spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

phase space networks of the six vertex model under different boundary conditions the six vertex model is mapped to three dimensional sphere stacks and different boundary conditions corresponding to different containers the shape of the container provides a qualitative visualization of the boundary effect based on the sphere stacking picture we map the phase spaces of the six vertex models to discrete networks a node in the network represents a state of the system and an edge between two nodes represents a zero energy spin flip which corresponds to adding or removing a sphere the network analysis shows that the phase spaces of systems with different boundary conditions share some common features we derived a few formulas for the number and the sizes of the disconnected phase space subnetworks under the periodic boundary conditions the sphere stacking provides new challenges in combinatorics and may cast light on some two dimensional models

uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

localization transition lifschitz tails and rare region effects in network models effects of heterogeneity in the suspected infected susceptible model on networks are investigated using quenched mean field theory the emergence of localization is described by the distributions of the inverse participation ratio and compared with the rare region effects appearing in simulations and in the lifschitz tails the latter in the linear approximation is related to the spectral density of the laplacian matrix and to the time dependent order parameter i show that these approximations indicate correctly griffiths phases both on regular one dimensional lattices and on small world networks exhibiting purely topological disorder i discuss the localization transition that occurs on scale free networks at 3 degree exponent

the higher order spectrum of simplicial complexes a renormalization group approach network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research the discrete topological objects that are investigated in network topology are simplicial complexes simplicial complexes generalize networks by not only taking pairwise interactions into account but also taking into account many body interactions between more than two nodes higher order laplacians are topological operators that describe higher order diffusion on simplicial complexes and constitute the natural mathematical objects that capture the interplay between network topology and dynamics we show that higher order up and down laplacians can have a finite spectral dimension characterizing the long time behaviour of the diffusion process on simplicial complexes that depends on their order m we provide a renormalization group theory for the calculation of the higher order spectral dimension of two deterministic models of simplicial complexes the apollonian and the pseudo fractal simplicial complexes we show that the rg flow is affected by the fixed point at zero mass which determines the higher order spectral dimension d s of the up laplacians of order m with m geq 0

spectral density of random graphs with topological constraints the spectral density of random graphs with topological constraints is analysed using the replica method we consider graph ensembles featuring generalised degree degree correlations as well as those with a community structure in each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question we highlight the effect of these topological constraints on the resulting spectral density

localizations on complex networks we study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix the probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the euclidean distance is invalid several quantities are used to describe the localization properties of the representative states such as the participation ratio the structural entropy and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks whole cell networks in the real world and the watts strogatz small world and barabasi albert scale free networks are considered the networks have nontrivial localization properties due to the nontrivial topological structures it is found that the ascending order ranked series of the occurrence probabilities at the nodes behave generally multifractally this characteristic can be used as a structural measure of complex networks

quantifying randomness in protein protein interaction networks of different species a random matrix approach abstract we analyze protein protein interaction networks for six different species under the framework of random matrix theory nearest neighbor spacing distribution of the eigenvalues of adjacency matrices of the largest connected part of these networks emulate universal gaussian orthogonal statistics of random matrix theory we demonstrate that spectral rigidity which quantifies long range correlations in eigenvalues for all protein protein interaction networks follow random matrix prediction up to certain ranges indicating randomness in interactions after this range deviation from the universality evinces underlying structural features in network

eigenvalue spectra of complex networks we examine the eigenvalue spectrum of the adjacency matrix of a random scale free network with an average of p edges per vertex using the replica method we show how in the dense limit when p one can obtain two relatively simple coupled equations whose solution yields for an arbitrary complex network for scale free graphs with degree distribution exponent we obtain an exact expression for the eigenvalue spectrum when 3 and show that 1 2 1 for large in the limit we recover known results for the erdos renyi random graph

