:theorem-connectivity}.\n\\end{proof}\nThis result is shown in the sense of convergence in probability by a straightforward application of Lemma  \\ref{Chernoff} and \\ref{Chernof-binomial} on the random variable $\\mathrm{L}_{i}$, then by applying Lemma \\ref{Chebyschev-inequality} and \\ref{variance-edges} to $\\xi_{n}.$\n\nIn what follows, we provide the eigenvalues of $\\mathbf{A}(\\mathcal{D}_{n})$ which approximates the eigenvalues of $\\mathbf{A}(\\mathcal{X}_{n})$ for $n$ sufficiently large.\n\n\\begin{lemma}\n\\label{corollary1:density}\n For $d \\geq 1$ and using the $\\ell_{\\infty}$-metric, the eigenvalues of $\\mathbf{A}(\\mathcal{D}_{n})$ are given by\n \\begin{equation}\n \\label{equation5}\n\\lambda_{m_{1},...,m_{d}}= \\prod_{s=1}^d  \\dfrac{\\sin(\\frac{m_{s} \\pi}{\\mathrm{N}}(a'_{n}+1)^{1\/d})}{\\sin(\\frac{m_{s} \\pi}{\\mathrm{N}})}  -1,\n \\end{equation}\nwhere,  $m_{1},...,m_{d}$ $\\in$ $\\lbrace 0,...\\mathrm{N}-1 \\rbrace$, $a'_{n}=(2k_{n}+1)^d-1, k_{n}= \\left\\lfloor \\mathrm{N}r_{n} \\right\\rfloor \\ \\ \\mathrm{and} \\ \\ n=\\mathrm{N}^{d}.$ The term $ \\left\\lfloor x \\right\\rfloor$ is the integer part, i.e.,  the greatest integer less than or equal to $x$.\n\\end{lemma}\n\\begin{proof}\n See Appendix  \\ref{app:lemma3}. \n\\end{proof}\nThe proof utilizes the result in \\cite{nyberg2014laplacian} which shows that the eigenvalues of the adjacency matrix of a \\gls{DGG} in $\\mathbb{T}^d$ are found by taking the $d$-dimensional DFT of an $\\mathrm{N}^{d}$ tensor of rank $d$ obtained from the first block row of $\\mathbf{A}(\\mathcal{D}_{n})$.\n\nFor $\\epsilon >0$, Theorem \\ref{theorem-connectivity} shows that when $a_{n}$ scales as $\\Omega (\\log^{\\epsilon}(n)\\sqrt{n})$ for $d=1$ and  as $\\Omega (\\log^{2}(n))$ when $d\\geq 2$, the \\gls{test} of the adjacency matrix of an \\gls{RGG} concentrate around the \\gls{test} of the adjacency matrix of a \\gls{DGG} as $n \\rightarrow \\infty$.   Therefore, for $n$ sufficiently large, the eigenvalues of the \\gls{DGG} given in (\\ref{equation5}) approximate very well the eigenvalues of the \\gls{DGG}.\n\n\n\n\\section{\\bf  Numerical Results}\n\\label{sec:results}\nWe present simulations to validate the results obtained in Section \\ref{sec:systemmodel}. More specifically, we corroborate our results on the spectrum of the adjacency matrix of \\glspl{RGG} in the connectivity regime by comparing the simulated and the analytical results.\n\nFig. \\ref{figg}(a) shows the cumulative distribution functions of the eigenvalues of the adjacency matrix of an \\gls{RGG} realization and the analytical spectral distribution in the connectivity regime. We notice that  for the chosen average vertex degree $a_{n}= \\log(n)\\sqrt{n}$ and $d=1$, the curves corresponding to the eigenvalues of the \\gls{RGG} and the \\gls{DGG} fit very well for a large value of $n$.\n\n\\captionsetup[figure]{labelfont={bf},labelformat={default},labelsep=period,name={Fig.}}\n\\begin{figure}\n\\begin{minipage}[b]{0.45\\textwidth}\n\\begin{subfigure}[b]{\\linewidth}\n\\includegraphics[width=\\linewidth]{RGG}\n\\caption{Connectivity regime, $r_{n}=\\frac{\\log(n)}{\\sqrt{n}}$, $n=2000$.}\n\\end{subfigure}\n\n\\end{minipage}\n\\caption{An illustration of the cumulative distribution function of the eigenvalues of an \\gls{RGG}. }\n\\label{figg}\n\\end{figure}\n\n\\section{\\bf Conclusion}\n\\label{sec:conclusion}\nIn this work, we study the spectrum of the adjacency matrix of \\glspl{RGG} in the connectivity regime.  Under some conditions on the average vertex degree $a_{n}$, we show that the \\glspl{test} of the adjacency matrices of an \\gls{RGG} and a \\gls{DGG} converge to the same limit as $n \\rightarrow \\infty$. Then,  based on the regular structure of the \\gls{DGG}, we approximate the  eigenvalues of $\\mathbf{A}(\\mathcal{X}_{n})$ by the eigenvalues of $\\mathbf{A}(\\mathcal{D}_{n})$ by taking the $d$-dimensional DFT of an $\\mathrm{N}^{d}$ tensor of rank $d$ obtained from the first block row of $\\mathbf{A}(\\mathcal{D}_{n})$.\n\n\\section{\\bf Acknowledgement}\nThis research was funded by the French Government through the Investments for\nthe Future Program with Reference: Labex UCN@Sophia-UDCBWN.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\\setcounter{equation}{0}\n\nOne of the fundamental question in physics is to\nunderstand the properties of matter at extreme density and temperature in\nthe first few microseconds after the big bang. Such a state of matter is\nknown as Quark-Gluon-Plasma (QGP) state where the quarks and the gluons are\nin deconfined state. A lot of progress has been made in undertanding the\nproperties and various aspects of the evolution of the strongly coupled QGP\nthrough the heavy-ion collision experiments at RHIC \n\\cite{Shuryak:2005, Kolb:2003KH} (see also \\cite{Tannenbaum:2006}) as well\nas in LHC (see \\cite{Florkowski:2017} for a comprehensive review).\nHydrodynamics plays an important role after the system undergoes a rapid \nthermalization and local thermal equilibrium is reached. Since it \nis difficult to solve the strongly coupled QCD, the qualitative \nfeatures of the hydrodynamics regime in the evolution of QGP has been \nextensively studied by using AdS\/CFT duality. This has provided \nan important framework for studying\nthe strongly coupled dynamics in a class of superconformal field theories,\nin particular, ${\\cal N} = 4$  super Yang-Mills theory\nand the corresponding gravity dual description in AdS space-time\n\\cite{Malda:1998, WittenGubser:1998}. In the context of heavy ion\ncollisions, AdS\/CFT correspondence has led to interesting results\nlike computation of shear viscosity of\nfinite temperature ${\\cal N} = 4$ Supersymmetric Yang-Mills theory\n\\cite{Policastro:2001}, viscosity from gravity dual description involving\nblack holes in AdS space  \\cite{Kovtun:2005} etc.\nAs we know, perfect local equilibrium system is described by ideal\nfluid dynamics. For small departures from equilibrium, the system is\ndescribed by dissipative fluid dynamics. The nonrelativistic \nlimit of hydrodynamics can be described by \nNavier-Stokes equations involving the energy density, pressure of the fluid \nas well as the shear and bulk viscosities. These equations are relevant \nfor describing the fluid at low energy.\nHowever, in the context of \nultrarelativistic heavy-ion collisions at RHIC and LHC, one needs \nto use relativistic hydrodynamics to study the evolution and properties \nof QGP describing fluid at high energy. \n\nThe microscopic dynamics describing \nprocesses characterized by $kl_{mfp}$ ($k$ being the momentum scale \nand $l_{mfp}$ is the mean free path of the system) has been studied \nin great detail using the relativistic hydrodynamics setup. \nIn the framework of gradient expansion in relativistic dissipative \nhydrodynamics, \nthe zeroth order theory is described by ideal hydrodynamics. \nFirst order theory is described by relativisitc Navier-Stokes \nequations. In first order hydrodynamics, due to the \nlack of an initial value formulation, \nsignals can be transmitted with arbitrarily high speed thereby violating \ncausality. The system in this case is described by parabolic equations. \nThe first order theory has\nbeen extended by M\\\"uller \\cite{Muller:67} and independently \nby Israel and Stewart(IS)\n\\cite{Israel:197679IS} by including the second order gradient terms thereby\npreserving causality in the resulting relativistic hydrodynamics equations.\nThe set of transport coefficients are extended in the second order\nhydrodynamics and the resulting equations become hyperbolic. However, the  \nM\\\"uller-Israel-Stewart theory does not contain all the corrections \nto second order in gradient. Subsequently, \nthe authors in refs \n\\cite{Baier:2007BRSSS, Sayantani:2007BHMR} obtained additional terms in \nthe stress-energy tensor by \nutilising the conformal symmetries of the theory and the new transport \ncoefficients were explicitly determined. The relaxation time \n(a transport coefficient \nin second order viscous hydrodynamics) has also been computed from the \nanalysis of the regularity of the dual geometry \\cite{Heller:2007HJ}. \n\nIn the second order Israel-Stewart (IS) theory, the equations in relativistic \ndissipative fluid dynamics have been obtained using the second \nmoment of Boltzmann equation in the kinetic theory description and \nGrad's 14-moment expansion \\cite{Grad} for the phase-space distribution \nfunction. \nHowever, 14-moment approximation used in IS theory does not provide\na unique theory as it leads to multiple fluid dynamical equations\nwith similar general structure but different transport coefficients\n\\cite{Denicol:2010xn, Denicol:2012}. For one dimensional Bjorken \nexpansion \\cite{Bjorken:1983}, it has been discussed  \nthat the second order \nIS theory has also some unphysical features like reheating of the \nexpanding medium \\cite{Muronga:2003ta}, negative pressure \n\\cite{Martinez:2009mf} for large viscosity and small initial expansion time; \nlarge ratio of shear viscosity to entropy \ndensity \n\\cite{El:2008yy} etc.\nIn order to address these problems with IS theory, higher order \ncorrections in the dissipative hydrodynamics were considered to study \ntheir effects \\cite{El:2009vj}.\nThough the resulting evolution equation \ndid not contain all possible terms in the second order dissipative \nhydrodynamics, the solutions with higher order correction indicated better\nagreement with the results from kinetic transport theory.   \n\nSecond-order dissipative equations have been \nderived from relativistic Boltzmann equation (BE)  \nusing gradient expansion of the distribution function thereby including\nnonlocal effects in the collision term \\cite{Jaiswal:2012qm}. In the above \nsetup, while deriving the evolution equations, the authors use the \ndefinition of dissipative current directly instead of \nusing the second moment of BE and it is important \nto note that the evolution equations included all possible second \norder terms allowed by the symmetry. Subsequently considering the nonlocal \ncollision term, the relativistic \nthird order dissipative evolution equation for the shear stress tensor has \nbeen derived from kinetic theory without using Grad's \n14-moment approximation and second moment of BE \\cite{AJ}. \nIn the above formalism, BE has been solved \niteratively in the relaxation \ntime approximation (RTA) for the collision term \nand the nonequilibrium phase space distribution function has been obtained\n(see also \\cite{Jaiswal:2013npa} for the discussion in\nthe context of first order and second order evolution equations).