thresholds for epidemic spreading in networks we study the threshold of epidemic models in quenched networks with degree distribution given by a power law forthesusceptible infected susceptiblemodeltheactivitythresholdc vanishesinthelargesize limitonanynetworkwhosemaximumdegree kmax divergeswiththesystemsize atoddswithheterogeneous mean field hmf theory the vanishing of the threshold has nothing to do with the scale free nature of the networkbutstemsinsteadfromthelargesthubinthesystembeingactiveforanyspreadingrate 1 ffiffiffiffiffiffiffiffiffimax p and playing the role of a self sustained source that spreads the infection to the rest of the system the susceptible infected removedmodel displays insteadagreement with hmftheory and afinite threshold for scale rich networks we conjecture that on quenched scale rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state

evaluation of an analytic approximate formula for the time varying sis prevalence in different networks abstract one of the most important quantities of the exact markovian sis epidemic process is the time dependent prevalence which is the average fraction of infected nodes unfortunately the markovian sis epidemic model features an exponentially increasing computational complexity with growing network size n in this paper we evaluate a recently proposed analytic approximate prevalence function introduced in van mieghem 2016 we compare the approximate function with the n intertwined mean field approximation nimfa and with simulation of the markovian sis epidemic process the results show that the new analytic prevalence function is comparable with other approximate methods

low density series expansions for directed percolation on square and triangular lattices greatly extended series have been derived for moments of the pair connectedness for bond and site percolation on the directed square and triangular lattices the length of the various series has been at least doubled to more than 110 100 terms for the square lattice bond site problem and more than 55 terms for the bond and site problems on the triangular lattice analysis of the series leads to very accurate estimates for the critical parameters and generally seems to rule out simple rational values for the critical exponents the values of the critical exponents for the average cluster size parallel and perpendicular connectedness lengths are estimated by d 2 277 69 4 k d 1 733 825 25 and d 1 096 844 14 respectively an improved estimate for the percolation probability exponent is obtained from the scaling relation d k c 2 d 0 276 49 4 in all cases the leading correction to scaling term is analytic

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

moment ratios for absorbing state phase transitions we determine the first through fourth moments of the order parameter and various ratios for several one and two dimensional models with absorbing state phase transitions we perform a detailed analysis of the system size dependence of these ratios and confirm that they are indeed universal for three models the contact process the a model and the pair contact process belonging to the directed percolation universality class our studies also yield a refined estimate for the critical point of the pair contact process

monotonicity for excited random walk in high dimensions we prove that the drift d for excited random walk in dimension d is monotone in the excitement parameter beta in 0 1 when d is sufficiently large we give an explicit criterion for monotonicity involving random walk green s functions and use rigorous numerical upper bounds provided by hara private communication 2007 to verify the criterion for d 9

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

on the speed of a cookie random walk we consider the model of the one dimensional cookie random walk when the initial cookie distribution is spatially uniform and the number of cookies per site is finite we give a criterion to decide whether the limiting speed of the walk is non zero in particular we show that a positive speed may be obtained for just three cookies per site we also prove a result on the continuity of the speed with respect to the initial cookie distribution

time dependent perturbation theory for nonequilibrium lattice models we develop a time dependent perturbation theory for nonequilibrium interacting particle systems we focus on models such as the contact process which evolve via destruction and autocatalytic creation of particles at a critical value of the destruction rate there is a continuous phase transition between an active steady state and the vacuum state which is absorbing we present several methods for deriving series for the evolution starting from a single seed particle including expansions for the ultimate survival probability in the super and subcritical regions expansions for the average number of particles in the subcritical region and short time expansions algorithms for computer generation of the various expansions are presented rather long series 24 terms or more and precise estimates of critical parameters are presented

decreasing the spectral radius of a graph by link removals the decrease of the spectral radius an important characterizer of network dynamics by removing links is investigated the minimization of the spectral radius by removing m links is shown to be an np complete problem which suggests considering heuristic strategies several greedy strategies are compared and several bounds on the decrease of the spectral radius are derived the strategy that removes that link l i j with largest product x1 i x1 j of the components of the eigenvector x1 belonging to the largest adjacency eigenvalue is shown to be superior to other strategies in most cases furthermore a scaling law where the decrease in spectral radius is inversely proportional to the number of nodes n in the graph is deduced another sublinear scaling law of the decrease in spectral radius versus the number m of removed links is conjectured

