synchronization in weighted scale free networks with degree degree correlation abstract we study synchronization phenomena in scale free networks of asymmetrically coupled dynamical systems featuring degree degree correlation in the topology of the connection wiring we show that when the interaction is dominant from the high degree to the low degree or from the low degree to the high degree nodes synchronization is enhanced for assortative or disassortative degree degree correlations

langevin approach for the dynamics of the contact process on annealed scale free networks we study the dynamics of the contact process one of the simplest nonequilibrium stochastic processes taking place on a scale free network we consider the network topology as annealed i e all links are rewired at each microscopic time step so that no dynamical correlation can build up this is a practical implementation of the absence of correlations assumed by mean field approaches we present a detailed analysis of the contact process in terms of a langevin equation including explicitly the effects of stochastic fluctuations in the number of particles in finite networks this allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs the fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has a hard upper bound when the upper bound is soft the presence of outliers with huge connectivity perturbs the picture even more inducing an apparent shift of the critical point in light of these findings recent theoretical and numerical results in the literature are critically reviewed

unified index to quantifying heterogeneity of complex networks abstract although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function a unified definition of the heterogeneity of a network with any degree distribution is absent in this paper we define a heterogeneity index 0 h 1 to quantify the degree heterogeneity of any given network we analytically show the existence of an upper bound of h 0 5 for exponential networks thus explain why exponential networks are homogeneous on the other hand we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0 5 if and only if its degree exponent is between 2 and 2 5 we further show that for any power law network with a degree exponent greater than 2 5 there always exists an exponential network such that both networks have the same heterogeneity index this may help to explain why 2 5 is a critical degree exponent for some dynamic behaviors on power law networks

self avoiding walks on scale free networks several kinds of walks on complex networks are currently used to analyze search and navigation in different systems many analytical and computational results are known for random walks on such networks self avoiding walks saw s are expected to be more suitable than unrestricted random walks to explore various kinds of real life networks here we study long range properties of random saw s on scale free networks characterized by a degree distribution p k ensuremath sim k ensuremath ensuremath gamma in the limit of large networks system size n ensuremath rightarrow ensuremath infty the average number s n of saw s starting from a generic site increases as ensuremath mu n with ensuremath mu k 2 k ensuremath 1 for finite n s n is reduced due to the presence of loops in the network which causes the emergence of attrition of the paths for kinetic growth walks the average maximum length l increases as a power of the system size l ensuremath sim n ensuremath alpha with an exponent ensuremath alpha increasing as the parameter ensuremath gamma is raised we discuss the dependence of ensuremath alpha on the minimum allowed degree in the network a similar power law dependence is found for the mean self intersection length of nonreversal random walks simulation results support our approximate analytical calculations

belief propagation algorithm and the ising model on networks with arbitrary distributions of motifs we generalize the belief propagation algorithm to sparse random networks with arbitrary distributions of motifs triangles loops etc each vertex in these networks belongs to a given set of motifs generalization of the configuration model these networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs here a hypergraph is a generalization of a graph where a hyperedge can connect any number of vertices these uncorrelated hypergraphs are treelike hypertrees which crucially simplifies the problem and allows us to apply the belief propagation algorithm to these loopy networks with arbitrary motifs as natural examples we consider motifs in the form of finite loops and cliques we apply the belief propagation algorithm to the ferromagnetic ising model with pairwise interactions on the resulting random networks and obtain an exact solution of this model we find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size the critical temperature increases compared to ordinary treelike complex networks however weak clustering does not change the critical behavior qualitatively our solution also gives the birth point of the giant connected component in these loopy networks

biased percolation on scale free networks biased degree dependent percolation was recently shown to provide strategies for turning robust networks fragile and vice versa here we present more detailed results for biased edge percolation on scale free net works we assume a network in which the probability for an edge between nodes i and j to be retained is proportional to kikjwith ki and kj the degrees of the nodes we discuss two methods of network recon struction sequential and simultaneous and investigate their properties by analytical and numerical means the system is examined away from the percolation transition where the size of the giant cluster is obtained and close to the transition where nonuniversal critical exponents are extracted using the generating functions method the theory is found to agree quite well with simulations by presenting an extension of the fortuin kasteleyn construction we find that biased percolation is well described by the q 1 limit of the q state potts model with inhomogeneous couplings