a new method optimizing the subgraph centrality of large networks abstract since many realistic networks such as wireless sensor ad hoc networks usually do not agree very well with the basic network models such as small word and scale free models we often need to obtain some expected structural features such as a small average path length and a regular degree distribution while optimizing the connectivity of these networks although a minor addition of links for optimizing network connectivity is not likely to change the structural properties of a network it is necessary to investigate the impact of link addition on network properties as the number of the added links increases however to the best of our knowledge the study of that problem has not been found so far furthermore two closely related questions to that problem i e how to measure and how to improve network connectivity have not been studied carefully enough yet to address the three problems above the authors derive a better measure of network connectivity for large networks and a new strategy that can increase decrease network connectivity the most and propose a spectral density algorithm optimizing the connectivity of large networks which is able to indicate the impact on the structural properties of a network while increasing decreasing its connectivity providing us a guided optimization of network connectivity in other words our algorithm can optimize not only the connectivity of a large network but also its structural features meanwhile our new findings about spectral density are also concluded in this paper in addition we may also apply this algorithm to solve all eigenvalues of an n n matrix with a low complexity of o n 2 at most

spectral characteristics of network redundancy many real world complex networks contain a significant amount of structural redundancy in which multiple vertices play identical topological roles such redundancy arises naturally from the simple growth processes which form and shape many real world systems since structurally redundant elements may be permuted without altering network structure redundancy may be formally investigated by examining network automorphism symmetry groups here we use a group theoretic approach to give a complete description of spectral signatures of redundancy in undirected networks in particular we describe how a network s automorphism group may be used to directly associate specific eigenvalues and eigenvectors with specific network motifs

spectra of modular random graphs we compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure modules are initially introduced as fully connected sub units of a graph by contrast inter module connectivity is taken to be incomplete two different types of inter module connectivities are considered one where the number of intermodule connections per node diverges and one where this number remains finite in the infinite module size limit in the first case the results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules in the second case matters can be more involved and depend in detail on inter module connectivities for suitable parameters we even find near triangular shaped spectral densities similar to those observed in certain scale free networks in a system consisting of just two coupled modules analytic results are presented for the infinite module size limit they are well corroborated by numerical simulations

scaling invariance in spectra of complex networks a diffusion factorial moment approach a new method called diffusion factorial moment is used to obtain scaling features embedded in the spectra of complex networks for an erdos renyi network with connecting probability p mathit er l1 n the scaling parameter is ensuremath delta 0 51 while for p mathit er ensuremath geqslant 1 n the scaling parameter deviates from it significantly for ws small world networks in the special region p r 0 05 0 2 typical scale invariance is found for growing random networks in the range of ensuremath theta 0 33 049 we have ensuremath delta 0 6 ifmmode pm else textpm fi 0 1 and the value of ensuremath delta oscillates around ensuremath delta 0 6 abruptly in the range of ensuremath theta 0 54 1 we have basically ensuremath delta g0 7 scale invariance is one of the common features of the three kinds of networks which can be employed as a global measurement of complex networks in a unified way

spectral theory of sparse non hermitian random matrices sparse non hermitian random matrices arise in the study of disordereda 13 physical systems with asymmetric local interactions and have applications ranginga 13 from neural networks to ecosystem dynamics the spectral characteristics of thesea 13 matrices provide crucial information on system stability and susceptibility however a 13 their study is greatly complicated by the twin challenges of a lack of symmetry anda 13 a sparse interaction structure in this review we provide a concise and systematica 13 introduction to the main tools and results in this field we show how the spectraa 13 of sparse non hermitian matrices can be computed via an analogy with infinitea 13 dimensional operators obeying certain recursion relations with reference to threea 13 illustrative examples adjacency matrices of regular oriented graphs adjacencya 13 matrices of oriented erd s renyi graphs and adjacency matrices of weighted orienteda 13 erd s renyi graphs we demonstrate the use of these methods to obtain both analytica 13 and numerical results for the spectrum the spectral distribution the location of outliera 13 eigenvalues and the statistical properties of eigenvectors a 13