contact processes on random graphs with power law degree distributions have critical value 0 if we consider the contact process with infection rate on a random graph on n vertices with power law degree distributions mean field calculations suggest that the critical value c of the infection rate is positive if the power 3 physicists seem to regard this as an established fact since the result has recently been generalized to bipartite graphs by gomez gardenes et al proc natl acad sci usa 105 2008 1399 1404 here we show that the critical value c is zero for any value of 3 and the contact process starting from all vertices infected with a probability tending to 1 as n maintains a positive density of infected sites for time at least exp n 1 for any 0 using the last result together with the contact process duality we can establish the existence of a quasi stationary distribution in which a randomly chosen vertex is occupied with probability it is expected that c as 0 here we show that 1 2 3 and so 2 for 3 thus even though the graph is locally tree like does not take the mean field critical value 1

multi excited random walks on integers we introduce a class of nearest neighbor integer random walks in random and non random media which includes excited random walks considered in the literature at each site the random walker has a drift to the right the strength of which depends on the environment at that site and on how often the walker has visited that site before we give exact criteria for recurrence and transience and consider the speed of the walk

epidemic threshold in directed networks epidemics have so far been mostly studied in undirected networks however many real world networks such as the online social network twitter and the world wide web on which information emotion or malware spreads are directed networks composed of both unidirectional links and bidirectional links we define the directionality as the percentage of unidirectional links the epidemic threshold c for the susceptible infected susceptible sis epidemic is lower bounded by 1 1 in directed networks where 1 also called the spectral radius is the largest eigenvalue of the adjacency matrix in this work we propose two algorithms to generate directed networks with a given directionality the effect of on the spectral radius 1 principal eigenvector x 1 spectral gap 1 2 and algebraic connectivity n 1 is studied important findings are that the spectral radius 1 decreases with the directionality whereas the spectral gap and the algebraic connectivity increase with the directionality the extent of the decrease of the spectral radius depends on both the degree distribution and the degree degree correlation d hence in directed networks the epidemic threshold is larger and a random walk converges to its steady state faster than that in undirected networks with the same degree distribution

epidemic thresholds of the susceptible infected susceptible model on networks a comparison of numerical and theoretical results recent work has shown that different theoretical approaches to the dynamics of the susceptible infectedsusceptible sis model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks here we present large scale numerical simulations of the sis dynamics on various types of networks allowing the precise determination of the effective threshold for systems of finite size n we compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean field theory and of the quenched mean field theory we show that the latter is in general more accurate scaling with n with the correct exponent but often failing to capture the correct prefactor

approximate formula and bounds for the time varying susceptible infected susceptible prevalence in networks based on a recent exact differential equation the time dependence of the sis prevalence the average fraction of infected nodes in any graph is first studied and then upper and lower bounded by an explicit analytic function of time that new approximate tanh formula obeys a riccati differential equation and bears resemblance to the classical expression in epidemiology of kermack and mckendrick proc r soc london a 115 700 1927 1364 502110 1098 rspa 1927 0118 but enhanced with graph specific properties such as the algebraic connectivity the second smallest eigenvalue of the laplacian of the graph we further revisit the challenge of finding tight upper bounds for the sis and sir epidemic threshold for all graphs we propose two new upper bounds and show the importance of the variance of the number of infected nodes finally a formula for the epidemic threshold in the cycle or ring graph is presented

critical exponents for the restricted sandpile i report large scale monte carlo studies of a one dimensional height restricted stochastic sandpile using the quasistationary simulation method results for systems of up to 50 000 sites yield estimates for critical exponents that differ significantly from those obtained using smaller systems but are consistent with recent predictions derived from a langevin equation for stochastic sandpiles ramasco phys rev e 69 045105 r 2004 this suggests that apparent violations of universality in one dimensional sandpiles are due to strong corrections to scaling and finite size effects

how to simulate the quasistationary state for a large class of processes with an absorbing state statistical properties of the surviving sample attain time independent values in the quasistationary qs regime we propose a practical simulation method for studying quasistationary properties based on the equation of motion governing the qs distribution in applications to the contact process the method is shown to reproduce exact results for the process on a complete graph and known scaling behavior to high precision at the critical point our method is about an order of magnitude more efficient than conventional simulation