exact solution of ising model on a small world network also the complete thermodynamics at all temperatures as a function of p and of the strength of long range interactions i though in this paper we only present the results for the free energy and specific heat in zero field the form of finite size scaling corrections to the thermodynamics can also be derived from our solution the following sec ii presents a description of our specific model and the basic algorithm for its exact solution the combinatorial treatment of its constituents is given is sec iii in sec iv we analyze the resulting thermodynamics and conclude on the mean field character of the ordering transition section v shows how the finite size scaling properties can be obtained within the same approach finally the relation between our results and those of dorogovtsev et al is discussed in sec vi

ferromagnetic transition in a simple variant of the ising model on multiplex networks abstract multiplex networks consist of a fixed set of nodes connected by several sets of edges which are generated separately and correspond to different networks layers here a simple variant of the ising model on multiplex networks with two layers is considered with spins located in the nodes and edges corresponding to ferromagnetic interactions between them critical temperatures for the ferromagnetic transition are evaluated for the layers in the form of random erdos renyi graphs or heterogeneous scale free networks using the mean field approximation and the replica method from the replica symmetric solution both methods require the use of different partial magnetizations associated with different layers of the multiplex network and yield qualitatively similar results if the layers are strongly heterogeneous the critical temperature differs noticeably from that for the ising model on a network being a superposition of the two layers evaluated in the mean field approximation neglecting the effect of the underlying multiplex structure on the correlations between the degrees of nodes the critical temperature evaluated from the replica symmetric solution depends sensitively on the correlations between the degrees of nodes in different layers and shows satisfactory quantitative agreement with that obtained from monte carlo simulations the critical behavior of the magnetization for the model with strongly heterogeneous layers can depend on the distributions of the degrees of nodes and is then determined by the properties of the most heterogeneous layer

correlative amplitude operational phase entanglement embodied by the epr pair eigenstate η after a penetrating analysis of the known number phase commutative relations we find that the common eigenvector of two particles relative coordinate and total momentum also embodies the entanglement with respect to the correlative amplitude operational phase based on the fact that the number difference operator is the canonical conjugate to the noh fougeres mandel operational phase operator the corresponding entanglement is also briefly discussed

ising models on power law random graphs we study a ferromagnetic ising model on random graphs with a power law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite degree exponent 2 for which the random graph has a tree like structure for this we closely follow the analysis by dembo and montanari ann appl probab 20 2 565 592 2010 which assumes finite variance degrees 3 adapting it when necessary and also simplifying it when possible our results also apply in cases where the degree distribution does not obey a power law

comment on correlative amplitude operational phase entanglement embodied by the epr pair eigenstate η in a recent paper in this journal fan fan h 2002 j phys a math gen 35 1007 discards the possibility of using a genuine phase difference operator to investigate number phase entanglement because of the lack of unitarity of the susskind glogower phase operators however fan overlooked the existence of a bona fide unitary operator exponential of the phase difference here we find the amplitude phase maximally entangled states as the simultaneous eigenstates of the total number and the phase difference operators

equivalence after extension and matricial coupling coincide with schur coupling on separable hilbert spaces it is known that two banach space operators that are schur coupled are also equivalent after extension or equivalently matricially coupled the converse implication that operators which are equivalent after extension or matricially coupled are also schur coupled was only known for fredholm hilbert space operators and fredholm banach space operators with index 0 we prove that this implication also holds for hilbert space operators with closed range generalizing the result for fredholm operators and banach space operators that can be approximated in operator norm by invertible operators the combination of these two results enables us to prove that the implication holds for all operators on separable hilbert spaces

schur coupling and related equivalence relations for operators on a hilbert space abstract for operators on hilbert spaces of any dimension we show that equivalence after extension coincides with equivalence after one sided extension thus obtaining a proof of their coincidence with schur coupling we also provide a concrete description of this equivalence relation in several cases in particular for compact operators

synchronization landscapes in small world connected computer networks motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small world networks embedded in one and two dimensions we find that the synchronization landscape corresponding to the progress of the individual processors exhibits kardar parisi zhang like kinetic roughening on regular networks with short range communication links although the processors on average progress at a nonzero rate their spread the width of the synchronization landscape diverges with the number of nodes desynchronized state hindering efficient data management when random communication links are added on top of the one and two dimensional regular networks resulting in a small world network large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system size limit synchronized state in the resulting synchronization scheme the processors make close to uniform progress with a nonzero rate without global intervention we obtain our results by simulating the simulations based on the exact algorithmic rules supported by coarse grained arguments