constrained markovian dynamics of random graphs we introduce a statistical mechanics formalism for the study of constrained graph evolution as a markovian stochastic process in analogy with that available for spin systems deriving its basic properties and highlighting the role of the mobility the number of allowed moves for any given graph as an application of the general theory we analyze the properties of degree preserving markov chains based on elementary edge switchings we give an exact yet simple formula for the mobility in terms of the graph s adjacency matrix and its spectrum this formula allows us to define acceptance probabilities for edge switchings such that the markov chains become controlled glauber type detailed balance processes designed to evolve to any required invariant measure representing the asymptotic frequencies with which the allowed graphs are visited during the process as a corollary we also derive a condition in terms of simple degree statistics sufficient to guarantee that in the limit where the number of nodes diverges even for state independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform we test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology showing explicitly that for instances where the simple edge swap dynamics fails to converge to the uniform measure a suitably modified markov chain instead generates the correct phase space sampling

normal mode analysis of spectra of random networks abstract several spectral fluctuation measures of random matrix theory rmt have been applied in the study of spectral properties of networks however the calculation of those statistics requires performing an unfolding procedure which may not be an easy task in this work network spectra are interpreted as time series and we show how their short and long range correlations can be characterized without implementing any previous unfolding in particular we consider three different representations of erd s renyi er random networks standard er networks er networks with random weighted self edges and fully random weighted er networks in each case we apply singular value decomposition svd such that the spectra are decomposed in trend and fluctuation normal modes we obtain that the fluctuation modes exhibit a clear crossover between the poisson and the gaussian orthogonal ensemble statistics when the average degree of er networks changes moreover by using the trend modes we perform a data adaptive unfolding to calculate for comparison purposes traditional fluctuation measures such as the nearest neighbor spacing distribution number variance 2 as well as 3 and n statistics the thorough comparison of rmt short and long range correlation measures make us identify the svd method as a robust tool for characterizing random network spectra

spectral perturbation and reconstructability of complex networks in recent years many network perturbation techniques such as topological perturbations and service perturbations were employed to study and improve the robustness of complex networks however there is no general way to evaluate the network robustness in this paper we propose a global measure for a network the reconstructability coefficient defined as the maximum number of eigenvalues that can be removed subject to the condition that the adjacency matrix can be reconstructed exactly our main finding is that a linear scaling law e an seems universal in that it holds for all networks that we have studied

laplacian spectra as a diagnostic tool for network structure and dynamics we examine numerically the three way relationships among structure laplacian spectra and frequency synchronization dynamics on complex networks we study the effects of clustering degree distribution and a particular type of coupling asymmetry input normalization all of which are known to have effects on the synchronizability of oscillator networks we find that these topological factors produce marked signatures in the laplacian eigenvalue distribution and in the localization properties of individual eigenvectors using a set of coordinates based on the laplacian eigenvectors as a diagnostic tool for synchronization dynamics we find that the process of frequency synchronization can be visualized as a series of quasi independent transitions involving different normal modes particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes for example there are important partially synchronized states in which a set of low lying modes remain unlocked while those in the main spectral peak are locked we find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue

anticipating stochastic differential equations regularity of the law abstract using malliavin calculus we give sufficient conditions ensuring the smoothness of the density for the law of the solution to a stratonovich stochastic differential equation with anticipating initial condition we study hormander type conditions restricted and unrestricted as well as a highly degenerate case

random matrix analysis of localization properties of gene coexpression network we analyze gene coexpression network under the random matrix theory framework the nearest neighbor spacing distribution of the adjacency matrix of this network follows gaussian orthogonal statistics of random matrix theory rmt spectral rigidity test follows random matrix prediction for a certain range and deviates afterwards eigenvector analysis of the network using inverse participation ratio suggests that the statistics of bulk of the eigenvalues of network is consistent with those of the real symmetric random matrix whereas few eigenvalues are localized based on these ipr calculations we can divide eigenvalues in three sets a the nondegenerate part that follows rmt b the nondegenerate part at both ends and at intermediate eigenvalues which deviates from rmt and expected to contain information about important nodes in the network c the degenerate part with zero eigenvalue which fluctuates around rmt predicted value we identify nodes corresponding to the dominant modes of the corresponding eigenvectors and analyze their structural properties