\nThe formulation of RTA for the collision term was proposed in a very \ninteresting paper by Anderson and Witting \\cite{Anderson-Witting} \nfor solving the relativistic BE as a power series in the relaxation \ntime and the transport coefficients \nfor a single component gas were obtained and compared with the relativistic \nGrad's moment method. \nIterative solution method has also been useful in obtaining \nhigher order corrections to entropy four current \\cite{CJPR:2014}.\nHydrodynamics gradient expansion in higher orders has been discussed \nin ref.\\cite{Grozdanov:2015}, where, third order corrections in conformal as \nwell as nonconformal hydrodynamics of neutral fluids have been investigated. \nIn the case of nonrelativistic systems, \nhigher order constitutive equations from kinetic theory have been discussed \nearlier in ref.\\cite{Cha}. \nThere has been progress in hydrodynamical \nformulations from various other approaches \n\\cite{Manes:2022, Cantarutti:2020, Jaiswal-rev:2021}.\nIn this work, we consider a generalization of Bjorken's one dimensional \nexpansion to three dimensional anisotropic expansion of the relativistic \nfluid, where the local rest frame (LRF) of the \nanisotropically expanding fluid is described by time \ndependent Kasner space-time. \nFor time dependent AdS\/CFT correspondence, Kasner \nspace-time has been studied earlier in the \ncontext of anisotropic expansion of the RHIC and LHC fireball, where \nexplicit expressions for the hydrodynamic quantities have been obtained \nin first and second order relativistic viscous hydrodynamics \nand gravity dual description has been studied \n\\cite{Nakamura:2006SNK, ppsm:2020}. Collisionless Boltzmann \nequation in Kasner space-time and its relation to \nanisotropic hydrodynamics has been discussed in ref.\\cite{Jaiswal:2017}.\nThough Kasner space-time is a curved space-time, Sin {\\it et al} \n\\cite{Nakamura:2006SNK} have shown that under a well controlled \napproximation, it can be considered as the LRF of the anisotropically \nexpanding fluid on Minkowski space-time. \nHere, we study the relativistic BE in RTA for the collision term. \nWe extend the results of \nref. \\cite{AJ} to three dimensional expansion case. Using the \niterative solution of BE for the nonequilibrium distribution function,\nwe obtain the dissipative evolution equations for the shear stress tensor \nand energy density to second and third order in gradients from kinetic \ntheory in Kasner space-time. These expressions have not \nbeen obtained before. We show that our results reduce to that of one\ndimensional expansion case under suitable conditions for the Kasner \nparameters. \n\nThe paper is organized as follows: Section 1 contains the introduction \nand motivation for the present study. Section 2 deals with the basic \nformalism for solving the relativistic BE \niteratively in RTA for the collision term. \nWe discuss the basic set-up in Minkowski space-time.  \nIn section 3, we generalize the \none dimensional Bjorken expansion to three dimensional anisotropic \nexpansion by considering Kasner space-time. Using the iterative \nsolutions of the Boltzmann equation for the nonequilibrium distribution\nfunction, we obtain the second and third order evolution equations \nfor the shear stress tensor ind energy density in terms of Kasner \nparameters. We also show that our evolution equations  \nagree with the one dimensional Bjorken expansion case \nin the appropriate limit of the Kasner parameters. We summarise \nand discuss the future perspective in section 4. \n\n\n\\section{Kinetic theory and iterative solution of relativistic \nBoltzmann equation}\n\\label{sec:Iterative solution}\n\\setcounter{equation}{0}\n\nAs we know, the macroscopic state of a system in relativistics \nfluid dynamics is described by the energy-momentum tensor and \nthe corresponding conservation law plays an important role in \nthe hydrodynamic evolution of a system. The conserved energy-momentum \ntensor $T^{\\mu\\nu}$ can be decomposed as, \n\\begin{eqnarray}\n\\label{NTD}\nT^{\\mu\\nu} = \\epsilon u^{\\mu} u^{\\nu} - P \\Delta^{\\mu\\nu} + \\pi^{\\mu\\nu},\n\\end{eqnarray}\nwhere, $\\epsilon$, $P$ and $\\pi^{\\mu\\nu}$ represent the energy density, \npressure and shear stress tensor respectively. The bulk viscosity \nvanishes for a system of massless particles and the corresponding \ntheory is conformal. We work in the Landau frame.  \nThe projection operator $\\Delta^{\\mu\\nu}$ defined on the space \northogonal to the fluid velocity is given by: \n$\\Delta^{\\mu\\nu} = g^{\\mu\\nu}-u^{\\mu} u^{\\nu}$, where the metric \ntensor $g^{\\mu\\nu} = diag (+, -, -, -)$ and $u^{\\mu}$ is the \nfluid 4-velocity. $u^{\\mu}$ is an eigen vector of the energy-momentum \ntensor ($T^{\\mu\\nu} u_{\\nu} = \\epsilon u^{\\mu}$) and it satisfies the \nfollowing properties: $\\Delta^{\\mu\\nu} u_{\\mu} = 0 = \\Delta^{\\mu\\nu} u_{\\nu}; \n\\,\\, \\Delta^{\\mu\\nu} \\Delta^{\\alpha}_{\\nu} = \\Delta^{\\mu\\alpha}$. \nIn the local rest frame of the fluid, the 4-velocity is given by \n$u^{\\mu} = (1, 0, 0, 0)$. \nProjecting the energy-momentum tensor conservation equation  \nin the direction parallel and perpendicular \nto the fluid 4-velocity results in the following fundamental \nequations for a relativistic viscous fluid: \n\n\\begin{eqnarray}\\label{evol}\n\\dot\\epsilon + (\\epsilon+P)\\theta - \\pi^{\\mu\\nu} \\sigma_{\\mu\\nu} &=0,\\nonumber\\\\\n(\\epsilon+P)\\dot u^\\alpha - \\nabla^\\alpha P + \\Delta^\\alpha_\\mu \\partial_\\nu\n\\pi^{\\mu\\nu} &= 0.\n\\end{eqnarray}\nWe use the notations of ref.\\cite{AJ}. Here\n$\\dot \\epsilon = u^\\mu\\partial_\\mu \\epsilon $ denotes the comoving\nderivative of the energy density, $\\theta \n\\equiv \\nabla_\\mu u^\\mu$ is the expansion\nscalar, $\\nabla^\\alpha\\equiv \\Delta^{\\mu\\alpha}\\partial_\\mu$ denotes \nthe space-like derivative and $\\sigma^{\\mu\\nu} \\equiv \n\\Delta^{\\langle\\mu}u^{\\nu\\rangle} = \n\\Delta^{\\mu\\nu}_{\\alpha\\beta} \\nabla^{\\alpha} u^{\\beta}$. \nSymmetrization is defined by $A^{(\\mu}B^{\\nu )}\n\\equiv \\frac{1}{2} (A^\\mu B^\\nu + A^\\nu B^\\mu)$. \n\n\nRelativistic hydrodynamics can be derived from kinetic theory where \nthe fundamental equation is Boltzmann equation. \nRelativistic BE in kinetic theory is given by\n\\cite{deGroot}, \n\\begin{equation}\\label{BE}\np^{\\mu} \\partial_{\\mu} f(x, p) =  {\\cal C}[f](x, p)\n\\end{equation}\nwhere $p^{\\mu}$ is the particle 4-momentum, $p^{\\mu} = \n(p^0, {\\bf p}$) with $p^0 = {\\sqrt {{\\bf p}^2 + m^2}}$, $f(x, p)$ is the \none particle phase space distribution function and \n${\\cal C}[f](x, p)$ is the collision term. \nRight hand side of the \nabove equation becomes zero if the collision between particles is neglected. \nIn the kinetic theory, the energy-momentum tensor is expressed in terms\nof the distribution function $f(x, p)$ and particle 4-momentum $p^{\\mu}$:\n\n\\begin{eqnarray}\\label{NTD}\nT^{\\mu\\nu} &= \\!\\int\\! dp \\ p^\\mu p^\\nu\\, f(x,p)\n\\end{eqnarray}\nwhere $dp\\equiv g d{\\bf p}\/[(2 \\pi)^3|\\bf p|]$ and g is the number of\ninternal degrees of freedom and we are considering a system of massless\nparticles.\n\nIn the RTA, the collision term is given by\n\\cite{Anderson-Witting}, \n\\begin{equation}\\label{collision}\n{\\cal C}[f](x, p) = \\frac{u \\cdot p}{\\tau_R} (f(x,p) - f_{eq}(x,p))\n\\end{equation}\nwhere, $u\\cdot p = u_{\\mu} p^{\\mu}$, $\\tau_R$ is the relaxation time, \n$f_{eq}(x, p)$ is the equilibrium distribution function and \nthe deviation $\\delta f$ from the equilibrium value is assumed to be \nsmall. We write,\n\\begin{equation}\\label{deviation} \nf(x, p) = f_{eq} (x, p) +  \\delta f(x, p)\n\\end{equation}\n\nThe shear stress tensor $\\pi^{\\mu\\nu}$ in the decomposition of $T^{\\mu\\nu}$ \ncan be calculated in terms of $\\delta f$ which is the deviation from \nthe equilibrium distribution function. \n$\\pi^{\\mu\\nu}$ can be written as, \n\\begin{eqnarray}\\label{FSE}\n\\pi^{\\mu\\nu} &= \\Delta^{\\mu\\nu}_{\\alpha\\beta} \\int dp \\, p^\\alpha p^\\beta\\, \n\\delta f,\n\\end{eqnarray}\n\nwhere,\n\\begin{equation}\\label{sym-proj-op}\n\\Delta^{\\mu\\nu}_{\\alpha\\beta}\\equiv \n\\frac{1}{2}(\\Delta^{\\mu}_{\\alpha}\\Delta^{\\nu}_{\\beta} + \n\t\\Delta^{\\mu}_{\\beta}\\Delta^{\\nu}_{\\alpha} - \n\t\\frac{2}{3}\\Delta^{\\mu\\nu}\\Delta_{\\alpha\\beta}) \n\\end{equation}\nis a traceless symmetric projection operator orthogonal to \n$u^\\mu$ and it satisfies the following properties: \n$\\Delta^{\\mu\\nu}_{\\alpha\\beta} \\Delta_{\\mu\\nu} = 0 = \n\\Delta^{\\mu\\nu}_{\\alpha\\beta} \\Delta^{\\alpha\\beta}$, \n$\\Delta^{\\mu\\nu}_{\\rho\\sigma}\\Delta^{\\rho\\sigma}_{\\alpha\\beta} \n= \\Delta^{\\mu\\nu}_{\\alpha\\beta}$.\n\nIn order to obtain the expression for the nonequilibrium part  $\\delta f$ \nappearing above in the expression for $\\pi^{\\mu\\nu}$, one solves \nthe BE iteratively in RTA \\cite{Jaiswal:2013npa}.  \nIn this formalism, one writes the deviation $\\delta f$ in a gradient \nexpansion \\cite{Chapman}:\n$\\delta f = \\delta f^{(1)} + \\delta f^{(2)} + \\delta f^{(3)} + \\cdots $,\nwhere $\\delta f^{(1)}$ is first order in gradient, $\\delta f^{(2)}$, \n$\\delta f^{(3)}$ are second and third order in gradients respectively. \n\nUsing the expression for the collision functional in the RTA \n\\cite{Anderson-Witting},\n\\begin{eqnarray}\\label{collision-RTA}\n{\\cal C}[f] = -u\\!\\cdot\\! p\\frac{\\delta f}{\\tau_R},\n\\end{eqnarray}\nthe relativistic BE has been solved iteratively \n\\cite{Jaiswal:2013npa}, where,   \n\\begin{eqnarray}\\label{F1F2}\nf_1 &= f_{eq} -\\frac{\\tau_R}{u\\!\\cdot\\! p} \\, p^\\mu \\partial_\\mu f_0, \\quad \n\\nonumber \\\\\nf_2 &= f_{eq} -\\frac{\\tau_R}{u\\!\\cdot\\! p} \\, p^\\mu \\partial_\\mu f_1, \n~~\\, \\cdots\n\\end{eqnarray}\nwith the notation $f_n = f_{eq} + \\delta f^{(1)} + \n\\delta f^{(2)} + \\cdot + \\delta f^{(n)}$.\nFrom here, one obtains the expressions for the deviation $\\delta f$ \nin an gradient expansion as \\cite{AJ}: \n\\begin{eqnarray}\n\\delta f^{(1)} &= -\\frac{\\tau_R}{u\\!\\cdot\\! p} \\, p^\\mu \\partial_\\mu f_{eq} \n\\label{FOC} \\\\\n\\delta f^{(2)} &= \\frac{\\tau_R}{u\\!\\cdot\\! p}p^\\mu p^\\nu\n\\partial_\\mu\\Big(\\frac{\\tau_R}{u\\!\\cdot\\! p} \n\\partial_\\nu f_{eq}\\Big). \\label{SOC}\n\\end{eqnarray}\nThe shear stress tensor to first order in gradients can be calculated \nby using the expression for $\\delta f^{(1)}$ given above \n[eqn.(\\ref{FOC})] in eqn.(\\ref{FSE}) resulting  \n$\\pi^{\\mu\\nu} = 2\\tau_R\\beta_\\pi\\sigma^{\\mu\\nu}, \n\\quad\\beta_\\pi = \\frac{4}{5}P$.  \nUsing the above results, one can obtain the expressions \nfor the shear stress tensor and its evolution to higher orders \nin derivatives.   \n\nThe above hydrodynamics set up has been dicussed in \nMinkowski space-time.\nIn the next section, we shall discuss the anisotropic expansion \nof the fluid, in which, Kasner space-time has been considered as the \nlocal rest frame of the fluid.  \nIt is important to note that though Kasner space-time is a curved \nspace-time, Sin {\\it et al} \\cite{Nakamura:2006SNK} have shown that \nunder a well controlled\napproximation, it can be considered as the LRF of the anisotropically\nexpanding fluid on Minkowski space-time. The classical Boltzmann equation in\ncurved space-time involves the Christoffel symbol, which is given by,\n\\begin{equation}\n\t(p^{\\mu} \\partial_{\\mu}-\n\\Gamma^{\\lambda}_{\\mu \\nu} \n\tp^{\\mu} p^{\\nu} \\partial_{\\lambda}) f(p,x)= {\\mathcal C} [f]\n\\end{equation}\nThe stress energy tensor is defined as, \n\\begin{equation}\nT^{\\mu\\nu} = \\int {\\sqrt{-g}}\\frac{d^3p}{p^0} p^{\\mu}p^{\\nu} f(x, p)\n\\end{equation}\nwhere, $g$ is the determinant of the metric tensor $g_{\\mu\\nu}$. \nOne obtains the hydrodynamic equations by taking the moments {\\it w.r.t.}\nto the particle momentum.  \nBaier {\\it et al} have discussed the set up in curved background in \nref.\\cite{Baier:2007BRSSS}. Hence instead of repeating, \nWe refer to the readers ref. \\cite{Baier:2007BRSSS} \nfor the discussion in a general curved background.\nIn the present framework, the effect of a general background \nspace-time \ncan be accounted for by replacing the partial derivative \n$\\partial_\\mu$ with the covariant derivative $D_\\mu$ throughout. \nFor example, we have, \n$\\dot u^{\\alpha} = u^{\\mu} D_{\\mu} u^{\\alpha}$ and  \nthe covariant derivative of a vector is given by ,\n\\begin{eqnarray}\\label{covariant-d}\nD_{\\mu} A^{\\nu} = \\partial_\\mu A^{\\nu} + \\Gamma^{\\nu}_{\\mu\\rho} A^{\\rho}; \n\\,\\,\\,\nD_{\\mu} A_{\\nu} = \\partial_\\mu A_{\\nu} - \\Gamma^{\\rho}_{\\mu\\nu} A_{\\rho}\n\\end{eqnarray} \nIn the next section, we use the covariant derivative for \ncomputing various expressions where \nwe consider Kasner space-time as the local rest frame \n(LRF) of the fluid.\n\\section{Evolution equations for shear stress tensor and Kasner space-time}\n\nIn order to obtain the evolution equation for the shear stress tensor\nto higher orders, \none needs to compute the comoving derivative of $\\delta f$ \n(which is the deviation from the equilibrium distribution function) \nexpanded in powers of space-time derivatives. In particular, one has,   \n\\begin{equation}\n\\dot\\pi^{\\langle\\mu\\nu\\rangle} = \\Delta^{\\mu\\nu}_{\\alpha\\beta} \n\\int dp\\, p^\\alpha p^\\beta\\, \\delta\\dot f, \\label{SSE}\n\\end{equation}\nwhere we have used the standard notation $A^{\\langle\\mu\\nu\\rangle}\\equiv \n\\Delta^{\\mu\\nu}_{\\alpha\\beta}A^{\\alpha\\beta}$. To second order in \ngradients, the evolution equation for the shear tensor is given by \n\\cite{Jaiswal:2013npa}, \n\\begin{equation}\\label{SOSHEAR}\n\\dot{\\pi}^{\\langle\\mu\\nu\\rangle} \\!+ \\frac{\\pi^{\\mu\\nu}}{\\tau_\\pi}\\!= \n2\\beta_{\\pi}\\sigma^{\\mu\\nu}\n\\!+2\\pi_\\gamma^{\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\n\\!-\\frac{10}{7}\\pi_\\gamma^{\\langle\\mu}\\sigma^{\\nu\\rangle\\gamma} \n\\!-\\frac{4}{3}\\pi^{\\mu\\nu}\\theta,\n\\end{equation}\nwhere $\\omega^{\\mu\\nu}\\equiv \\frac{1}{2}(\\nabla^\\mu u^\\nu-\\nabla^\\nu u^\\mu)$\nis the fluid vorticity.\nIn the derivation of this equation, one uses the expression for \n$\\delta f^{(1)}$ in the expansion of the particle distribution \nfunction $f$ and keeps terms upto quadratic in the gradient \nexpansion. In the above, the Boltzmann relaxation time $\\tau_R$ has \nbeen replaced by the shear relaxation time $\\tau_{\\pi}$ which is \na second order transport coefficient and can be expressed as \n$\\tau_{\\pi} = \\frac{\\eta}{\\beta_{\\pi}}$ where $\\eta$ is the first order \ntransport coefficient and $\\beta_{\\pi}$ is related to the pressure \n$P$ of the fluid. \n\nSimilarly, using the expressions for $\\delta f^{(1)}$ (eqn.\\ref{FOC}), \n$\\delta f^{(2)}$ (eqn.\\ref{SOC}), computing their comoving derivatives \nand keeping terms upto cubic order in derivatives, the \nthird order evolution equation for $\\pi^{\\mu\\nu}$ has \nbeen obtained by Jaiswal \\cite{AJ}:\n\n\\begin{eqnarray}\\label{TOSHEAR}\n\\dot{\\pi}^{\\langle\\mu\\nu\\rangle} =& -\\frac{\\pi^{\\mu\\nu}}{\\tau_\\pi}\n+2\\beta_\\pi\\sigma^{\\mu\\nu}\n+2\\pi_{\\gamma}^{\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\n-\\frac{10}{7}\\pi_\\gamma^{\\langle\\mu}\\sigma^{\\nu\\rangle\\gamma}  \n-\\frac{4}{3}\\pi^{\\mu\\nu}\\theta\n+\\frac{25}{7\\beta_\\pi}\\pi^{\\rho\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\pi_{\\rho\\gamma}\n-\\frac{1}{3\\beta_\\pi}\\pi_\\gamma^{\\langle\\mu}\\pi^{\\nu\\rangle\\gamma}\\theta \\nonumber \\\\\n&-\\frac{38}{245\\beta_\\pi}\\pi^{\\mu\\nu}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}\n-\\frac{22}{49\\beta_\\pi}\\pi^{\\rho\\langle\\mu}\\pi^{\\nu\\rangle\\gamma}\\sigma_{\\rho\\gamma} \n-\\frac{24}{35}\\nabla^{\\langle\\mu}\\left(\\pi^{\\nu\\rangle\\gamma}\\dot u_\\gamma\\tau_\\pi\\right)\n+\\frac{4}{35}\\nabla^{\\langle\\mu}\\left(\\tau_\\pi\\nabla_\\gamma\\pi^{\\nu\\rangle\\gamma}\\right) \\nonumber \\\\\n&-\\frac{2}{7}\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\langle\\mu}\\pi^{\\nu\\rangle\\gamma}\\right)\n+\\frac{12}{7}\\nabla_{\\gamma}\\left(\\tau_\\pi\\dot u^{\\langle\\mu}\\pi^{\\nu\\rangle\\gamma}\\right) \n-\\frac{1}{7}\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle\\mu\\nu\\rangle}\\right)\n+\\frac{6}{7}\\nabla_{\\gamma}\\left(\\tau_\\pi\\dot u^{\\gamma}\\pi^{\\langle\\mu\\nu\\rangle}\\right) \\nonumber \\\\\n&-\\frac{2}{7}\\tau_\\pi\\omega^{\\rho\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\pi_{\\rho\\gamma}\n-\\frac{2}{7}\\tau_\\pi\\pi^{\\rho\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\omega_{\\rho\\gamma} \n-\\frac{10}{63}\\tau_\\pi\\pi^{\\mu\\nu}\\theta^2\n+\\frac{26}{21}\\tau_\\pi\\pi_\\gamma^{\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\theta.\n\\end{eqnarray}\n\nNote that the above equation was formulated within the \nframework of kinetic theory for a system of massless particles which has \nconformal symmetry. For such a system, dissipation due to bulk viscosity \nand heat current can be neglected.\nUsing entropy current and \nsecond law of thermodynamics, El {\\it etal} (ref.\\cite{El:2009vj})\nhave obtained the evolution equation before. However, in the context \nof one dimensional Bjorken expansion, the evolution equation \nobtained there misses out many more terms. \nNext, we study the evolution equation to second \nand third order for the shear stress tensor and energy density \nin the context of three dimensional anisotropic\nexpansion of the conformal fluid. \nWe consider the generalisation of Bjorken's 1-dimensional expansion to \n3-dimensional expansion of the fluid in order to \nconnect to the realistic description of the RHIC and LHC fireball and  \nfor this, we consider Kasner space-time as the LRF of the fluid\n\\cite{Nakamura:2006SNK, ppsm:2020, Jaiswal:2017}.\nThe metric is \\cite{Kasner:1921}: \n\\begin{eqnarray}\\label{Kasner}\nds^{2} = (d \\tau)^{2} - \\tau^{2a}(dx_{1})^{2} - \\tau^{2b}(dx_{2})^{2}\n- \\tau^{2c}(dx_{3})^{2}\n\\end{eqnarray}\n\nHere $x_1, x_2, x_3$ are the comoving coordinates, $\\tau$ is \nthe proper time and $a, b, c$ are constants known as Kasner parameters. \nThe Kasner parameters satisfy\nthe conditions,\n\\begin{eqnarray}\\label{Kasner-condn}\na + b + c = 1, \\hspace{20pt}a^2 + b^2 + c^2 = 1 \n\\end{eqnarray}\nThe above metric is an exact solution of vacuum Einstein's\nequation and it describes a homogeneous and anisotropic\nexpansion of the Universe. The physical quantities are assumed to \ndepend only on proper time $\\tau$.\nThe nonzero components of the affine connection \nfor the Kasner metric are given by, \n\\begin{eqnarray}\\label{affine}\n\\Gamma^{\\tau}_{x_1 x_1} = a \\tau^{2 a -1},\\, \\Gamma^{\\tau}_{x_2 x_2}\n= b \\tau^{2 b -1},\\, \\Gamma^{\\tau}_{x_3 x_3} = c \\tau^{2 c -1},\\nonumber \\\\ \n\\Gamma^{x_1}_{x_1 \\tau} = \\frac{a}{\\tau}, \\,\n\\Gamma^{x_2}_{x_2 \\tau} = \\frac{b}{\\tau}, \\,\\Gamma^{x_3}_{x_3 \\tau} = \n\\frac{c}{\\tau}\n\\end{eqnarray}\n\nThe Ricci tensor turns out\nto be zero upon using Kasner conditions $a + b + c =1$; $a^2 + b^2 + c^2 =1$.\nRicci tensor for Kasner space-time is give by,\n\\begin{eqnarray}\nR_{00} & = & \\frac{1}{\\tau^2} [ (a + b + c) - (a^2 + b^2 + c^2)] \\\\ \\nonumber\nR_{11} & = & a [(a + b + c) - 1] \\tau^{2a-2} \\\\ \\nonumber\nR_{22} & = & b [(a + b + c) - 1] \\tau^{2b-2} \\\\ \\nonumber\nR_{33} & = & c [(a + b + c) - 1] \\tau^{2c-2} \n\\end{eqnarray}\nwhere, $1, 2, 3$ corresponds to $x_1, x_2, x_3$ coordinates, and\n$0$ corresponds to $\\tau$ coordinate. As one can see, the components\nvanish upon using Kasner conditions. The nonzero components of Riemann tensor\nare given by,\n\\begin{eqnarray}        \nR_{0101} &=& (1-a)a \\tau^{2a-2}, \\, R_{0202} = (1-b) b\\tau^{2b-2} \n        \\\\ \\nonumber \n        R_{0303} &=& (1-c) c \\tau^{2c-2}, \\,\n        R_{1212} = a b \\tau^{2a+2b-2} \\\\ \\nonumber\n        R_{1313} &=& a c \\tau^{2a + 2c-2} , \\, R_{2323} = \n        b c \\tau^{2b + 2c -2} \n\\end{eqnarray}\nThese expressions become zero for the Bjorken case corresponding to\n$a=1, b=0, c=0$. So the terms involving $R^{\\mu\\nu\\rho\\sigma}$\ncould contribute to the shear tensor $\\pi^{\\mu\\nu}$ with a coefficient\n$\\kappa$ (we refer to eqn. 3.12 in ref. \\cite{Baier:2007BRSSS}).\nHowever, the general expression for the shear stress tensor $\\pi^{\\mu\\nu}$\nwhen derived from kinetic theory does not contain the $\\kappa$ term which\nwould have involved the Riemann tensor (we refer to eqn. 5.23 of\nthe above reference \\cite {Baier:2007BRSSS} and subsequent discussion). \nBoltzmann equation does not contain the term involving $\\kappa$. \nHence, there is no inconsistency in the order of the derivative expansion.\n\nWe compute various quantities appearing in the second and third order \nevolution equations for the shear stress tensor. \nAs mentioned in the previous section, here\nwe use the notations involving covariant \nderivative $D_{\\mu}$ (as defined in eqn. (\\ref {covariant-d})) instead \nof partial \nderivative $\\partial_{\\mu}$.