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

understanding the temporal pattern of spreading in heterogeneous networks theory of the mean infection time for a reliable prediction of an epidemic or information spreading pattern in complex systems well defined measures are essential in the susceptible infected model on heterogeneous networks the cluster of infected nodes in the intermediate time regime exhibits too large fluctuation in size to use its mean size as a representative value the cluster size follows quite a broad distribution which is shown to be derived from the variation of the cluster size with the time when a hub node was first infected on the contrary the distribution of the time taken to infect a given number of nodes is well concentrated at its mean suggesting the mean infection time is a better measure we show that the mean infection time can be evaluated by using the scaling behaviors of the boundary area of the infected cluster and use it to find a nonexponential but algebraic spreading phase in the intermediate stage on strongly heterogeneous networks such slow spreading originates in only small degree nodes left susceptible while most hub nodes are already infected in the early exponential spreading stage our results offer a way to detour around large statistical fluctuations and quantify reliably the temporal pattern of spread under structural heterogeneity

robustness and fragility of the susceptible infected susceptible epidemic models on complex networks we analyze two alterations of the standard susceptible infected susceptible sis dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree all models have the same epidemic thresholds in mean field theories but depending on the network properties simulations yield a dual scenario in which the epidemic thresholds of the modified sis models can be either dramatically altered or remain unchanged in comparison with the standard dynamics for uncorrelated synthetic networks having a power law degree distribution with exponent 5 2 differences are more remarkable for 3 where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one this duality is elucidated in terms of epidemic lifespan on star graphs we verify that the activation of the modified sis models is triggered in the innermost component of the network given by a k core decomposition for 3 the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard sis the duality also appears in the finite size scaling of the critical quantities where mean field behaviors are observed for the modified but not for the original dynamics our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models

flux distribution of metabolic networks close to optimal biomass production we study a statistical model describing the steady state distribution of the fluxes in a metabolic network the resulting model defined on continuous variables can be solved by the cavity method in particular analytical tractability is possible solving the cavity equation over an ensemble of networks with the same degree distribution as a real metabolic network the flux distribution that optimizes production of biomass has a fat tail with a power law exponent independent of the structural properties of the underlying network these results are in complete agreement with the flux balance analysis outcome and in qualitative agreement with the experimental results

reply to the comment on correlative amplitude operational phase entanglement embodied by the epr pair eigenstate ηrangle we compare and contrast our amplitude phase entanglement with that of luis in his comment luis s entangled state is defined in a finite fock space his comment on the operational phase operator seems to be contradicting the original meaning of mandel et al

spin models on random graphs with controlled topologies beyond degree constraints we study ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution p k can be chosen arbitrarily but which allows for further fine tuning of the topology via preferential attachment of edges on the basis of an arbitrary function q k k of the degrees of the vertices involved we solve these models using finite connectivity equilibrium replica theory within the replica symmetric ansatz in our ensemble of graphs phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution but also on the choice made for q k k the increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real world interacting particle systems by spin systems on suitably defined random graphs

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

partition function zeros for the ising model on complete graphs and on annealed scale free networks we analyse the partition function of the ising model on graphs of two different types complete graphs wherein all nodes are mutually linked and annealed scale free networks for which the degree distribution decays as p k k we are interested in zeros of the partition function in the cases of complex temperature or complex external field fisher and lee yang zeros respectively for the model on an annealed scale free network we find an integral representation for the partition function which in the case 5 reproduces the zeros for the ising model on a complete graph for 3 5 we derive the dependent angle at which the fisher zeros impact onto the real temperature axis this in turn gives access to the dependent universal values of the critical exponents and critical amplitudes ratios our analysis of the lee yang zeros reveals a difference in their behaviour for the ising model on a complete graph and on an annealed scale free network when 3 5 whereas in the former case the zeros are purely imaginary they have a non zero real part in latter case so that the celebrated lee yang circle theorem is violated

number of cliques in random scale free network ensembles abstract in this paper we calculate the average number of cliques in random scale free networks in the limit of large network size n 1 we consider first the hidden variable ensemble and subsequently the molloy reed ensemble in both cases we find that cliques i e fully connected subgraphs appear also when the average degree is finite this is in contrast to what happens in erdos and renyi graphs in which diverging average degree is required to observe cliques of size c 3 moreover we show that in random scale free networks the clique number i e the size of the largest clique present in the network diverges with the system size