spectral analysis and slow spreading dynamics on complex networks the susceptible infected susceptible sis model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous annihilation reactions the effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean field qmf approximations in scale free networks qmf can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix therefore it can provide predictions on possible rare region effects thus on the occurrence of slow dynamics i compare qmf results of sis with simulations on various large dimensional graphs in particular i show that for erd s renyi graphs this method predicts correctly the occurrence of rare region effects it also provides a good estimate for the epidemic threshold in case of percolating graphs griffiths phases emerge if the graph is fragmented or if we apply a strong exponentially suppressing weighting scheme on the edges the latter model describes the connection time distributions in the face to face experiments in case of a generalized barabasi albert type of network with aging connections strong rare region effects and numerical evidence for griffiths phase dynamics are shown the dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices

quasistationary analysis of the contact process on annealed scale free networks we present an analysis of the quasistationary qs state of the contact process cp on annealed scale free networks using a mapping of the cp dynamics in a one step process and analyzing numerically and analytically the corresponding master equation the relevant qs quantities determined via the master equation exhibit an excellent agreement with direct qs stochastic simulations of the cp the high accuracy of the resulting data allows a probe of the strong corrections to scaling present in both the critical and supercritical regions corrections that mask the correct finite size scaling obtained analytically by applying an exact heterogeneous mean field approach our results represent a promising starting point for a deeper understanding of the contact process and absorbing phase transitions on real quenched complex networks

on a general many dimensional excited random walk in this paper we study a substantial generalization of the model of excited random walk introduced in electron commun probab 8 2003 86 92 by benjamini and wilson we consider a discrete time stochastic process xn n 0 1 2 taking values on zd d 2 described as follows when the particle visits a site for the first time it has a uniformly positive drift in a given direction l when the particle is at a site which was already visited before it has zero drift assuming uniform ellipticity and that the jumps of the process are uniformly bounded we prove that the process is ballistic in the direction l so that lim infn xn ln 0 a key ingredient in the proof of this result is an estimate on the probability that the process visits less than n1 2 distinct sites by time n where is some positive number depending on the parameters of the model this approach completely avoids the use of tan points and coupling methods specific to the excited random walk furthermore we apply this technique to prove that the excited random walk in an i i d random environment satisfies a ballistic law of large numbers and a central limit theorem

extinction dynamics of a discrete population in an oasis understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology the first theoretical approach to the problem dates back to the 1950s with the kierstead slobodkin and skellam kiss model namely a continuous reaction diffusion equation for a population growing on a patch of finite size l surrounded by a deadly environment with infinite mortality i e an oasis in a desert the main outcome of the model is that only patches above a critical size allow for population persistence here we introduce an individual based analog of the kiss model to investigate the effects of discreteness and demographic stochasticity in particular we study the average time to extinction both above and below the critical patch size of the continuous model and investigate the quasistationary distribution of the number of individuals for patch sizes above the critical threshold

sampling methods for the quasistationary regime of epidemic processes on regular and complex networks a major hurdle in the simulation of the steady state of epidemic processes is that the system will unavoidably visit an absorbing disease free state at sufficiently long times due to the finite size of the networks where epidemics evolves in the present work we compare different quasistationary qs simulation methods where the absorbing states are suitably handled and the thermodynamical limit of the original dynamics can be achieved we analyze the standard qs sqs method where the sampling is constrained to active configurations the reflecting boundary condition rbc where the dynamics returns to the pre absorbing configuration and hub reactivation hr where the most connected vertex of the network is reactivated after a visit to an absorbing state we apply the methods to the contact process cp and susceptible infected susceptible sis models on regular and scale free networks the investigated methods yield the same epidemic threshold for both models for cp that undergoes a standard collective phase transition the methods are equivalent for sis whose phase transition is ruled by the hub mutual reactivation the sqs and hr methods are able to capture localized epidemic phases while rbc is not we also apply the autocorrelation time as a tool to characterize the phase transition and observe that this analysis provides the same finite size scaling exponents for the critical relaxation time for the investigated methods finally we verify the equivalence between rbc method and a weak external field for epidemics on networks

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

second order mean field susceptible infected susceptible epidemic threshold given the adjacency matrix a of a network we present a second order mean field expansion that improves on the first order n intertwined susceptible infected susceptible sis epidemic model unexpectedly we found that in contrast to first order second order mean field theory is not always possible the network size n should be large enough under the assumption of large n we show that the crucial and characterizing quantity the sis epidemic threshold ensuremath tau c obeys an eigenvalue equation more complex than the one in the first order n intertwined model however the resulting epidemic threshold is more accurate ensuremath tau c 2 ensuremath tau c 1 o frac ensuremath tau c 1 n where the first order epidemic threshold is ensuremath tau c 1 frac 1 ensuremath lambda 1 a and where ensuremath lambda 1 a is the spectral radius of the adjacency matrix a