discrete time quantum walk on the apollonian network this work discusses the discrete time quantum walk dtqw on the first five generations of the apollonian network this structure is constructed in a geometric recurrent way and is characterized by among other features a scale free distribution of node degrees the dtqw formalism requires a node dependent coin operator that for each node has as many different output states as that node degree so that each local coin operator is expressed by the fourier operator the dtqw time evolution matrix has a larger rank than that of the network adjacency matrix or the matrix representation of the continuous time quantum walk ctqw results for the time evolution return time to specific nodes and asymptotic probability of site occupancy are obtained such dtqw specific features are discussed and compared to those obtained for the classical random walk and ctqw on the same lattice

collective chaos induced by structures of complex networks abstract mapping a complex network of n coupled identical oscillators to a quantum system the nearest neighbor level spacing nnls distribution is used to identify collective chaos in the corresponding classical dynamics on the complex network the classical dynamics on an erdos renyi network with the wiring probability p er 1 n is in the state of collective order while that on an erdos renyi network with p er 1 n in the state of collective chaos the dynamics on a ws small world complex network evolves from collective order to collective chaos rapidly in the region of the rewiring probability p r 0 0 0 1 and then keeps chaotic up to p r 1 0 the dynamics on a growing random network grn is in a special state deviates from order significantly in a way opposite to that on ws small world networks each network can be measured by a couple values of two parameters

spectral similarity for barabasi albert and chung lu models abstract in the paper we have analyzed spectral similarity between barabasi albert and chung lu models we have shown the similarity of spectral distribution for sufficiently large barabasi albert parameter value contrary extreme eigenvalues and principal eigenvector are not similar for those model we provide applications of obtained results related to the spectral graph theory and efficiency of quantum spatial search

geometry of complex networks and topological centrality abstract we explore the geometry of complex networks in terms of an n dimensional euclidean embedding represented by the moore penrose pseudo inverse of the graph laplacian l the squared distance of a node i to the origin in this n dimensional space l i i yields a topological centrality index defined as c i 1 l i i in turn the sum of reciprocals of individual node centralities i 1 c i i l i i or the trace of l yields the well known kirchhoff index k an overall structural descriptor for the network to put into context this geometric definition of centrality we provide alternative interpretations of the proposed indices that connect them to meaningful topological characteristics first as forced detour overheads and frequency of recurrences in random walks that has an interesting analogy to voltage distributions in the equivalent electrical network and then as the average connectedness of i in all the bi partitions of the graph these interpretations respectively help establish the topological centrality c i of node i as a measure of its overall position as well as its overall connectedness in the network thus reflecting the robustness of i to random multiple edge failures through empirical evaluations using synthetic and real world networks we demonstrate how the topological centrality is better able to distinguish nodes in terms of their structural roles in the network and along with kirchhoff index is appropriately sensitive to perturbations re wirings in the network

rotationally invariant family of levy like random matrix ensembles we introduce a family of rotationally invariant random matrix ensembles characterized by a parameter while 1 corresponds to well known critical ensembles we show that 1 describes levy like ensembles characterized by power law eigenvalue densities for 1 the density is bounded as in gaussian ensembles but 1 describes ensembles characterized by densities with long tails in particular the model allows us to evaluate in terms of a novel family of orthogonal polynomials the eigenvalue correlations for levy like ensembles these correlations differ qualitatively from those in either the gaussian or the critical ensembles

the convergence of diagonal pade approximants and the pade conjecture questions related to the convergence problem of diagonal pade approximants are discussed a central place is taken by the pade conjecture also known as the baker gammel wills conjecture partial results concerning this conjecture are reviewed and weaker and more special versions of the conjecture are formulated and their plausibility is investigated great emphasis is given to the role of spurious poles of the approximants a conjecture by nuttall 1970 about the number and distribution of such poles is stated and its importance for the pade conjecture is analyzed