\nWe have, \n\\begin{eqnarray}\n\\nabla_{\\mu}u_{\\nu} = \\Delta^{\\rho}_{\\mu} D_{\\rho}u_{\\nu},\n\\nonumber \\\\\n\\sigma^{\\mu\\nu}  = \\Delta^{\\mu\\nu}_{\\alpha\\beta} \\nabla^{\\alpha} u^{\\beta}=\n\\Delta^{\\mu\\nu}_{\\alpha\\beta} \\Delta^{\\alpha\\rho} D_{\\rho}u^{\\beta}\n\\end{eqnarray}\n\nWe obtain the \ncomponents of the projection operator as,\n\\begin{equation}\\label{projector}\t\n\\Delta ^{\\mu \\nu}\\equiv\\mathrm{diag}(0, \n\t- \\frac{1}{\\tau^{2a}},\n\t- \\frac{1}{\\tau^{2b}}, - \\frac{1}{\\tau^{2c}})\n\\end{equation}\nComponents of the shear tensor are obtained as,  \n\n\\begin{eqnarray}\\label{secondterm}\n\\sigma^{\\tau\\tau} = 0,\\,\n\\sigma^{x_1 x_1} = \\frac{-2a+b+c}{3}\\tau^{-2a-1},\\,\\nonumber\\\\\n\\sigma^{x_2 x_2} = \\frac{a-2b+c}{3}\\tau^{-2b-1},\\,\\nonumber\\\\\n\\sigma^{x_3 x_3} = \\frac{a+b-2c}{3}\\tau^{-2c-1}\n\\end{eqnarray}\nwhere we have used Kasner conditions and,\n\n\\begin{equation}\\label{theta}\n\\theta\\equiv \\nabla_\\mu u^\\mu = \\Delta^{\\nu}_{\\mu}D_{\\nu}u^{\\mu} \n=\\frac{a+b+c}{\\tau}.\n\\end{equation}\n\nUpon using Kasner condition, $\\theta$ becomes $\\frac{1}{\\tau}$ \nwhich is the same as in one dimensional Bjorken expansion case. \nThe shear stress tensor is diagonal. \nWe assume that it can be \ncharacterized by a function $\\pi$ and the  Kasner parameters in the \nfollowing form:   \n$\\pi^{\\mu\\nu}\\equiv\\mathrm{diag}(0,-\\pi\\tau^{-2a},\n\\frac{\\pi}{2}\\tau^{-2b},\\frac{\\pi}{2}\\tau^{-2c})$. One can check \nthat $\\pi^{\\mu\\nu}$ is traceless \nas we are considering a conformal fluid. \n\nThe components of $\\dot{\\pi}^{\\langle \\mu\\nu \\rangle}$ are given by, \n\\begin{eqnarray}{\\label{zeroterm}}\n\\dot{\\pi}^{\\langle \\tau\\tau \\rangle}= 0, \\,\n\\dot{\\pi}^{\\langle x_1x_1 \\rangle}=\\frac{-1}{\\tau^{2a}}\\frac{d\\pi}{d\\tau}, \\,\\nonumber\\\\\n\\dot{\\pi}^{\\langle x_2x_2 \\rangle}=\\frac{1}{2\\tau^{2b}}\\frac{d\\pi}{d\\tau}, \\,\n\\dot{\\pi}^{\\langle x_3x_3 \\rangle}=\\frac{1}{2\\tau^{2c}}\\frac{d\\pi}{d\\tau}\n\\end{eqnarray}\n\nComponents of other terms appearing in the second order \nevolution equation are obtained as,\n\\begin{eqnarray}{\\label{fourthterm}}\n\\pi_\\gamma^{\\langle \\tau}\\sigma^{\\tau\\rangle\\gamma}=0,\n\\pi_\\gamma^{\\langle x_1}\\sigma^{x_1\\rangle\\gamma}=\\frac{\\pi(b+c-2a)}\n{6 \\tau^{2a+1}},\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_2}\\sigma^{x_2 \\rangle\\gamma}=\n\\frac{\\pi(a+b-2c)}{6 \\tau^{2b+1}},\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_3}\\sigma^{x_3 \\rangle\\gamma}=\\frac{\\pi(a+c-2b)}\n{6 \\tau^{2c+1}}\n\\end{eqnarray}\nSubstituing the above expressions, the second order evolution equations\nare given by\n\\begin{eqnarray}\n\\frac{d\\epsilon}{d\\tau} &= -\\frac{1}{\\tau}\\left(\\epsilon + P \\right)  \n-\\frac{\\pi(b+c-2a)}{2\\tau} \\label{BED1}, \\\\\n\\frac{d\\pi}{d\\tau} &= - \\frac{\\pi}{\\tau_\\pi} - \n\\frac{2\\beta_\\pi (-2a+b+c)}{3\\tau} - \\frac{\\pi(38a+23b+23c)}{21\\tau}, \n\\label{Bshear1}\n\\end{eqnarray}\nNote that, adding the three equations for the nonzero components of\n$\\dot{\\pi}^{\\langle \\mu\\nu \\rangle}$, we get only one independent \nequation as given above in eqn.(\\ref{Bshear1}). \nIt is important to note that the Kasner metric in the limit of \n$a=1, b=0, c=0$ reduces to that of the Minkowski metric in \nMilne coordinates. In this limit, the above second order evolution \nequations reduce to the evolution equations in the one dimensional \nBjorken expansion case \\cite{AJ}:\n\\begin{eqnarray}\n\\frac{d\\epsilon}{d\\tau} &= -\\frac{1}{\\tau}\\left(\\epsilon + P \\right)  \n+\\frac{\\pi}{\\tau} \\label{BED2} \\\\\n\\frac{d\\pi}{d\\tau} &= - \\frac{\\pi}{\\tau_\\pi} + \\frac{4\\beta_{\\pi}}{3\\tau} \n- \\frac{38\\pi}{21\\tau}  \\label{Bshear2}\n\\end{eqnarray}\n\nNext, we compute the other terms appearing in the third order \nevolution equation of $\\pi^{\\mu\\nu}$ and give the explicit expressions \nfor the various terms. Components of $\\pi_\\gamma^{\\langle \\tau}\\pi^{\\tau \n\\rangle\\gamma}\\theta$ are given by, \n\n\\begin{eqnarray}{\\label{seventhterm}}\n\\pi_\\gamma^{\\langle \\tau}\\pi^{\\tau \\rangle\\gamma}\\theta=0,\n\\pi_\\gamma^{\\langle x_1}\\pi^{x_1 \\rangle\\gamma}\\theta=\n\\frac{\\pi^{2}(a+b+c)}{2\\tau^{2a+1}},\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_2}\\pi^{x_2 \\rangle\\gamma}\\theta=\n\\frac{\\pi^{2}(a+b+c)}{4\\tau^{2b+1}}\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_3}\\pi^{x_3 \\rangle\\gamma}\\theta=\n\\frac{\\pi^{2}(a+b+c)}{4\\tau^{2c+1}}\n\\end{eqnarray}\n\nComponents of $\\pi^{\\mu\\nu}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}$ are : \n\\begin{eqnarray}{\\label{eightterm}}\n\\pi^{\\tau\\tau}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=0,\\nonumber\\\\\n\\pi^{x_1x_1}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{-\\pi^{2}(2a-b-c)}{2\\tau^{2a+1}},\\nonumber\\\\\n\\pi^{x_2x_2}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{\\pi^{2}(2a-b-c)}{4\\tau^{2b+1}}\\nonumber\\\\\n\\pi^{x_3x_3}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{\\pi^{2}(2a-b-c)}{4\\tau^{2c+1}}\n\\end{eqnarray}\n\nComponents of $\\pi^{\\rho\\langle \\mu}\\pi^{\\nu \\rangle\\gamma}\\sigma_{\\rho\\gamma}$\nare given by, \n\\begin{eqnarray}{\\label{ninethterm}}\n\\pi^{\\rho\\langle \\tau}\\pi^{\\tau \\rangle\\gamma}\\sigma_{\\rho\\gamma}=0,\\nonumber\\\\\n\\pi^{\\rho\\langle x_1}\\pi^{x_1\\rangle\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{\\pi^{2}(-2a+b+c)}{4\\tau^{2a+1}},\\nonumber\\\\\n\\pi^{\\rho\\langle x_2}\\pi^{x_2 \\rangle\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{\\pi^{2}(a-b)}{4\\tau^{2b+1}}\\nonumber\\\\\n\\pi^{\\rho\\langle x_3}\\pi^{x_3 \\rangle\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{\\pi^{2}(a-c)}{4\\tau^{2c+1}}\n\\end{eqnarray}\n\nFor $\\nabla^{\\langle \\mu}\\left(\\nabla_\\gamma\\pi^{\\nu \\rangle\\gamma}\\right)$\nwe have, \n\\begin{eqnarray}{\\label{eleventhterm}}\n\\nabla^{\\langle \\tau}\\left(\\nabla_\\gamma\\pi^{\\tau \\rangle\\gamma}\\right)=0\\nonumber\\\\\n\\nabla^{\\langle x_1}\\left(\\nabla_\\gamma\\pi^{x_1 \\rangle\\gamma}\\right) =\\frac{\\pi}{6\\tau^{2a+2}}(8a^{2}+2b^{2}+2c^{2}\n-4ab+2bc-4ac)\\nonumber\\\\\n\\nabla^{\\langle x_2}\\left(\\nabla_\\gamma\\pi^{x_2\\rangle\\gamma}\\right) =\n\\frac{\\pi}{6\\tau^{2b+2}}(-4a^{2}-4b^{2}+2c^{2}\n+5ab-bc-ac)\\nonumber\\\\\n\\nabla^{\\langle x_3}\\left(\\nabla_\\gamma\\pi^{x_3\\rangle\\gamma}\\right) =\n\\frac{\\pi}{6\\tau^{2c+2}}(-4a^{2}+2b^{2}-4c^{2}\n-ab-bc+5ac)\n\\end{eqnarray}\n\nFor $\\nabla_{\\gamma}\\left(\\nabla^{\\langle \\mu}\\pi^{\\nu \\rangle\\gamma}\\right)$,\nwe get, \n\\begin{eqnarray}{\\label{twelveterm}}\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle \\tau}\\pi^{\\tau \\rangle\\gamma}\\right)=0\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle x_1}\\pi^{x_1\\rangle\\gamma}\\right) =\n\\frac{\\pi}{6\\tau^{2a+2}}(8a^{2}+2b^{2}+2c^{2}\n+5ab+2bc+5ac)\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle x_2}\\pi^{x_2 \\rangle\\gamma}\\right) =\n\\frac{\\pi}{6\\tau^{2b+2}}(-4a^{2}-4b^{2}+2c^{2}\n-4ab-bc-ac)\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle x_3}\\pi^{x_3 \\rangle\\gamma}\\right) =\n\\frac{\\pi}{6\\tau^{2c+2}}(-4a^{2}+2b^{2}-4c^{2}\n-ab-bc-4ac)\n\\end{eqnarray}\n\nComponents of $\\pi^{\\mu\\nu}\\theta^2$ are given by, \n\\begin{eqnarray}{\\label{eighteenterm}}\n\\pi^{\\tau \\tau}\\theta^2=0,\n\\pi^{x_1x_1}\\theta^2=\\frac{-\\pi}{\\tau^{2a}}\\left(\\frac{a+b+c}{\\tau}\\right)^{2} \\nonumber\\\\\n\\pi^{x_2x_2}\\theta^2=\\frac{\\pi}{2\\tau^{2b}}\\left(\\frac{a+b+c}{\\tau}\\right)^{2}\n\\nonumber\\\\\n\\pi^{x_3x_3}\\theta^2=\\frac{\\pi}{2\\tau^{2c}}\\left(\\frac{a+b+c}{\\tau}\\right)^{2}\n\\end{eqnarray}\n\nFor $\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle \\mu\\nu \\rangle}\n\\right)$, we obtain, \n\\begin{eqnarray}{\\label{fourteenterm}}\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle \\tau\\tau \\rangle}\n\\right)=0,\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle x_1x_1 \\rangle}\\right)\n=\\frac{\\pi(4a^{2}+b^{2}+c^{2})}{3\\tau^{2a+2}},\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle x_2x_2 \\rangle}\\right)\n=\\frac{\\pi(-2a^{2}-2b^{2}+c^{2})}{3\\tau^{2b+2}}\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle x_3x_3 \\rangle}\\right)\n=\\frac{\\pi(-2a^{2}+b^{2}-2c^{2})}{3\\tau^{2c+2}}\n\\end{eqnarray}\n\nAll these expressions can be simplified further by using Kasner \nconditions. Here we have \n$\\omega^{\\mu\\nu} = \\dot {u}^{\\mu} = \\nabla^{\\mu} \\tau_{\\pi} = 0$.\nThe other terms in the evolution equations as given below become zero, namely, \n\\begin{eqnarray}{\\label{zero-terms}}\n\\pi_{\\gamma}^{\\langle\\mu}\\omega^{\\nu\\rangle\\gamma} & =0 \\\\ \n\\nonumber\n\\pi^{\\rho\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\pi_{\\rho\\gamma} & =0 \\\\\n\\nonumber\n\\nabla^{\\langle\\mu}\\left(\\pi^{\\nu\\rangle\\gamma}\\dot u_\\gamma\\tau_\\pi\\right) \n&=0 \\\\\n\\nonumber\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\dot u^{\\langle\\mu}\\pi^{\\nu\\rangle\\gamma}\\right)\n& =0 \\\\\n\\nonumber\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\dot u^{\\gamma}\\pi^{\\langle\\mu\\nu\\rangle}\\right)\n&=0 \\\\\n\\nonumber\n\\tau_\\pi\\omega^{\\rho\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\pi_{\\rho\\gamma}&=0 \\\\\n\\nonumber\n\\tau_\\pi\\pi^{\\rho\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\omega_{\\rho\\gamma} &=0 \\\\\n\\nonumber\n\\tau_\\pi\\pi_\\gamma^{\\langle\\mu}\\omega^{\\nu\\rangle\\gamma}\\theta &=0 \n\\end{eqnarray}\n\nSubstituting the expressions for the above nonzero terms, the third order \nevolution equations for $x_1x_1$, $x_2x_2$, $x_3x_3$ components of the \nshear stress tensor are obtained as,  \n\\begin{eqnarray}{\\label{x-x term}}\n\\frac{d\\pi}{d\\tau} &=-\\frac{\\pi}{\\tau_\\pi}-2\\beta_\\pi\\frac{(-2a+b+c)}{3\\tau}+\\frac{\\pi(-38a-23b-23c)}{21\\tau}\n+\\frac{\\pi^{2}(-803a+34b+34c)}{1470\\beta_{\\pi}\\tau}\\nonumber\\\\\n&+\\frac{\\pi^{2}(106a^{2}-11b^{2}-11c^{2}-13ab-13ac-76bc)}{420\\beta_{\\pi}\\tau}  \n\\end{eqnarray}\n\\begin{eqnarray}{\\label{y-y term}}\n\\frac{d\\pi}{d\\tau} &= - \\frac{\\pi}{\\tau_\\pi} + 4\\beta_\\pi\\frac{(a-2b+c)}{3\\tau}+\\frac{\\pi(-38a-38b-8c)}{21\\tau}\n+\\frac{\\pi^{2}(-803a+199b-131c)}{1470\\beta_{\\pi}\\tau}\\nonumber\\\\\n&+\\frac{\\pi^{2}(53a^{2}+53b^{2}-64c^{2}+25ab-38ac-38bc)}{210\\beta_{\\pi}\\tau}  \n\\end{eqnarray}\n\\begin{eqnarray}{\\label{z-z term}}\n\\frac{d\\pi}{d\\tau} &= - \\frac{\\pi}{\\tau_\\pi} + 4\\beta_\\pi\\frac{(a+b-2c)}{3\\tau}+\\frac{\\pi(-38a-8b-38c)}{21\\tau}\n+\\frac{\\pi^{2}(-803a-131b+199c)}{1470\\beta_{\\pi}\\tau}\\nonumber\\\\\n&+\\frac{\\pi^{2}(53a^{2}-64b^{2}+53c^{2}+25ac-38ab-38bc)}{210\\beta_{\\pi}\\tau}  \n\\end{eqnarray}\n\nThese equations can be further simplified by using Kasner conditions.  \nOne can check that adding the equations for the $x_1x_1$, \n$x_2x_2$ and $x_3x_3$ components, \none gets only one independent equation, which is eq.(\\ref{x-x term}). \nThe evolution equation for the energy density $\\epsilon$ is obtained as, \n\\begin{eqnarray}\\label{epsilon}\n\\frac{d\\epsilon}{d\\tau} = -\\frac{1}{\\tau} (\\epsilon + P){\\tau} \n- \\frac{\\pi (b + c -2a)}{2\\tau}\n\\end{eqnarray}\n\nIn the limit $a=1, b=0, c=0$ for the Kasner parameters, \nthe evolution equation for the energy density reduces\nto that of the one dimensional Bjorken expansion case \\cite{AJ}, \n\\begin{eqnarray}\\label{epsilon-1}\n\\frac{d\\epsilon}{d\\tau} &= -\\frac{1}{\\tau} (\\epsilon + P -\\pi)\n\\end{eqnarray}\nand the third order evolution equation for the shear stress tensor \n$\\pi^{\\mu\\nu}$ reduces to \nthat of the equation for the Bjorken's one dimensional \nexpansion case, namely, \n\\begin{eqnarray}\n\\frac{d\\pi}{d\\tau} &= - \\frac{\\pi}{\\tau_\\pi} + \n\\frac{4 \\beta_\\pi}{3\\tau} - \n\\frac{38}{21}\\frac{\\pi}{\\tau} - \\frac{72}{245}\\frac{\\pi^2}{\\beta_\\pi\\tau}. \n\\end{eqnarray}\nBy comparing the above equation with the third order evolution equation \nin the one dimensional expansion case \\cite{AJ}\n\\begin{eqnarray}\n\\frac{d\\pi}{d\\tau} &= - \\frac{\\pi}{\\tau_\\pi} +\n\\frac{4 \\beta_\\pi}{3\\tau} -\n\\lambda \\frac{\\pi}{\\tau} - \\chi \\frac{\\pi^2}{\\beta_\\pi\\tau},\n\\end{eqnarray}\nthe transport coefficients are given by, \n\\begin{equation}\\label{BTC}\n\\tau_\\pi = \\frac{\\eta}{\\beta_\\pi}, \\quad \\beta_\\pi = \\frac{4P}{5}, \n\\quad \\lambda = \\frac{38}{21}, \\quad \\chi = \\frac{72}{245}.\n\\end{equation}\nwhich matches with the results in the one dimensional expansion case. \n\nHere we would like to point out that one could have \nintroduced three \nindependent fields $\\pi_i (i=1, 2, 3)$ for the shear stress tensor \n$\\pi^{\\mu\\nu}$ instead of a\nsingle function $\\pi$, as has been introduced in the above discussion \nto charataterize $\\pi^{\\mu\\nu}$. We have\nexplicitly checked that by making a general ansatz\nfor the shear stress tensor $\\pi^{\\mu\\nu} = diag (0, \\pi_1 \\tau^{-2a},\n\\pi_2 \\tau^{-2b}, \\pi_3 \\tau^{-2c})$ by introducing three independent\nfunctions $\\pi_1, \\pi_2, \\pi_3$, we get three independent third order\nevolution equations (also second order) for the components of the shear \nstress tensor \ninvolving $\\pi_1, \\pi_2$ and $\\pi_3$. These are given by, \n\\begin{eqnarray}\n\\frac{d\\pi_{1}}{d\\tau} &= - \\frac{\\pi_{1}}{\\tau_\\pi}+\n2\\beta_\\pi\\frac{(-2a+b+c)}{3\\tau}+\\frac{\\pi_{1}(-124a-64b-64c)}{63\\tau}\n+\\frac{\\pi_{2}(-10a+20b-10c)}{63\\tau}+\n\\frac{\\pi_{3}(-10a-10b+20c)}{63\\tau}\\nonumber\\\\\n&-\\frac{1}{735\\beta_{\\pi}\\tau}[\\pi_{1}^{2}(-386a-52b-52c)\n+\\pi_{2}^{2}(45a+155b+45c)+\\pi_{3}^{2}(45a+45b+155c)\\nonumber\\\\\n&+\\pi_{1}\\pi_{2}(38a-76b+38c)+\\pi_{1}\\pi_{3}(38a+38b-76c)]\\nonumber\\\\\n&+\\frac{\\tau_{\\pi}}{630 \\tau^{2}}[\\pi_{1}(212a^{2}-100b^{2}-100c^{2}-68ab-68ac\\nonumber\\\\\n&-200bc)+\\pi_{2}(-156b^{2}-84ab-48bc)\n+\\pi_{3}(-156c^{2}-84ac-48bc)]\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{d\\pi_{2}}{d\\tau} &=-\\frac{\\pi_{2}}{\\tau_\\pi}+\n2\\beta_\\pi\\frac{(a-2b+c)}{3\\tau}+\\frac{\\pi_{1}(20a-10b-10c)}{63\\tau}\n+\\frac{\\pi_{2}(-64a-124b-64c)}{63\\tau}+\\frac{\\pi_{3}(-10a-10b+20c)}{63\\tau}\\nonumber\\\\\n&-\\frac{1}{735\\beta_{\\pi}\\tau}[\\pi_{1}^{2}(155a+45b+45c)\n+\\pi_{2}^{2}(-52a-386b-52c)+\\pi_{3}^{2}(45a+45b+155c)\\nonumber\\\\\n&+\\pi_{1}\\pi_{2}(-76a+38b+38c)+\\pi_{2}\\pi_{3}(38a+38b-76c)]\\nonumber\\\\\n&+\\frac{\\tau_{\\pi}}{315\\tau^{2}}[\\pi_{1}(-78a^{2}-42ab-24ac)+\\pi_{2}(-50a^{2}+106b^{2}\\nonumber\\\\\n&-50c^{2}-34ab-34bc\n-100ac)+\\pi_{3}(-78c^{2}-24ac-42bc)]\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{d\\pi_{3}}{d\\tau} &=-\\frac{\\pi_{3}}{\\tau_\\pi} +\n2\\beta_\\pi\\frac{(a+b-2c)}{3\\tau}+\\frac{\\pi_{1}(20a-10b-10c)}{63\\tau}\n+\\frac{\\pi_{2}(-10a+20b-10c)}{63\\tau}+\\frac{\\pi_{3}(-64a-64b-124c)}{63\\tau}\\nonumber\\\\\n&-\\frac{1}{735\\beta_{\\pi}\\tau}[\\pi_{1}^{2}(155a+45b+45c)\n+\\pi_{2}^{2}(45a+155b+45c)+\\pi_{3}^{2}(-52a-52b-386c)\\nonumber\\\\\n&+\\pi_{1}\\pi_{3}(-76a+38b+38c)+\\pi_{2}\\pi_{3}(38a-76b+38c)]\\nonumber\\\\\n&+\\frac{\\tau_{\\pi}}{315\\tau^{2}}[\\pi_{1}(-78a^{2}-24ab-42ac)\n+\\pi_{2}(-78b^{2}-24ab-42bc)+\\pi_{3}(-50a^{2}-50b^{2}\\nonumber\\\\\n&+106c^{2}-34ac-34bc-100ab)]\n\\end{eqnarray}\nSince $\\pi^{\\mu\\nu}$ is traceless, for the Kasner metric $g_{\\mu\\nu}$, \nwe get the condition\n\\begin{equation}\n\\pi_1 + \\pi_2 + \\pi_3 = 0\n\\end{equation}\nUsing this tracelessness condition, we find\nthat adding the equations for $\\pi_2$ and $\\pi_3$, we get precisely the\nequation for $\\pi_1$ (where, $\\dot\\pi^{\\langle 1 1 \\rangle} = \n\\frac{-1}{\\tau^{2 a}} \\frac{d\\pi_1} {d\\tau}$). Below we show this \nexplicitly. For making this check, \nwe have also computed all the relevant terms appearing in the third order\nevolution equations in term of $\\pi_1, \\pi_2, \\pi_3$ (we give these \nexpressions in the appendix).  \nAdding the equations for $\\frac{d\\pi_{2}}{d\\tau}$ and $\\frac{d\\pi_{3}}{d\\tau}$,\nand using $\\pi_2 + \\pi_3 = -\\pi_1$, we obtain,\n\\begin{eqnarray}{\\label{11-final-term}}\n\\frac{d\\pi_{1}}{d\\tau} &= - \\frac{\\pi_{1}}{\\tau_\\pi}+ 2\\beta_\\pi\\frac{(-2a+b+c)}{3\\tau}+\\frac{\\pi_{1}(-40a+20b+20c)}{63\\tau}\n+\\frac{\\pi_{2}(74a+104b+74c)}{63\\tau}+\\frac{\\pi_{3}(74a+74b+104c)}{63\\tau}\\nonumber\\\\\n&-\\frac{1}{735\\beta_{\\pi}\\tau}[\\pi_{1}^{2}(-310a-90b-90c)\n+\\pi_{2}^{2}(7a+231b+7c)+\\pi_{3}^{2}(7a+7b+231c)\\nonumber\\\\\n&+\\pi_{1}\\pi_{2}(76a-38b-38c)+\\pi_{1}\\pi_{3}(76a-38b-38c)\\nonumber\\\\\n&+\\pi_{2}\\pi_{3}(-76a+38b+38c)]\n+\\frac{\\tau_{\\pi}}{315 \\tau^{2}} [\\pi_{1}(156 a^{2} +\n66 ab + 66ac) + \\pi_2 (50 a^2 -28 b^2 \\nonumber\\\\\n&+ 50c^2+58ab + 76 bc + 100 ac)+\\pi_3 (50 a^2\\nonumber\\\\\n& + 50 b^2 -28 c^2 + 76 bc + 58 ac + 100ab)]\n\\end{eqnarray}\nThis can be simplified to,\n\\begin{eqnarray}{\\label{eta-eta term}}\n\\frac{d\\pi_{1}}{d\\tau} &= - \\frac{\\pi_{1}}{\\tau_\\pi}+ 2\\beta_\\pi\\frac{(-2a+b+c)}{3\\tau}+\\frac{\\pi_{1}(-124a-64b-64c)}{63\\tau}\n+\\frac{\\pi_{2}(-10a+20b-10c)}{63\\tau}\n+\\frac{\\pi_{3}(-10a-10b+20c)}{63\\tau}\\nonumber\\\\\n&-\\frac{1}{735\\beta_{\\pi}\\tau}[\\pi_{1}^{2}(-386a-52b-52c)\n+\\pi_{2}^{2}(45a+155b+45c)+\\pi_{3}^{2}(45a+45b+155c)\\nonumber\\\\\n&+\\pi_{1}\\pi_{2}(38a-76b+38c)+\\pi_{1}\\pi_{3}(38a+38b-76c)]\\nonumber\\\\\n&+\\frac{\\tau_{\\pi}}{630 \\tau^{2}}[\\pi_{1}(212a^{2}-100b^{2}-100c^{2}-68ab-68ac-200bc)+\\pi_{2}(-156b^{2}\n\\nonumber\\\\\n&-84ab-48bc)+\\pi_{3}(-156c^{2}\n-84ac-48bc)]+\\frac{(\\pi_{1}+\\pi_{2}+\\pi_{3})(84a+84b+84c)}{63\\tau} \\nonumber\\\\\n&+ \\frac{1}{735\\beta_{\\pi}\\tau}[(-38a-38b-38c)\n(\\pi_1+\\pi_2+\\pi_3)^2 \n+ 114 (a \\pi_1 + b \\pi_{2} + c \\pi_{3})(\\pi_{1}+\\pi_{2}+\\pi_{3})] \\nonumber\\\\\n&+\\frac{\\tau_{\\pi}}{630 \\tau^{2}}\n(\\pi_{1}+\\pi_{2}+\\pi_{3})(100a^2 + 100b^2+100c^2\n+200ab+200bc+200ac)\n\\end{eqnarray}\nwhere the last three terms are zero as $\\pi_1 + \\pi_2 + \\pi_3 = 0$.\nRest of\nthe terms are precisely the expression for the $\\frac{d\\pi_1}{d\\tau}$\nequation. Hence there is no inconsistency and there is only one\nindependent ordinary differential equation (ODE).\nHence, for simplicity, we have made the assumption that\n$\\pi_{\\mu\\nu}$\ncould be characterized by a single function $\\pi$ and the three ODEs\ninvolving $x_1x_1, x_2x_2, x_3x_3$ components of the shear tensor \nare not totally\nindependent, rather adding the $x_2x_2$ and $x_3x_3$ equations ,\nit reduces to the equation for the $x_1x_1$ component. This we have \nalready discussed with reference to equations (40), (41) and (42) \nin this section. We have shown above \nthat this is also the case for the general ansatz in terms\nof $\\pi_1, \\pi_2$ and $\\pi_3$. We have also checked that putting\n$\\pi_1 = - \\pi, \\pi_2 = \\frac{\\pi}{2}, \\pi_3 = \\frac{\\pi}{2}$ in the\nODEs, we get back the earlier equations in terms of $\\pi$, namely, \nequations (40), (41) and (42) (where we have used the first order \nrelation $\\pi = \\frac{4\\beta_{\\pi}\\tau_{\\pi}}{3 \\tau}$ for rewriting \nthe last term in terms of the coefficient $\\frac{\\pi^2}{\\beta_{\\pi}\\tau}$. \nAll these expressions also reduce to the\nBjorken case for $a=1, b=0=c$.\nHence the ansatz is consistent. This is also true for the second\norder evolution equation.  \n\n\n\\section{Summary and discussion}\n\\label{sec:Summary}\n\\setcounter{equation}{0}\n\nIn this work, we have studied relativistic viscous hydrodynamics \nfrom kinetic theory to second and third order in gradient expansion. \nWe have considered Kasner space-time as the LRF\nof the three dimensional anisotropic expansion of a conformal fluid \nand have obtained the second and \nthird order evolution equations for the shear stress tensor and energy \ndensity. \nWe have used the iterative solutions of the Boltzmann \nequation for the nonequilibrium distribution function in RTA. \nWe have shown that our \nresults for the three dimensional expansion agree with the one dimensional \nBjorken expansion case in appropriate limit of the anisotropy parameters. \nFor a general \nfluid in Kasner space-time, the system may not be conformal, in which \ncase the stress energy tensor will not be traceless. It is a challenging \ntopic to study the evolution of the shear stress tensor for a nonconformal \nsystem with nonzero bulk viscosity in a general curved background. \nNonconformal fluid models contain a much larger number of transport \ncoefficients. There has been some progress in related aspects from various \napproaches (see for example, \n\\cite{Romatschke:2009, Denicol:2014, Finazzo:2014}).\nIt will be certainly interesting to explore the dissipative evolution \nequations in a higher order gradient expansion involving nonconformal \nfluid in Kasner space-time.\nOne can also work out the higher order corrections to entropy four current \nin our setup of anisotropic expansion. It will be interesting to \nrelate our results for anisotropic \nspace-time to anisotropic hydrodynamics \n(see ref.\\cite{Strickland:2014} for a review on anisotropic hydrodynamics). \nIt will be worth exploring higher order dissipative hydrodynamics for \nGubser flow \\cite{Chandrodoy:2018} and relate it to anisotropic hydrodynamics \nin the present scenario. \nIn the context of anisotropic expansion of the fluid, it will also be\ninteresting to study the higher order corrections in relativistic \nhydrodynamics in extended relaxation time approximation \n\\cite{Rocha-etal:2021, Jaiswal-etal:2021}, where the \nrelaxation time depends on particle energy. We hope to report \non these issues in future. \n\n\\vskip 10pt\n\\noindent\n{\\bf Acknowledgements} \\\\\nWe would like to thank Amaresh Jaiswal for helpful discussions.