a law of large numbers for random walks in random environment we derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of kalikow the approach is based on the existence of a renewal structure under an assumption of transience in the direction l this extends to a multidimensional context previous work of kesten our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution

high prevalence regimes in the pair quenched mean field theory for the susceptible infected susceptible model on networks reckoning of pairwise dynamical correlations significantly improves the accuracy of mean field theories and plays an important role in the investigation of dynamical processes in complex networks in this work we perform a nonperturbative numerical analysis of the quenched mean field theory qmf and the inclusion of dynamical correlations by means of the pair quenched mean field pqmf theory for the susceptible infected susceptible model on synthetic and real networks we show that the pqmf considerably outperforms the standard qmf theory on synthetic networks of distinct levels of heterogeneity and degree correlations providing extremely accurate predictions when the system is not too close to the epidemic threshold while the qmf theory deviates substantially from simulations for networks with a degree exponent 2 5 the scenario for real networks is more complicated still with pqmf significantly outperforming the qmf theory however despite its high accuracy for most investigated networks in a few cases pqmf deviations from simulations are not negligible we found correlations between accuracy and average shortest path while other basic network metrics seem to be uncorrelated with the theory accuracy our results show the viability of the pqmf theory to investigate the high prevalence regimes of recurrent state epidemic processes in networks a regime of high applicability

laplacian spectra of and random walks on complex networks are scale free architectures really important we study the laplacian operator of an uncorrelated random network and as an application consider hopping processes diffusion random walks signal propagation etc on networks we develop a strict approach to these problems we derive an exact closed set of integral equations which provide the averages of the laplacian operator s resolvent this enables us to describe the propagation of a signal and random walks on the network we show that the determining parameter in this problem is the minimum degree q m of vertices in the network and that the high degree part of the degree distribution is not that essential the position of the lower edge of the laplacian spectrum c appears to be the same as in the regular bethe lattice with the coordination number q m namely c 0 if q m 2 and c 0 if q m 2 in both of these cases the density of eigenvalues p 0 as c 0 but the limiting behaviors near c are very different in terms of a distance from a starting vertex the hopping propagator is a steady moving gaussian broadening with time this picture qualitatively coincides with that for a regular bethe lattice our analytical results include the spectral density p near c and the long time asymptotics of the autocorrelator and the propagator

susceptible infected susceptible epidemics on the complete graph and the star graph exact analysis since mean field approximations for susceptible infected susceptible sis epidemics do not always predict the correct scaling of the epidemic threshold of the sis metastable regime we propose two novel approaches a an sis generalized model and b a modified sis model that prevents the epidemic from dying out i e without the complicating absorbing sis state both adaptations of the sis model feature a precisely defined steady state that corresponds to the sis metastable state and allow an exact analysis in the complete and star graph consisting of a central node and n leaves the n intertwined mean field approximation nimfa is shown to be nearly exact for the complete graph but less accurate to predict the correct scaling of the epidemic threshold c in the star graph which is found as c c 1 where 1 2logn 3 2loglogn and where c 1 1 n c is the first order epidemic threshold for the star in nimfa and equal to the inverse of the spectral radius of the star s adjacency matrix

perturbative series expansion for the subcritical stationary properties of the contact process the subcritical stationary properties of the contact process in one dimension are studied by perturbative series expansions in powers of the creation rate to avoid entering the absorbing state and thus obtaining a nonzero stationary distribution of particles three small modifications on the rules of the contact process are considered the series expansions for the average number of particles and the spatial correlation length are analysed by pade approximants from which the values of the critical creation rate and the critical exponents are obtained

finite size scaling in complex networks a finite size scaling fss theory is proposed for various models in complex networks in particular we focus on the fss exponent which plays a crucial role in analyzing numerical data for finite size systems based on the droplet excitation hyperscaling argument we conjecture the values of the fss exponents for the ising model the susceptible infected susceptible model and the contact process all of which are confirmed reasonably well in numerical simulations

excited brownian motions as limits of excited random walks we obtain the convergence in law of a sequence of excited also called cookies random walks toward an excited brownian motion this last process is a continuous semi martingale whose drift is a function say of its local time it was introduced by norris rogers and williams as a simplified version of brownian polymers and then recently further studied by the authors to get our results we need to renormalize together the sequence of cookies the time and the space in a convenient way the proof follows a general approach already taken by toth and his coauthors in multiple occasions which goes through ray knight type results namely we first prove when is bounded and lipschitz that the convergence holds at the level of the local time processes this is done via a careful study of the transition kernel of an auxiliary markov chain which describes the local time at a given level then we prove a tightness result and deduce the convergence at the level of the full processes