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

sparse random matrices the eigenvalue spectrum revisited we revisit the derivation of the density of states of sparse random matrices we derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit using the iterative scheme introduced by biroli and monasson 1999 j phys a math gen 32 l255 we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration the combination of the two methods yields a very simple geometric interpretation of the tails of the spectrum we test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum

spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities we study structure eigenvalue spectra and random walk dynamics in a wide class of networks with subgraphs modules at mesoscopic scale the networks are grown within the model with three parameters controlling the number of modules their internal structure as scale free and correlated subgraphs and the topology of connecting network within the exhaustive spectral analysis for both the adjacency matrix and the normalized laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees the minimally connected nodes the clustering and the average connectivity affect the central part of the spectrum the number of distinct modules leads to an extra peak at the lower part of the laplacian spectrum in cyclic graphs such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs we also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda l 2 of the laplacian further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs the distribution of first return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1 3 for trees and sigma approximately 2 3 for cyclic graphs which is independent of their mesoscopic and global structure

exactly solvable scale free network model we study a deterministic scale free network recently proposed by barabasi ravasz and vicsek we find that there are two types of nodes the hub and rim nodes which form a bipartite structure of the network we first derive the exact numbers p k of nodes with degree k for the hub and rim nodes in each generation of the network respectively using this we obtain the exact exponents of the distribution function p k of nodes with k degree in the asymptotic limit of k infinity we show that the degree distribution for the hub nodes exhibits the scale free nature p k proportional to k gamma with gamma ln 3 ln 2 1 584 962 while the degree distribution for the rim nodes is given by p k proportional to e gamma k with gamma ln 3 2 0 405 465 second we analytically calculate the second order average degree of nodes d third we numerically as well as analytically calculate the spectra of the adjacency matrix a for representing topology of the network we also analytically obtain the exact number of degeneracies at each eigenvalue in the network the density of states i e the distribution function of eigenvalues exhibits the fractal nature with respect to the degeneracy fourth we study the mathematical structure of the determinant of the eigenequation for the adjacency matrix fifth we study hidden symmetry zero modes and its index theorem in the deterministic scale free network finally we study the nature of the maximum eigenvalue in the spectrum of the deterministic scale free network we will prove several theorems for it using some mathematical theorems thus we show that most of all important quantities in the network theory can be analytically obtained in the deterministic scale free network model of barabasi ravasz and vicsek therefore we may call this network model the exactly solvable scale free network

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

general connectivity distribution functions for growing networks with preferential attachment of fractional power we study the general connectivity distribution functions for growing networks with preferential attachment pa of fractional power i k using simon s method we first show that the heart of the previously known methods of the rate equations for the connectivity distribution functions is nothing but simon s method for word problem secondly we show that for the case of fractional the z transformation of the rate equation provides a fractional differential equation of a new type which coincides with that for pa with linear power when 1 we show that to solve such a fractional differential equation we need to define a transcendental function a b c z that we call upsilon function most of all the previously known results are obtained consistently in the framework of a unified theory

explicit construction of the eigenvectors and eigenvalues of the graph laplacian on the cayley tree abstract a generalized fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph laplacian using the symmetries of the cayley tree we recursively construct the family of eigenvectors with exponentially growing eigenspaces associated with eigenvalues in the lower part of the spectrum the spectral gap decays exponentially with the tree size for large trees the eigenvalues and eigenvectors obey recursion relations which arise from the nested geometry of the tree

random matrix theory models of electric grid topology abstract the random matrix theory is useful in the study of large systems such as electric grids these transmission systems can be modeled as complex networks with high voltage lines the edges that connect nodes representing power plants and substations we draw upon established literature of complex systems theory and introduce methods from nuclear and statistical physics to identify new characteristics of these networks we show that most grids can be characterized by the gaussian orthogonal ensemble an indicator of chaos in many complex systems under certain circumstances however grids may be described by poisson statistics an indicator of regularity we use the random matrix formalism to describe the interconnection of multiple grids and construct a simple model of a distributed grid