\n\\vskip 10pt\n\n\\section{Appendix}\n\\label{sec:Appendix}\n\\setcounter{equation}{0}\nWe assume that the shear stress tensor is diagonal and \nis characterized \nby three functions $\\pi_1, \\pi_2$ and $\\pi_3$, namely,  \n$\\pi^{\\mu\\nu}\\equiv\\mathrm{diag}(0,\\pi_{1}\\tau^{-2a},\n\\pi_{2}{\\tau^{-2b}},\\pi_{3}{\\tau^{-2c}})$.\nThe components of $\\dot{\\pi}^{\\langle \\mu\\nu \\rangle}$ are \ngiven by, \n\\begin{eqnarray}{\\label{zeroterm}}\n\\dot{\\pi}^{\\langle \\tau\\tau \\rangle}= 0, \\,\n\\dot{\\pi}^{\\langle x_1x_1 \\rangle}=\\frac{1}{\\tau^{2a}}\\frac{d\\pi_1}{d\\tau}, \\,\n\\nonumber\\\\\n\\dot{\\pi}^{\\langle x_2x_2 \\rangle}=\\frac{1}{\\tau^{2b}}\\frac{d\\pi_2}{d\\tau}, \\,\n\\dot{\\pi}^{\\langle x_3x_3 \\rangle}=\\frac{1}{\\tau^{2c}}\\frac{d\\pi_3}{d\\tau}\n\\end{eqnarray}\n\nComponents of other terms appearing in the third order \nevolution equation are obtained as,\n\\begin{eqnarray}{\\label{fourthterm}}\n\\pi_\\gamma^{\\langle \\tau}\\sigma^{\\tau\\rangle\\gamma}= 0\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_{1}}\\sigma^{x_{1}\\rangle\\gamma}=\\frac{1}\n{9 \\tau^{2a+1}}(-2\\pi_{1}(-2a+b+c)\\nonumber\\\\\n+\\pi_{2}(a-2b+c)+\\pi_{3}(a+b-2c)\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_{2}}\\sigma^{x_{2} \\rangle\\gamma}=\n\\frac{1}{9 \\tau^{2b+1}}(\\pi_{1}(-2a+b+c)\\nonumber\\\\\n-2\\pi_{2}(a-2b+c)+\\pi_{3}(a+b-2c)\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_{3}}\\sigma^{x_{3} \\rangle\\gamma}=\\frac{1}\n{9 \\tau^{2c+1}}(\\pi_{1}(-2a+b+c)\\nonumber\\\\\n+\\pi_{2}(a-2b+c)-2\\pi_{3}(a+b-2c)\n\\end{eqnarray}\n\nComponents of $\\pi_\\gamma^{\\langle \\tau}\\pi^{\\tau \n\\rangle\\gamma}\\theta$ are given by, \n\\begin{eqnarray}{\\label{seventhterm}}\n\\pi_\\gamma^{\\langle \\tau}\\pi^{\\tau \\rangle\\gamma}\\theta=0\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_{1}}\\pi^{x_{1} \\rangle\\gamma}\\theta=\n\\frac{(-2\\pi_{1}^{2}+\\pi_{2}^{2}+\\pi_{3}^{2})(a+b+c)}{3\\tau^{2a+1}}\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_{2}}\\pi^{x_{2} \\rangle\\gamma}\\theta=\\frac{(\\pi_{1}^{2}-\n2\\pi_{2}^{2}+\\pi_{3}^{2})(a+b+c)}{3\\tau^{2b+1}}\\nonumber\\\\\n\\pi_\\gamma^{\\langle x_{3}}\\pi^{x_{3} \\rangle\\gamma}\\theta=\\frac{(\\pi_{1}^{2}+\n\\pi_{2}^{2}-2\\pi_{3}^{2})(a+b+c)}{3\\tau^{2c+1}}\n\\end{eqnarray}\n\nComponents of $\\pi^{\\mu\\nu}\\pi^{\\rho\\gamma}\n\\sigma_{\\rho\\gamma}$ are : \n\\begin{eqnarray}{\\label{eightterm}}\n\\pi^{\\tau\\tau}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=0\\nonumber\\\\\n\\pi^{x_{1}x_{1}}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=\\frac{\\pi_{1}}{3\\tau^{2a+1}}[(\\pi_{1}(-2a+b+c)\\nonumber\\\\\n+\\pi_{2}(a-2b+c)+\\pi_{3}(a+b-2c)]\\nonumber\\\\\n\\pi^{x_{2}x_{2}}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=\\frac{\\pi_{2}}{3\\tau^{2b+1}}[(\\pi_{1}(-2a+b+c)\\nonumber\\\\\n+\\pi_{2}(a-2b+c)+\\pi_{3}(a+b-2c)]\\nonumber\\\\\n\\pi^{x_{3}x_{3}}\\pi^{\\rho\\gamma}\\sigma_{\\rho\\gamma}=\\frac{\\pi_{3}}{3\\tau^{2c+1}}[(\\pi_{1}(-2a+b+c)\\nonumber\\\\\n+\\pi_{2}(a-2b+c)+\\pi_{3}(a+b-2c)]\n\\end{eqnarray}\n\nComponents of $\\pi^{\\rho\\langle \\mu}\\pi^{\\nu \\rangle\\gamma}\n\\sigma_{\\rho\\gamma}$ are given by, \n\\begin{eqnarray}{\\label{ninethterm}}\n\\pi^{\\rho\\langle \\tau}\\pi^{\\tau \\rangle\\gamma}\\sigma_{\\rho\\gamma}=0\\nonumber\\\\\n\\pi^{\\rho\\langle x_{1}}\\pi^{x_{1}\\rangle\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{1}{9\\tau^{2a+1}}[(2\\pi_{1}^{2}(-2a+b+c)\\nonumber\\\\\n-\\pi_{2}^{2}(a-2b+c)-\\pi_{3}^{2}(a+b-2c)]\\nonumber\\\\\n\\pi^{\\rho\\langle x_{2}}\\pi^{x_{2} \\rangle\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{1}{9\\tau^{2b+1}}[-\\pi_{1}^{2}(-2a+b+c)\\nonumber\\\\\n+2\\pi_{2}^{2}(a-2b+c)-\\pi_{3}^{2}(a+b-2c)]\\nonumber\\\\\n\\pi^{\\rho\\langle x_{3}}\\pi^{x_{3} \\rangle\\gamma}\\sigma_{\\rho\\gamma}=\n\\frac{1}{9\\tau^{2c+1}}[-\\pi_{1}^{2}(-2a+b+c)\\nonumber\\\\\n-\\pi_{2}^{2}(a-2b+c)+2\\pi_{3}^{2}(a+b-2c)]\n\\end{eqnarray}\n\nFor $\\nabla^{\\langle \\mu}\\left(\\nabla_\\gamma\\pi^{\\nu \n\\rangle\\gamma}\\right)$ we have, \n\\begin{eqnarray}{\\label{eleventhterm}}\n\\nabla^{\\langle \\tau}\\left(\\nabla_\\gamma\\pi^{\\tau \\rangle\\gamma}\\right)=0\n\\nonumber\\\\\n\\nabla^{\\langle x_{1}}\\left(\\nabla_\\gamma\\pi^{x_{1} \\rangle\\gamma}\\right)=\n\\frac{1}{3\\tau^{2a+2}}(\\pi_{1}(-4a^{2}+ab+ac)\\nonumber\\\\\n+\\pi_{2}(2b^{2}-2ab+bc)+\\pi_{3}(2c^{2}+bc-2ac))\\nonumber\\\\\n\\nabla^{\\langle x_{2}}\\left(\\nabla_\\gamma\\pi^{x_{2} \\rangle\\gamma}\\right)=\n\\frac{1}{3\\tau^{2b+2}}(\\pi_{1}(2a^{2}-2ab+ac)\\nonumber\\\\+\\pi_{2}(-4b^{2}+ab+bc)+\\pi_{3}(2c^{2}-2bc+ac))\\nonumber\\\\\n\\nabla^{\\langle x_{3}}\\left(\\nabla_\\gamma\\pi^{x_{3} \\rangle\\gamma}\\right)=\n\\frac{1}{3\\tau^{2c+2}}(\\pi_{1}(2a^{2}+ab-2ac)\\nonumber\\\\+\\pi_{2}(2b^{2}+ab-2bc)\n+\\pi_{3}(-4c^{2}+bc+ac))\n\\end{eqnarray}\n\nFor $\\nabla_{\\gamma}\\left(\\nabla^{\\langle \\mu}\\pi^{\\nu \n\\rangle\\gamma}\\right)$, we get, \n\\begin{eqnarray}{\\label{twelveterm}}\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle \\tau}\\pi^{\\tau \\rangle\\gamma}\\right)=0\n\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle x_{1}}\\pi^{x_{1} \\rangle\\gamma}\\right)=\n\\frac{1}{3\\tau^{2a+2}}(\\pi_{1}(-4a^{2}-2ab-2ac)\\nonumber\\\\\n+ \\pi_{2}(2b^{2}+ab+bc)+\\pi_{3}(2c^{2}+bc+ac))\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle x_{2}}\\pi^{x_{2} \\rangle\\gamma}\\right)=\n\\frac{1}{3\\tau^{2b+2}}(\\pi_{1}(2a^{2}+ab+ac)\\nonumber\\\\\n+\\pi_{2}(-4b^{2}-2ab-2bc)+\\pi_{3}(2c^{2}+bc+ac))\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\nabla^{\\langle x_{3}}\\pi^{x_{3} \\rangle\\gamma}\\right)=\n\\frac{1}{3\\tau^{2c+2}}(\\pi_{1}(2a^{2}+ab+ac)\\nonumber\\\\\n+\\pi_{2}(2b^{2}+ab+bc)+\\pi_{3}(-4c^{2}-2bc-2ac))\n\\end{eqnarray}\n\nFor $\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\n\\pi^{\\langle \\mu\\nu \\rangle} \\right)$, we obtain, \n\\begin{eqnarray}{\\label{fourteenterm}}\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle \\tau\\tau \\rangle}\n\\right)=0\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle x_{1}x_{1} \\rangle}\n\\right)=\\frac{2\\tau_\\pi(-2a^{2}\\pi_{1}+b^{2}\\pi_{2}+c^{2}\\pi_{3})}{3\\tau^{2a+2}}\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle x_{2}x_{2} \\rangle}\n\\right)=\\frac{2\\tau_\\pi(a^{2}\\pi_{1}-2b^{2}\\pi_{2}+c^{2}\\pi_{3})}{3\\tau^{2b+2}}\n\\nonumber\\\\\n\\nabla_{\\gamma}\\left(\\tau_\\pi\\nabla^{\\gamma}\\pi^{\\langle x_{3}x_{3} \\rangle}\n\\right)=\\frac{2\\tau_\\pi(a^{2}\\pi_{1}+b^{2}\\pi_{2}-2c^{2}\\pi_{3})}{3\\tau^{2c+2}}\n\\end{eqnarray}\n\nComponents of $\\pi^{\\mu\\nu}\\theta^2$ are given by, \n\\begin{eqnarray}{\\label{eighteenterm}}\n\\pi^{\\tau \\tau}\\theta^2=0\\nonumber\\\\\n\\pi^{x_{1}x_{1}}\\theta^2=\\frac{\\pi_{1}}{\\tau^{2a}}\\left(\\frac{a+b+c}{\\tau}\n\\right)^{2}\\nonumber\\\\\n\\pi^{x_{2}x_{2}}\\theta^2=\\frac{\\pi_{2}}{\\tau^{2b}}\\left(\\frac{a+b+c}{\\tau}\n\\right)^{2}\\nonumber\\\\\n\\pi^{x_{3}x_{3}}\\theta^2=\\frac{\\pi_{3}}{\\tau^{2c}}\\left(\\frac{a+b+c}{\\tau}\n\\right)^{2}\n\\end{eqnarray}\nThese expressions have beeen used to obtain the differential equations \nfor the components of $\\pi^{\\mu\\nu}$ in terms of \n$\\pi_1, \\pi_2, \\pi_3$. All these expressions reduce to our earlier \nresults where we have considered a single function $\\pi$ to characterize\nthe shear stress tensor $\\pi^{\\mu\\nu}$.  \n\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe cyclic sieving phenomenon for rectangular standard tableaux was first\nproved in \\cite{Rhoades2010}. This proof used the Khazdhan-Lusztig\nbasis of the Hecke algebras. This result has been reproved in \\cite{Purbhoo2013} using the geometry of the Wronksian and has been generalised in using \\cite{Fontaine2014} using the geometry\nof the affine Grassmannian and in \\cite{Westbury2016} using Lusztig's\nbased modules. Here we give a new proof which is relatively self-contained\nusing the action of the cactus groups in the seminormal bases for the\nirreducible representations of the Hecke algebras.\n\nOur main theorem has a straightforward statement.\nLet $\\lambda$ be a partition of size $r$ and let $U_\\lambda$ be the\nassociated irreducible representation of the symmetric group, $\\mathfrak{S}_r$,\nover the rational field, $\\mathbb{Q}$. Let $c_1$ be the matrix representing the\nlong cycle with respect to a chosen basis.\nLet $c_0$ be the permutation matrix of the jeu-de-taquin promotion acting\non the set of standard tableaux of shape $\\lambda$.\nThen our main theorem is:\n\\begin{thm}\\label{thm:main} If the partition $\\lambda$ has rectangular shape then the matrices $c_1$ and $c_0$ are conjugate.\n\\end{thm}\n\nThe method of proof is to construct an interpolating matrix. This is a matrix, $c_q$, with entries in the field of rational functions, $\\mathbb{Q}(q)$, with\nthe properties:\n\\begin{itemize}\n\t\\item The evaluation of $c_q$ at $q=1$ is defined and gives $c_1$\n\t\\item The evaluation of $c_q$ at $q=0$ is defined and gives $c_0$\n\t\\item $c_q^r=1$\n\\end{itemize}\n\nThis proves Theorem~\\ref{thm:main} since $c_q$ is semisimple, the eigenvalues\nare $r$-th roots of unity and the eigenvalues are analytic functions of $q$.\nHence the eigenvalues and their multiplicities are independent of $q$.\n\n\\begin{ex}\nThe rotation matrix and its inverse are\n\\begin{equation*}\n\\left(\\begin{array}{rrrrr}\n\\frac{1}{3} & \\frac{4}{9} & 0 & \\frac{2}{3} & 0 \\\\\n\\frac{1}{2} & -\\frac{1}{12} & \\frac{3}{8} & -\\frac{1}{8} & \\frac{9}{16} \\\\\n1 & -\\frac{1}{6} & -\\frac{1}{4} & -\\frac{1}{4} & -\\frac{3}{8} \\\\\n0 & \\frac{1}{2} & \\frac{3}{4} & -\\frac{1}{4} & -\\frac{3}{8} \\\\\n0 & 1 & -\\frac{1}{2} & -\\frac{1}{2} & \\frac{1}{4}\n\\end{array}\\right)\n\\qquad\n\\left(\\begin{array}{rrrrr}\n\\frac{1}{3} & \\frac{4}{9} & \\frac{2}{3} & 0 & 0 \\\\\n\\frac{1}{2} & -\\frac{1}{12} & -\\frac{1}{8} & \\frac{3}{8} & \\frac{9}{16} \\\\\n0 & \\frac{1}{2} & -\\frac{1}{4} & \\frac{3}{4} & -\\frac{3}{8} \\\\\n1 & -\\frac{1}{6} & -\\frac{1}{4} & -\\frac{1}{4} & -\\frac{3}{8} \\\\\n0 & 1 & -\\frac{1}{2} & -\\frac{1}{2} & \\frac{1}{4}\n\\end{array}\\right)\n\\end{equation*}\n\nThe promotion matrix and its inverse are\n\\begin{equation*}\n\\left(\\begin{array}{rrrrr}\n0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0\n\\end{array}\\right)\n\\qquad\n\\left(\\begin{array}{rrrrr}\n0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0\n\\end{array}\\right)\n\\end{equation*}\n\nThe matrix which interpolates between promotion and rotation and its inverse are\n\\begin{equation*}\n\\left(\\begin{array}{rrrrr}\n\\frac{1}{[3]} & \\frac{[4]}{[3]^2} & 0 & \\frac{[4]}{[2][3]} & 0 \\\\\n\\frac{1}{[2]} & \\frac{-1}{[3][2]^2} & \\frac{[3]}{[2]^3} & \\frac{-1}{[2]^3} & \\frac{[3]^2}{[2]^4} \\\\\n1 & \\frac{-1}{[2][3]} & \\frac{-1}{[2]^2} & \\frac{-1}{[2]^2} & \\frac{-[3]}{[2]^3} \\\\\n0 & \\frac{1}{[2]} & \\frac{[3]}{[2]^2} & \\frac{-1}{[2]^2} & \\frac{-[3]}{[2]^3} \\\\\n0 & 1 & \\frac{-1}{[2]} & \\frac{-1}{[2]} & \\frac{1}{[2]^2}\n\\end{array}\\right)\n\\qquad\n\\left(\\begin{array}{rrrrr}\n\\frac{1}{[3]} & \\frac{[4]}{[3]^2} & \\frac{[2]}{[3]} & 0 & 0 \\\\\n\\frac{1}{[2]} & -\\frac{1}{[2]^2[3]} & -\\frac{1}{[2]^3} & \\frac{[3]}{[2]^3} & \\frac{[3]^2}{[2]^4} \\\\\n0 & \\frac{1}{[2]} & -\\frac{1}{[2]^2} & \\frac{[3]}{[2]^2} & -\\frac{[3]}{[2]^3} \\\\\n1 & -\\frac{1}{[2][3]} & -\\frac{1}{[2]^2} & -\\frac{1}{[2]^2} & -\\frac{[3]}{[2]^3} \\\\\n0 & 1 & -\\frac{1}{[2]} & -\\frac{1}{[2]} & \\frac{1}{[2]^2}\n\\end{array}\\right)\n\\end{equation*}\n\nIn this example the matrix which intertwines promotion and rotation can be given explicitly.