finite size scaling of directed percolation above the upper critical dimension critical phenomena such as second order phase transitions are characterized by singularities causing a discontinuous behavior of various quantities at the transition point e g the speciflc heat the susceptibility and the correlation length these singularities are described by power laws deflning the well known critical exponents studying the phase transition of a given system the aim of investigation is to identify the set of critical exponents which characterizes together with certain universal scaling functions the so called universality class since most systems are not analytically tractable their critical behavior is often investigated via numerical methods for example monte carlo simulations or transfer matrix calculations in these cases the obtained data are limited to flnite system sizes therefore flnite size scaling fss is widely used to extrapolate to the behavior of the inflnite systems in particular fss is an e cient method

contact process on a voronoi triangulation we study the continuous absorbing state phase transition in the contact process on the voronoi delaunay lattice the voronoi construction is a natural way to introduce quenched coordination disorder in lattice models we simulate the disordered system using the quasistationary simulation method and determine its critical exponents and moment ratios our results suggest that the critical behavior of the disordered system is unchanged with respect to that on a regular lattice i e that of directed percolation

brownian motion and random walk perturbed at extrema let bt be brownian motion we show there is a unique adapted process xt which satisfies dxt dbt except when xt is at a maximum or a minimum when it receives a push the magnitudes and directions of the pushes being the parameters of the process for some ranges of the parameters this is already known we show that if a random walk close to bt is perturbed properly its paths are close to those of xt

dynamical correlations and pairwise theory for the symbiotic contact process on networks the two species symbiotic contact process 2scp is a stochastic process in which each vertex of a graph may be vacant or host at most one individual of each species vertices with both species have a reduced death rate representing a symbiotic interaction while the dynamics evolves according to the standard single species contact process rules otherwise we investigate the role of dynamical correlations on the 2scp on homogeneous and heterogeneous networks using pairwise mean field theory this approach is compared with the ordinary one site theory and stochastic simulations we show that our approach significantly outperforms the one site theory in particular the stationary state of the 2scp model on random regular networks is very accurately reproduced by the pairwise mean field even for relatively small values of vertex degree where expressive deviations of the standard mean field are observed the pairwise approach is also able to capture the transition points accurately for heterogeneous networks and provides rich phase diagrams with transitions not predicted by the one site method our theoretical results are corroborated by extensive numerical simulations

absorbing state phase transitions exact solutions of small systems i derive precise results for absorbing state phase transitions using exact numerically determined quasistationary qs probability distributions for small systems analysis of the contact process on rings of 23 or fewer sites yields critical properties control parameter order parameter ratios and critical exponents z and ensuremath beta ensuremath nu ensuremath perp with an accuracy of 0 06 or better precise results are also obtained for the pair contact process the qs distribution yields insights on the statistical entropy of these models preliminary application to a model in the stochastic sandpile class is also described

continuous time vertex reinforced jump processes we study the continuous time integer valued process which jumps to each of its two nearest neighbors at the rate of one plus the total time the process has previously spent at that neighbor we show that the proportion of the time before t which this process spends at integers j converges to positive random variables v j which sum to one and whose joint distribution is explicitly described we also show

contact process in disordered and periodic binary two dimensional lattices the critical behavior of the contact process cp in disordered and periodic binary two dimensional 2d lattices is investigated numerically by means of monte carlo simulations as well as via an analytical approximation and standard mean field theory phase separation lines calculated numerically are found to agree well with analytical predictions around the homogeneous point for the disordered case values of static scaling exponents obtained via quasistationary simulations are found to change with disorder strength in particular the finite size scaling exponent of the density of infected sites approaches a value consistent with the existence of an infinite randomness fixed point as conjectured before for the 2d disordered cp at the same time both dynamical and static scaling exponents are found to coincide with the values established for the homogeneous case thus confirming that the contact process in a heterogeneous environment belongs to the directed percolation universality class

quasi stationary distributions for stochastic processes with an absorbing state we study the long time behaviour of stochastic models with an absorbing state conditioned on survival for a large class of processes in which saturation prevents unlimited growth statistical properties of the surviving sample attain time independent limiting values we may then define a quasi stationary probability distribution as one in which the ratios pn t pm t for any pair of non absorbing states n and m are time independent this is not a true stationary distribution since the overall normalization decays as probability flows irreversibly to the absorbing state we construct quasi stationary solutions for the contact process on a complete graph the malthus verhulst process

equivalence of conditional and external field ensembles in absorbing state phase transitions i comment on the relation between two sampling methods for absorbing state models it is shown that a certain ensemble without external field conditional to activity coincides with the unconditional ensemble for sufficiently small external field the actual physical processes involved are identical and the derivation of the identity of the scaling behavior relies on a single established scaling law while the conditional ensemble by construction does not contain information about the system with large external field it contains all the information about the limit of vanishing external field and about the vicinity of the critical point finite size scaling as well as critical scaling in the temperature like variable or in small external field