random networks eigenvalue spectra abstract we analyze the spectra of eigenvalues for random graphs with a local tree like structure the exact equations to the spectra of networks with a local tree like structure are presented we propose a simple approximation and in the framework of effective medium approximation calculate spectra of various graphs analytically we show that spectra of locally tree like random graphs gives a good description of the spectral properties of real life networks like the internet

random walks along the streets and canals in compact cities spectral analysis dynamical modularity information and statistical mechanics different models of random walks on the dual graphs of compact urban structures are considered analysis of access times between streets helps to detect the city modularity the statistical mechanics approach to the ensembles of lazy random walkers is developed the complexity of city modularity can be measured by an informationlike parameter which plays the role of an individual fingerprint of genius loci global structural properties of a city can be characterized by the thermodynamic parameters calculated in the random walk problem

spectra of modular and small world matrices we compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter and intra module degree distributions subject only to the constraint of finite mean connectivities we also evaluate spectra of a certain class of small world matrices generated from random graphs by introducing shortcuts via additional random connectivity components both adjacency matrices and the associated graph laplacians are investigated for the laplacians we find lifshitz type singular behaviour of the spectral density in a localized region of small values in the case of modular networks we can identify contributions of local densities of state from individual modules for small world networks we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states exhibiting only localized states in the band gaps results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances and with direct diagonalization results

spectra of random graphs with arbitrary expected degrees we study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices we give a complete prescription for calculating the spectra that is exact in the limit of large network size and large vertex degrees we also study the effect on the spectra of hubs in the network vertices of unusually high degree and show that these produce isolated eigenvalues outside the main spectral band akin to impurity states in condensed matter systems with accompanying eigenvectors that are strongly localized around the hubs we give numerical results that confirm our analytic expressions

scattering and transport properties of tight binding random networks we study numerically scattering and transport statistical properties of tight binding random networks characterized by the number of nodes n and the average connectivity we use a scattering approach to electronic transport and concentrate on the case of a small number of single channel attached leads we observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements the conductance probability distribution w t the average conductance the shot noise power p and the elastic enhancement factor f by varying from small 0 to large 1 values we also show that all these quantities are invariant for fixed n moreover we proposes a heuristic and universal relation between and p and the disorder parameter

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

dynamical scaling behavior of percolation clusters in scale free networks in this work we investigate the spectra of laplacian matrices that determine many dynamic properties of scale free networks below and at the percolation threshold we use a replica formalism to develop analytically based on an integral equation a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of treelike networks close to the percolation threshold we find characteristic scaling functions for the density of states rho lambda of scale free networks rho lambda shows characteristic power laws rho lambda approximately lambda alpha 1 or rho lambda approximately lambda d 2 for small lambda where alpha 1 holds below and alpha 2 at the percolation threshold in the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results

structural and spectral properties of a family of deterministic recursive trees rigorous solutions as one of the most significant models the uniform recursive tree urt has found many applications in a variety of fields in this paper we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees durts that are deterministic versions of urt firstly from the perspective of complex networks we investigate analytically the main structural characteristics of durts and obtain the accurate solutions for these properties which include degree distribution average path length distribution of node betweenness and degree correlations then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for durts our research may shed light on a better understanding of the features for urt also the analytical methods used here are capable of being extended to many other deterministic networks making the precise computation of their properties especially the full spectrum characteristics possible

phenomenological models of socioeconomic network dynamics we study a general set of models of social network evolution and dynamics the models consist of both a dynamics on the network and evolution of the network links are formed preferentially between similar nodes where the similarity is defined by the particular process taking place on the network the interplay between the two processes produces phase transitions and hysteresis as seen using numerical simulations for three specific processes we obtain analytic results using mean field approximations and for a particular case we derive an exact solution for the network in common with real world social networks we find coexistence of high and low connectivity phases and history dependence