\nThis matrix and its inverse are:\n\\begin{equation*}\n\\left(\\begin{array}{rrrrr}\n1 & -\\frac{[2]}{[3]} & 0 & 0 & -\\frac 1{[2]} \\\\\n0 & 1 & -\\frac 1{[2]} & -\\frac 1{[2]} & \\frac 1{[2]^2} \\\\ \n0 & 0 & 1 & 0 & -\\frac 1{[2]} \\\\\n0 & 0 & 0 & 1 & -\\frac 1{[2]} \\\\\n0 & 0 & 0 & 0 & 1\n\\end{array}\\right)\n\\qquad\n\\left(\\begin{array}{rrrrr}\n1 & \\frac{[2]}{[3]} & \\frac 1{[3]} & \\frac 1{[3]} & \\frac {[2]}{[3]} \\\\\n0 & 1 & \\frac 1{[2]} & \\frac 1{[2]} & \\frac 1{[2]^2} \\\\\n0 & 0 & 1 & 0 & \\frac 1{[2]} \\\\\n0 & 0 & 0 & 1 & \\frac 1{[2]} \\\\\n0 & 0 & 0 & 0 & 1\n\\end{array}\\right)\n\\end{equation*}\n\\end{ex}\n\nOur construction of the interpolating matrix uses the cactus group $\\mathfrak{C}_r$.\nThis group acts on the set of standard tableaux of shape $\\lambda$. This\naction is given explicitly in \\cite{Kirillov1995} (where the cactus groups are\nimplicit) and implicitly in \\cite{Henriques2006d} (where the cactus groups are\nexplicit). The relationship between these two papers was established in\n\\cite{Chmutov2016}.\n\nThe Hecke algebra, $H_r$, is a $q$-analogue of the symmetric group algebra $\\mathbb{Q}\\,\\mathfrak{S}_r$. As a $\\mathbb{Q}(q)$-algebra it is split semisimple and each \n$U_\\lambda$ has a $q$-analogue which is also absolutely irreducible.\nOur main technical tool is that, for $r>1$, there is a homomorphism\n$\\mathfrak{C}_r\\to H_r$. This homomorphism is constructed implictly using \nquantum Schur-Weyl duality and the result that the category of type I\nfinite dimensional representations of a quantised enveloping algebra is a\ncoboundary category; this is part of the construction in \\cite{Drinfelprimed1989}.\nThis implies that, for each $\\lambda\\vdash r$, there\nis an action of $\\mathfrak{C}_r$ on $U_\\lambda$.\nThis action was made explicit in \\cite{Westbury2018} in that the matrices\nrepresenting a set of generators with respect to the seminormal basis\nof $U_\\lambda$ are given explicitly. Here we make use of these matrices.\n\nThe contents of the sections are:\n\\begin{description}\n\t\\item[Cyclic sieving phenomenon] In this section we give the background\non the cyclic sieving phenomenon and deduce the cyclic sieving phenomenon\nfrom Theorem~\\ref{thm:main}.\n\\item[Cactus groups] In this section we define the cactus groups by\nfinite presentations and give the results that are needed in the proof of\nTheorem~\\ref{thm:main}.\n\\item[Hecke algebras] In this section we recall the construction of\nthe representations of the Hecke algebra in the seminormal basis and\nprove Theorem~\\ref{thm:main}.\n\\item[Conclusion] In this section we put the statement and method of proof\nof Theorem~\\ref{thm:main} in the context of the representation theory\nof quantised enveloping algebras and discuss the main difficulty in\ngeneralising Theorem~\\ref{thm:main} to this context.\n\\end{description}\n\n\\section{Cyclic sieving phenomenon}\nIn this section we explain how Theorem~\\ref{thm:main} gives the\ncyclic sieving phenomenon in \\cite{Rhoades2010}.\n\nLet $r>1$ and $\\omega$ be a primitive $r$-th root of unity. Recall the\ndefinition of the cyclic sieving phenomenon from \\cite{Reiner2004}.\n\\begin{defn}\nLet $X$ be a finite set and $c\\colon X\\to X$ a bijection that satisfies\n$c^r=1$. If the polynomial $P\\in \\mathbb{Z}[q]$ \\footnote{Note that the $q$ in this section is not the $q$ used in other sections.} satisfies\n\\begin{equation*}\n\tP(\\omega^k) = |\\Fix(c^k)|\n\\end{equation*}\nfor all $k$ where $\\Fix$ is the set of fixed points then the triple $(X,c,P)$\nexhibits the \\Dfn{cyclic sieving phenomenon}.\n\\end{defn}\nLet $\\lambda\\vdash r$ be a rectangular shape and let $X$ be the set of\nstandard tableaux of shape $\\lambda$. Denote the promotion operator\non $X$ by $p$. The problem is to determine a polynomial $P$ such that\n$(X,p,P)$ exhibits the cyclic sieving phenomenon.\n\nThe linear version of the cyclic sieving phenomenon is:\n\\begin{defn}\nLet $U$ be a finite dimensional vector space over $\\mathbb{Q}$ and $\\rho\\colon U\\to U$ an isomorphism that satisfies $\\rho^r=1$. A \\Dfn{character polynomial} is a \npolynomial $P\\in \\mathbb{Q}[q]$ that satisfies\n\\begin{equation*}\n\tP(\\omega^k) = \\tr(\\rho^k)\n\\end{equation*}\nfor all $k$.\n\\end{defn}\n\nThe two basic properties of the character polynomial are:\n\\begin{itemize}\n\\item If $\\rho$ is the permutation matrix of a bijection $c$ then $(X,c,P)$\nsatisfies the cyclic sieving phenomenon if and only if $P$ is a\ncharacter polynomial of $\\rho$.\n\n\\item Given $\\rho\\colon U\\to U$ with character polynomial $P$ and\n$\\rho'\\colon U'\\to U'$ with character polynomial $P'$ then \n$(U,\\rho)$ and $(U',\\rho')$ are isomorphic if and only if $P=P'$.\n\\end{itemize}\nIt then follows from Theorem~\\ref{thm:main} that $(X,p,P)$ exhibits the cyclic sieving phenomenon if and only if $P$ is a character polynomial for the\naction of the long cycle on $U_\\lambda$.\n\nThis character polynomial can be determined using:\n\\begin{prop}\\label{prop:fr} Let $U$ be a representation of $\\mathfrak{S}_r$. Then the principal specialisation of the Frobenius character of $U$ is a character polynomial for the action of the long cycle.\n\\end{prop}\n\n\\begin{lemma} Let $\\lambda\\vdash r$. Take $P$ to be the $q$-analogue of the hook-length formula\n\t\\begin{equation*}\n\t\t\\frac{[n]!}{\\prod_{(i,j)\\in \\lambda} [h(i,j)]}\n\t\\end{equation*}\n\twhere $h(i,j)$ is the hook length of the cell $(i,j)$ and $[n]$ is given by\n\t\\begin{equation*}\n\t\t[n] = \\frac{1-q^n}{1-q}\n\t\\end{equation*}\nThen $P$ is a character polynomial for the action of the long cycle on $U_\\lambda$.\n\\end{lemma}\n\n\\begin{proof} This is an application of Proposition~\\ref{prop:fr}.\nThe Frobenius character of $U_\\lambda$ is the Schur function $s_\\lambda$;\nand the principal specialisation of $s_\\lambda$ is given by\nthe $q$-analogue of the hook-length formula.\n\\end{proof}\n\nOther interpretations of this polynomial are available. For example,\nthis polynomial is the generating function for the statistic major index\non the set of standard tableaux of shape $\\lambda$.\n\nThe conclusion is that if $P$ is the $q$-analogue of the hook-length formula\nthen $(X,p,P)$ exhibits the cyclic sieving phenomenon. This is the main\ntheorem of \\cite{Rhoades2010}.\n\n\\section{Cactus group}\\label{sec:cactus}\nThe finite presentations of the cactus groups are:\n\\begin{defn}\\label{defn:cactus} The \\Dfn{$r$-fruit cactus group}, $\\mathfrak{C}_r$, has generators $s_{p,\\,q}$ for $1\\le p<q\\le r$ and defining relations\n\\begin{itemize}\n\t\\item $s_{p,\\,q}^2=1$\n\t\\item $s_{p,\\,q}\\, s_{k,\\,l}=s_{k,\\,l}\\, s_{p,\\,q}$ if $\\os{p,q}\\cap\\os{k,l}=\\emptyset$\n\t\\item $s_{p,\\,q}\\, s_{k,\\,l}=s_{p+q-l,\\,p+q-k}\\, s_{p,\\,q}$ if $\\os{k,l}\\subseteq\\os{p,q}$\n\\end{itemize}\n\\end{defn}\n\nLet $\\mathfrak{S}_r$ be the symmetric group on $r$ letters.\nThere is a homomorphism $\\mathfrak{C}_r\\rightarrow \\mathfrak{S}_r$ defined by $s_{p,\\,q}\\mapsto \\widehat{s}_{p,\\,q}$ where $\\widehat{s}_{p,\\,q}$ is the permutation\n\\begin{equation}\\label{eq:hom}\n\\widehat{s}_{p,\\,q}(i)=\\begin{cases}\np+q-i & \\text{if $p\\le i\\le q$} \\\\\ni & \\text{otherwise}\n\\end{cases}\n\\end{equation}\n\nNote that $\\mathfrak{C}_r$ is generated by $s_{1,q}$ for $2\\le q\\le r$, since\n\\begin{equation}\\label{eqn:gen}\ns_{p,\\,q}=s_{1,\\,q}s_{1,\\,q-p}s_{1,\\,q}\n\\end{equation}\n\nNext we introduce several sets of generators for the cactus groups,\nfollowing \\cite{Kirillov2001}.\n\nThe first set is $q_i=s_{1,\\,i}$ for $1\\leqslant i\\leqslant n$.\nThese are generators, as noted in \\eqref{eqn:gen}.\nThe second set of generators\nare $p_i$ given by the relations\n\\begin{equation*}\np_i=\\begin{cases}\nq_1 & \\text{if $i=1$}\\\\\nq_{i-1}\\,q_i & \\text{if $i>1$}\n\\end{cases} \\qquad\nq_i = p_1\\,p_2\\dotsb p_i\n\\end{equation*}\nThe third set of generators are $t_i$ given by\n\\begin{equation*}\nt_i=\\begin{cases}\np_1 & \\text{if $i=1$}\\\\\np_i\\,p_{i-1}^{-1} & \\text{if $i>1$}\n\\end{cases} \\qquad\np_i = t_i\\,t_{i-1}\\dotsb t_1\n\\end{equation*}\n\nThen we also have\n\\begin{equation*}\nt_i=\\begin{cases}\nq_1 & \\text{if $i=1$}\\\\\nq_1\\,q_2\\,q_1 & \\text{if $i=2$}\\\\\nq_{i-1}\\,q_i\\,q_{i-1}\\,q_{i-2} & \\text{if $i>2$}\n\\end{cases}\n\\end{equation*}\n\nThe images of these three sets of generators under the homomorphism to $\\mathfrak{S}_r$ are:\n\\begin{itemize}\n\\item the image of $t_i$ is the transposition $(i,i+1)$\n\\item the image of $p_i$ is the cycle $(i,1,2,\\dotsc,i-1)$\n\\item the image of $q_i$ is the involution\n\\begin{equation*}\nq_i(j)=\\begin{cases}\ni-j+1  & \\text{if $1\\leqslant j\\leqslant i$}\\\\\nj & \\text{if $j>i$}\n\\end{cases}\n\\end{equation*}\nIn particular, the image of $p_{r-1}\\in\\mathfrak{S}_r$ is the long cycle and\nthe image of $q_r\\in\\mathfrak{S}_r$ is the longest element.\n\\end{itemize}\n\nThere is a dual version of these generators.\n\\begin{align*}\n\tv_i &= t_i\\,t_{i+1}\\dotsc t_{r-1}\\\\\n\tw_i &= v_{r-1}\\,v_{r-2}\\dotsc v_{r-i}\n\\end{align*}\n\nThe following is \\cite[Proposition~1.4]{Kirillov1995} and is also clear from\nthe growth diagram.\n\\begin{lemma}\\label{lem:cyclic} For $r>1$,\n\t\\begin{equation*}\n\tp_{r-1}^r=w_{r-1}\\,q_{r-1}\n\t\\end{equation*}\n\\end{lemma}\n\nFor a standard tableau, $T$, let $s_iT$ be the\ntableau obtained from $T$ by interchanging $i$ and $i+1$.