criticality on networks with topology dependent interactions weighted scale free networks with topology dependent interactions are studied it is shown that the possible universality classes of critical behavior which are known to depend on topology can also be explored by tuning the form of the interactions at fixed topology for a model of opinion formation simple mean field and scaling arguments show that a mapping gamma gamma mu 1 mu describes how a shift of the standard exponent gamma of the degree distribution can absorb the effect of degree dependent pair interactions j ij proportional to k i k j mu where k i stands for the degree of vertex i this prediction is verified by extensive numerical investigations using the cavity method and monte carlo simulations the critical temperature of the model is obtained through the bethe peierls approximation and with the replica technique the mapping can be extended to nonequilibrium models such as those describing the spreading of a disease on a network

schur complements and state space realizations motivated by state space realizations of transfer functions from system theory a number of operations on schur complements are introduced and studied these operations are equivalence extension multiplication inversion and factorization together they form an algebraic framework which is of independent interest and also useful in solving problems in analysis

equivalence after extension for compact operators on banach spaces in recent years the coincidence of the operator relations equivalence after extension and schur coupling was settled for the hilbert space case by showing that equivalence after extension implies equivalence after one sided extension in the present paper we investigate consequences of equivalence after extension for compact banach space operators we show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension in analogy with the necessary and sufficient conditions for compact hilbert space operators to be equivalent after extension in terms of their singular values we prove under certain additional conditions the necessity of a similar relationship between the s numbers of two compact banach space operators that are equivalent after extension for arbitrary s functions we investigate equivalence after extension for operators on lp spaces we show that two operators that act on different lp spaces cannot be equivalent after one sided extension such operators can still be equivalent after extension for instance all invertible operators are equivalent after extension however if one of the two operators is compact then they cannot be equivalent after extension this contrasts the hilbert space case where equivalence after one sided extension and equivalence after extension are in fact identical relations finally for general banach spaces x and y we investigate consequences of an operator on x being equivalent after extension to a compact operator on y we show that in this case a closed finite codimensional subspace of y must embed into x and that certain general banach space properties must transfer from x to y we also show that no operator on x can be equivalent after extension to an operator on y if x and y are essentially incomparable banach spaces

statistically consistent coarse grained simulations for critical phenomena in complex networks we propose a degree based coarse graining approach that not just accelerates the evaluation of dynamics on complex networks but also satisfies the consistency conditions for both equilibrium statistical distributions and nonequilibrium dynamical flows for the ising model and susceptible infected susceptible epidemic model we introduce these required conditions explicitly and further prove that they are satisfied by our coarse grained network construction within the annealed network approximation finally we numerically show that the phase transitions and fluctuations on the coarse grained network are all in good agreements with those on the original one

first and second order phase transitions in scale free networks we study first and second order phase transitions of ferromagnetic lattice models on scale free networks with a degree exponent gamma using the example of the q state potts model we derive a general self consistency relation within the frame of the weiss molecular field approximation which presumably leads to exact critical singularities depending on the value of gamma we have found three different regimes of the phase diagram as a general trend first order transitions soften with decreasing gamma and the critical singularities at the second order transitions are gamma dependent

quenched central limit theorems for the ising model on random graphs the main goal of the paper is to prove central limit theorems for the magnetization rescaled by sqrt n for the ising model on random graphs with n vertices both random quenched and averaged quenched measures are considered we work in the uniqueness regime beta beta c or beta 0 and b ne 0 where beta is the inverse temperature beta c is the critical inverse temperature and b is the external magnetic field in the random quenched setting our results apply to general tree like random graphs as introduced by dembo montanari and further studied by dommers and the first and third author and our proof follows that of ellis in mathbb z d for the averaged quenched setting we specialize to two particular random graph models namely the 2 regular configuration model and the configuration model with degrees 1 and 2 in these cases our proofs are based on explicit computations relying on the solution of the one dimensional ising models

complementary schur complements abstract for square matrices the relationship is discussed between the notion of schur complement and the equivalent concepts of matricial coupling and equivalence after extension special attention is paid to the hermitian and skew hermitian cases

effects of local population structure in a reaction diffusion model of a contact process on metapopulation networks we investigate the effects of local population structure in reaction diffusion processes representing a contact process cp on metapopulations represented as complex networks considering a model in which the nodes of a large scale network represent local populations defined in terms of a homogeneous graph we show by means of extensive numerical simulations that the critical properties of the reaction diffusion system are independent of the local population structure even when this one is given by a ordered linear chain this independence is confirmed by the perfect matching between numerical critical exponents and the results from a heterogeneous mean field theory suited in principle to describe situations of local homogeneous mixing the analysis of several variations of the reaction diffusion process allows us to conclude the independence from population structure of the critical properties of cp like models on metapopulations and thus of the universality of the reaction diffusion description of this kind of models