universal hypotrochoidic law for random matrices with cyclic correlations the celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off diagonal pairs of elements having applications to a wide range of physical and biological systems here we investigate the generalization of this law to random matrices exhibiting higher order cyclic correlations between k tuples of matrix entries we show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k fold rotational symmetry this hypotrochoid law applies to full matrices as well as sparse ones and thereby holds with remarkable universality we further extend our analysis to matrices and graphs with competing cycle motifs which are described more generally by polytrochoid spectral boundaries

linear processes in high dimensions phase space and critical properties in this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality we consider in particular the vector autoregressive var model and the multivariate hawkes process we analyze both deterministic and random versions of these models showing the existence of a stable phase and an unstable phase we find that along the transition region separating the two regimes the correlations of the process decay slowly and we characterize the conditions under which these slow correlations are expected to become power laws we check our findings with numerical simulations showing remarkable agreement with our predictions we finally argue that real systems with a strong degree of self interaction are naturally characterized by this type of slow relaxation of the correlations

on p q rough paths we extend the work of t lyons 1998 and t lyons and z qian 2002 to define integrals and solutions of differential equations along product of p and q rough paths with 1 p 1 q 1 we use this to write an ito formula at the level of rough paths and to see that any rough path can always be interpreted as a product of a p geometric rough path and a p 2 geometric rough path

spectral analysis of deformed random networks we study spectral behavior of sparsely connected random networks under the random matrix framework subnetworks without any connection among them form a network having perfect community structure as connections among the subnetworks are introduced the spacing distribution shows a transition from the poisson statistics to the gaussian orthogonal ensemble statistics of random matrix theory the eigenvalue density distribution shows a transition to the wigner s semicircular behavior for a completely deformed network the range for which spectral rigidity measured by the dyson mehta ensuremath delta 3 statistics follows the gaussian orthogonal ensemble statistics depends upon the deformation of the network from the perfect community structure the spacing distribution is particularly useful to track very slight deformations of the network from a perfect community structure whereas the density distribution and the ensuremath delta 3 statistics remain identical to the undeformed network on the other hand the ensuremath delta 3 statistics is useful for the larger deformation strengths finally we analyze the spectrum of a protein protein interaction network for helicobacter and compare the spectral behavior with those of the model networks

spectral density of complex networks with two species of nodes the adjacency and laplacian matrices of complex networks with two species of nodes are studied and the spectral density is evaluated by using the replica method in statistical physics the network nodes are classified into two species a and b and connections are made only between the nodes of different species a static model of such bipartite networks with power law degree distributions is introduced by applying goh kahng and kim s method to construct scale free networks as a result the spectral density is shown to obey a power law in the limit of large mean degree

spectra of random graphs with community structure and arbitrary degrees using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure both reveal interesting spectral features including deviations from the wigner semicircle distribution and phase transitions in the spectra of community structured networks in this paper we generalize both calculations giving a prescription for calculating the spectrum of a network with both community structure and an arbitrary degree distribution in general the spectrum has two parts a continuous spectral band which can depart strongly from the classic semicircle form and a set of outlying eigenvalues that indicate the presence of communities

markov chain methods for analyzing urban networks complex transport networks abstracted as graphs undirected directed or multi component can be effectively analyzed by random walks or diffusions we have unified many concepts into one framework and studied in details the structural and spectral properties of spatial graphs for five compact urban patterns

spectral density of complex networks with a finite mean degree in order to clarify the statistical features of complex networks the spectral density of adjacency matrices has often been investigated adopting a static model introduced by goh kahng and kim we analyse the spectral density of complex scale free networks for this purpose we utilize the replica method and effective medium approximation ema in statistical mechanics as a result we identify a new integral equation which determines the asymptotic spectral density of scale free networks with a finite mean degree p in the limit p known asymptotic formulae are rederived moreover the 1 p corrections to known results are analytically calculated by a perturbative method

good rough path sequences and applications to anticipating stochastic calculus we consider anticipative stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed under a simple condition on the stochastic process we show that the unique solution of the above sde understood in the rough path sense is actually a stratonovich solution we then show that this condition is satisfied by the brownian motion as application we obtain rather flexible results such as support theorems large deviation principles and wong zakai approximations for sdes driven by brownian motion along anticipating vectorfields in particular this unifies many results on anticipative sdes