\nDefine an involution $t_i$ on standard tableaux by\n\\begin{equation*}\n\tt_i(T) = \\begin{cases}\n\t\ts_iT & \\text{if $s_iT$ is standard} \\\\\n\t\tT & \\text{otherwise}\n\t\\end{cases}\n\\end{equation*}\n\nThen these involutions generate an action of the cactus group $\\mathfrak{C}_r$\non standard tableaux of size $r$. The action of the element $p_{r-1}\\in\\mathfrak{C}_r$\nis the same as jeu-de-taquin promotion.\n\n\\begin{prop}\\label{prop:order} The operator $p_{r-1}$ acting on rectangular tableaux of size $r$\n\tsatisfies\n\t$$p_{r-1}^r=1$$\n\\end{prop}\n\n\\begin{proof}\n\tBy Lemma~\\ref{lem:cyclic} it is sufficient to show $w_{n-1} = q_{n-1}$.\n\tThis follows from two observations on the reverse-complement.\n\tRecall that reverse-complement is the involution given by rotating a rectangular shape\n\tstandard tableaux through a half-turn and reversing the numbering.\n\t\n\tThe first observation is that conjugating by reverse-complement interchanges the two actions of the cactus groups. In particular, $w_{r-1}$ is the conjugate of $q_{n-1}$.\n\tThe second observation is that $q_{r-1}$ commutes with reverse-complement.\n\\end{proof}\n\n\\section{Hecke algebras}\\label{sec:hecke}\nLet $\\mathbb{Q}(q)$ be the field of rational functions in an indeterminate $q$.\nThe quantum integers $[n]\\in \\mathbb{Q}(q)$ are defined by\n\\begin{equation*}\n[n] = \\frac{q^n-q^{-n}}{q-q^{-1}}\n\\end{equation*}\n\n\n\\begin{defn}\n\tThe Hecke algebra $H_r(q)$ is generated by $u_i$ for $1\\le i\\le r-1$\n\tand the defining relations are\n\t\\begin{align*}\n\tu_i^2 &= -[2]\\, u_i \\\\\n\tu_i\\,u_{i+1}\\,u_i-u_i&=u_{i+1}\\,u_i\\,u_{i+1}-u_{i+1} \\\\\n\tu_i\\,u_j&=u_j\\,u_i\\qquad\\text{for $|i-j|>1$}\n\t\\end{align*}\n\\end{defn}\n\nLet $\\sigma_i$ be the standard generators of the braid group, $B_r$.\nThen we have a homomorphism $B_r\\to H_r(q)$ given by\n\\begin{equation*}\n\t\\sigma_i^{\\pm 1} \\mapsto q^{\\pm1}  + u_i\n\\end{equation*}\nThe image of $\\sigma_i$ then satisfies\n\\begin{equation*}\n\t\\sigma_i-\\sigma_i^{-1} = q - q^{-1}\n\\end{equation*}\nYoung's seminormal forms are representations of the symmetric groups.\nThese were introduced in \\cite[Theorem~IV]{Young1932}. Here we give\nthe analogous construction for the Hecke algebras.\n\nFix a shape $\\lambda$ of size $r$. Then we construct a \nrepresentation of $H_r(q)$ on the vector space with basis the set\nof standard tableaux of shape $\\lambda$.\n\nThe \\Dfn{content vector} of a standard tableau $T$ of size $n$ is a function\n$\\mathbf{ct}_T\\colon [1,2,\\dotsc ,n]\\to\\mathbb{Z}$. The entry\n$\\mathbf{ct}_T(k)\\in\\mathbb{Z}$ is given by $\\mathbf{ct}_T(k)=j-i$ if $k$ is in box $(i,j)$ in $T$.\nThe content vector of $T$ determines $T$. The \\Dfn{axial distance}\nis $a_T(i) = \\mathbf{ct}_{T}(i+1)-\\mathbf{ct}_{T}(i)$.\n\nDefine a linear operator $\\overline{s}_i$ by\n\\begin{equation*}\n\t\\overline{s}_i(T) = \\begin{cases}\n\t\ts_iT & \\text{if $s_iT$ is standard} \\\\\n\t\t0 & \\text{otherwise}\n\t\\end{cases}\n\\end{equation*}\n\nThe following is \\cite[Theorem~3.22]{Ram2003}. The seminormal representations of\n$H_r(q)$ are defined by giving the matrices representing the generators.\n\\begin{prop}\n\tThe action of $u_i$ is given by\n\t\\begin{equation*} u_i\\:T = \n\t\\begin{cases}\n\t-\\frac{[a-1]}{[a]}\\:T + \\frac{[a-1]}{[a]}\\:\\overline{s}_iT &\\text{if $\\mathbf{ct}_T(i+1)>\\mathbf{ct}_T(i)$}\\\\\n\t-\\frac{[a-1]}{[a]}\\:T + \\frac{[a+1]}{[a]}\\:\\overline{s}_iT &\\text{if $\\mathbf{ct}_T(i)>\\mathbf{ct}_T(i+1)$}\t\n\t\\end{cases}\n\t\\end{equation*}\nwhere $a$ is the axial distance $a_T(i)$.\n\\end{prop}\n\nAssume $T$ and $s_iT$ are standard and $a_T(i)>1$. Then on the subspace\nwith ordered basis $(T,s_iT)$.\n\\begin{equation*}\nu_i =\t\\left[ \\begin {array}{cc} -\\frac {[a-1]}{ [a]  }& \\frac{[a-1]}{[a]}\n\\\\ \\noalign{\\medskip}{\\frac {[a+1]  }{\n\t\t[a]}}&-\\frac {[a+1]}{ [a]  }\\end {array} \\right]\n\t\\qquad\n\\sigma_i=\\left[ \\begin {array}{cc} \\frac {q^a}{ [a]  }&\\frac{[a-1]}{[a]}\\:\n\\\\ \\noalign{\\medskip}{\\frac {[a+1]  }{\n\t\t[a]}}&\\frac {-q^{-a}}{ [a]  }\\end {array} \\right]\n\\end{equation*}\n\nThe action of $\\mathfrak{C}_r$ is defined by giving the matrices representing the generators\n$\\{t_i\\}$. The following is \\cite[Theorem~4.8]{Westbury2018}.\n\\begin{thm} The action of $t^{(q)}_i$ is given by\n\t\\begin{equation*} t^{(q)}_i (T) = \n\t\\begin{cases}\n\t-\\frac{1}{[a]}\\:T + \\frac{[a-1]}{[a]}\\:s_iT &\\text{if $\\mathbf{ct}_T(i+1)>\\mathbf{ct}_T(i)$}\\\\\n\t-\\frac{1}{[a]}\\:T + \\frac{[a+1]}{[a]}\\:s_iT &\\text{if $\\mathbf{ct}_T(i)>\\mathbf{ct}_T(i+1)$}\t\n\t\\end{cases}\n\t\\end{equation*}\n\\end{thm}\n\nAssume $T$ and $s_iT$ are standard and $a_T(i)>1$. Then on the subspace\nwith ordered basis $(T,s_iT)$.\n\\begin{equation*}\nt^{(q)}_i=\\left[ \\begin {array}{cc} \\frac {1}{ [a]  }&\\frac{[a-1]}{[a]}\n\\\\ \\noalign{\\medskip}{\\frac {[a+1]  }{\n\t\t[a]}}&-\\frac {1}{ [a]  }\\end {array} \\right]\n\\end{equation*}\n\nLet $p^{(q)}_{r-1}$ be the matrix representing $p_{r-1}\\in\\mathfrak{C}_r$. Then we prove\nTheorem~\\ref{thm:main} by showing that a modification of $p^{(q)}_{r-1}$\nis an interpolating matrix.\n\n\\begin{prop} The operator $p^{(q)}_{r-1}$ acting on rectangular tableaux of size $r$\n\tsatisfies\n\t$$(p^{(q)}_{r-1})^r=1$$\n\\end{prop}\n\n\\begin{proof} The proof is the same as the proof of Proposition~\\ref{prop:order}.\n\\end{proof}\n\n\\begin{lemma} The evaluation of $p^{(q)}_{r-1}$ at $q=1$ is defined and gives the action of the long cycle. \n\\end{lemma}\n\n\\begin{proof} The action of $\\mathfrak{C}_r$ on $U_\\lambda$ can be evaluated at $q=1$.\nThis action factors through the homomorphism in \\eqref{eq:hom} to give the\nstandard action of $\\mathfrak{S}_r$ in the seminormal basis. In particular,\nthe homomorphism in \\eqref{eq:hom} maps $p^{(q)}_{r-1}$ to the long cycle in\n$\\mathfrak{S}_r$.\n\\end{proof}\n\nThe matrix $t^{(q)}_i$ is not regular at $q=0$ as $\\frac{[a+1]}{[a]}$ has a\nsimple pole at $q=0$. Let $D$ be the diagonal matrix whose diagonal entry\ncorresponding to $T$ is $q^{\\mathrm{inv}(T)}$ where $\\mathrm{inv}(T)$ is the\ninversion number of $T$. Let $\\widehat{t}^{(q)}_i$ be $t^{(q)}_i$ conjugated by $D$.\n\n\\begin{lemma}\nThe matrix $\\widehat{t}^{(q)}_i$ is regular at $q=0$ and the evaluation at $q=0$ is the matrix of the involution $t_i$.\n\\end{lemma}\n\\begin{proof} By inspection.\n\\end{proof}\n\nFor any $\\lambda\\vdash r$, the matrices $\\widehat{t}^{(q)}_i$ generate an action of $\\mathfrak{C}_r$ and the matrix\nof any element is regular at $q=0$. Let $\\widehat{p}^{(q)}_{r-1}$ be the matrix\nof $p_{r-1}\\in\\mathfrak{C}_r$. Then we have shown that if $\\lambda$ has rectangular shape then $\\widehat{p}^{(q)}_{r-1}$\nis an interpolating matrix and so have proved Theorem~\\ref{thm:main}.\n\n\\section{Conclusion}\nIn conclusion we put this result in the context of the representation theory\nof quantum groups and discuss the generalisation of the statement and method of proof of Theorem~\\ref{thm:main} in this context.\n\nLet $\\mathsf{C}$ be a finite type Cartan matrix and $U_q(\\mathsf{C})$ the associated\nquantised enveloping algebra. Let $\\varpi$ be a dominant weight in the\nweight lattice of $\\mathsf{C}$. Let $V_q(\\varpi)$ be the type I highest weight\nrepresentation of $U_q(\\mathsf{C})$ and let $C(\\varpi)$ be the crystal of $V_q(\\varpi)$.\n\nThe cactus group $\\mathfrak{C}_r$ acts on the crystal $\\otimes^r C(\\varpi)$.\nThis implies that $\\mathfrak{C}_r$ acts on the set of highest weight words\nin $\\otimes^r C(\\varpi)$, preserving the weight. \nIn particular $\\mathfrak{C}_r$ acts on $\\Inv_r(C(\\varpi))$, the set of highest weight\nwords of weight 0. Let $c_0$ be the permutation matrix of the action of\n$p_{r-1}\\in\\mathfrak{C}_r$.\n\nLet $L(\\mathsf{C})$ be the semisimple Lie algebra of $\\mathsf{C}$. Let $V(\\varpi)$ be the\nhighest weight representation of $L(\\mathsf{C})$. The symmetric group $\\mathfrak{S}_r$\nacts on the representation $\\otimes^r V(\\varpi)$. \nThis implies that $\\mathfrak{S}_r$ acts on the space of highest weight tensors\nin $\\otimes^r V(\\varpi)$, preserving the weight. \nIn particular $\\mathfrak{S}_r$ acts on $\\Inv_r(V(\\varpi))$, the space of invariant tensors.\nLet $c_1$ be the matrix representing the long cycle with respect to a\nchosen basis.\n\nThe generalisation of Theorem~\\ref{thm:main} is that the matrices\n$c_0$ and $c_1$ are conjugate. Theorem~\\ref{thm:main} is the case $\\mathsf{C}$\nhas type $A$ and $\\varpi$ is the first fundamental weight; so \n$L(\\mathsf{C})$ is $\\mathfrak{sl}(n)$ for some $n$ and $V(\\varpi)$ is the vector representation.\n\nNow we attempt to generalise the proof of Theorem~\\ref{thm:main}.\nThe cactus group $\\mathfrak{C}_r$ acts on the representation $\\otimes^r V_q(\\varpi)$.\nThis implies that $\\mathfrak{C}_r$ acts on the set of highest weight tensors\nin $\\otimes^r V_q(\\varpi)$, preserving the weight. \nIn particular $\\mathfrak{C}_r$ acts on $\\Inv_r(V_q(\\varpi))$, the space of invariant tensors. Let $c_q$ be the matrix representing $p_{r-1}\\in\\mathfrak{C}_r$. with respect to a\nchosen basis.\n\nThen $c_q$ has the following two of the three properties of an interpolating\nmatrix. These properties are independent of the choice of basis of\n$\\Inv_r(V_q(\\varpi))$.\n\\begin{itemize}\n\t\\item If the evaluation of $c_q$ at $q=1$ is defined then this gives the action of the long cycle.\n\t\\item $c_q^r=1$.\n\\end{itemize}\n\nHence to show that $c_q$ is an interpolating matrix it remains to show\nthat there exists a basis of $\\Inv_r(V_q(\\varpi))$ such that $c_q$ is\nregular at $q=0$ and the evaluation at $q=0$ is $c_0$. If\n$V(\\varpi)$ admits an invariant symplectic form then there are sign issues.\nThese sign issues can be dealt with by taking $V(\\varpi)$ to be an odd super vector\nspace so that the symplectic form becomes a symmetric inner product.\nThis is proved in \\cite{Westbury2016} using the theory of based modules\nin \\cite{Lusztig1993}.\nIt would be preferable to have a self-contained proof.\n\nIn the proof of Theorem~\\ref{thm:main} we showed that a modification of the seminormal basis has the desired property. The seminormal basis\nof $\\Inv_r(V_q(\\varpi))$ is defined for any $\\varpi$ such that all nonzero\nweight spaces of $V(\\varpi)$ are one dimensional. Examples are,\n\\begin{itemize}\n\t\\item All minuscule representations.\n\t\\item The symmetric powers of the vector representation of $\\mathfrak{sl}(n)$.\n\t\\item The fundamental representation of $G_2$.\n\\end{itemize}\nThis raises the question of whether a modification of the seminormal basis\nhas the desired property.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}