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

correlated random networks we develop a statistical theory of networks a network is a set of vertices and links given by its adjacency matrix c and the relevant statistical ensembles are defined in terms of a partition function z summation operator exp betah c the simplest cases are uncorrelated random networks such as the well known erdos renyi graphs here we study more general interactions h c which lead to correlations for example between the connectivities of adjacent vertices in particular such correlations occur in optimized networks described by partition functions in the limit beta infinity they are argued to be a crucial signature of evolutionary design in biological networks

modularity and extreme edges of the internet we study the spectral properties of a diffusion process taking place on the internet network focusing on the slowest decaying modes these modes identify an underlying modular structure roughly corresponding to individual countries for instance in the slowest decaying mode the diffusion current flows from russia to u s military sites quantitatively the modular structure manifests itself in a 10 times larger participation ratio of its slow decaying modes compared to a random scale free network we propose to use the fraction of nodes participating in slow decaying modes as a general measure of the modularity of a network for the 100 slowest decaying modes of the internet this fraction is 30 finally we suggest that the degree of isolation of an individual module can be assessed by comparing its participation in different diffusion modes

integrating cross frequency and within band functional networks in resting state meg a multi layer network approach neuronal oscillations exist across a broad frequency spectrum and are thought to provide a mechanism of interaction between spatially separated brain regions since ongoing mental activity necessitates the simultaneous formation of multiple networks it seems likely that the brain employs interactions within multiple frequency bands as well as cross frequency coupling to support such networks here we propose a multi layer network framework that elucidates this pan spectral picture of network interactions our network consists of multiple layers frequency band specific networks that influence each other via inter layer cross frequency coupling applying this model to meg resting state data and using envelope correlations as connectivity metric we demonstrate strong dependency between within layer structure and inter layer coupling indicating that networks obtained in different frequency bands do not act as independent entities more specifically our results suggest that frequency band specific networks are characterised by a common structure seen across all layers superimposed by layer specific connectivity and inter layer coupling is most strongly associated with this common mode finally using a biophysical model we demonstrate that there are two regimes of multi layer network behaviour one in which different layers are independent and a second in which they operate highly dependent results suggest that the healthy human brain operates at the transition point between these regimes allowing for integration and segregation between layers overall our observations show that a complete picture of global brain network connectivity requires integration of connectivity patterns across the full frequency spectrum

a note on the notion of geometric rough paths we use simple sub riemannian techniques to prove that every weak geometric p rough path a geometric p rough path in the sense of 20 is the limit in sup norm of a sequence of canonically lifted smooth paths uniformly bounded in p variation thus clarifying the two different definitions of a geometric p rough path our proofs are sufficiently general to include the case of holder and modulus type regularity this allows us to extend a few classical results on holder spaces and p variation spaces to the non commutative setting necessary for the theory of rough paths as an application we give a precise description of the support of enhanced fractional brownian motion and prove a conjecture by ledoux et al

scaling relation for earthquake networks abstract the scaling relation 2 1 for the exponents of the power law connectivity distribution and the power law eigenvalue distribution of the adjacency matrix is theoretically predicted to be fulfilled by a locally treelike scale free network in the effective medium approximation i e an analog of the mean field approximation here it is shown that such a relation holds well for the reduced simple earthquake networks i e the network without tadpole loops and multiple edges constructed from the seismic data taken from california and japan this validates the goodness of the effective medium approximation in the earthquake networks and is consistent with the hierarchical organization of the networks the present result may be useful for modeling seismicity on complex networks