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

nature of the epidemic threshold for the susceptible infected susceptible dynamics in networks we develop an analytical approach to the susceptible infected susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks we find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold in particular small world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit

numerical analysis of the master equation applied to the master equation the usual numerical integration methods such as runge kutta method become inefficient when the rates associated with various transitions differ by several orders of magnitude we introduce an integration scheme that remains stable with much larger time increments than can be used in standard methods when only the stationary distribution is required a direct iteration method is even more rapid this method may be extended to construct the quasistationary distribution of a process with an absorbing state applications to birth and death processes reveal gains in efficiency of two or more orders of magnitude

excited random walk against a wall we analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex a k a excited random walk we show that it is recurrent with an expected number of returns of sqrt log t

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

role of depletion on the dynamics of a diffusing forager we study the dynamics of a starving random walk in general spatial dimension d this model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources the forager depletes its environment by consuming resources and dies if it wanders too long without finding food in the exactly solvable case of one dimension we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation we also give a heuristic derivation for the averages of these two quantities we tackle the complex but ecologically relevant case of two dimensions by an approximation in which the depleted zone is assumed to always be circular and which grows incrementally each time the walk reaches the edge of this zone within this framework we derive a lower bound for the scaling of the average lifetime and number of distinct sites visited at starvation we also determine the asymptotic distribution of the number of distinct sites visited at starvation finally we solve the case of high spatial dimensions within a mean field approach

epidemics in networks with nodal self infection and the epidemic threshold since the susceptible infected susceptible sis epidemic threshold is not precisely defined in spite of its practical importance the classical sis epidemic process has been generalized to the sis model where a node possesses a self infection rate in addition to a link infection rate and a curing rate the exact markov equations are derived from which the steady state can be computed the major advantage of the sis model is that its steady state is different from the absorbing or overall healthy state and approximates for a certain range of small 0 the in reality observed phase transition also called the metastable state that is characterized by the epidemic threshold the exact steady state analysis for the complete graph illustrates the effect of small and the quality of the first order mean field approximation the n intertwined model proposed earlier apart from duality principles often used in the mathematical literature we present an exact recursion relation for the markov infinitesimal generator

metastable localization of diseases in complex networks we describe the phenomenon of localization in the epidemic susceptible infective susceptible model on highly heterogeneous networks in which strongly connected nodes hubs play the role of centers of localization we find that in this model the localized states below the epidemic threshold are metastable the longevity and scale of the metastable outbreaks do not show a sharp localization transition instead there is a smooth crossover from localized to delocalized states as we approach the epidemic threshold from below analyzing these long lasting local outbreaks for a random regular graph with a hub we show how this localization can be detected from the shape of the distribution of the number of infective nodes

sir model on a dynamical network and the endemic state of an infectious disease abstract in this work we performed a numerical study of an epidemic model that mimics the endemic state of whooping cough in the pre vaccine era we considered a stochastic sir model on dynamical networks that involve local and global contacts among individuals and analysed the influence of the network properties on the characterization of the quasi stationary state we computed probability density functions pdf for infected fraction of individuals and found that they are well fitted by gamma functions excepted the tails of the distributions that are q exponentials we also computed the fluctuation power spectra of infective time series for different networks we found that network effects can be partially absorbed by rescaling the rate of infective contacts of the model an explicit relation between the effective transmission rate of the disease and the correlation of susceptible individuals with their infective nearest neighbours was obtained this relation quantifies the known screening of infective individuals observed in these networks we finally discuss the goodness and limitations of the sir model with homogeneous mixing and parameters taken from epidemiological data to describe the dynamic behaviour observed in the networks studied

multiple transitions of the susceptible infected susceptible epidemic model on complex networks the epidemic threshold of the susceptible infected susceptible sis dynamics on random networks having a power law degree distribution with exponent 3 has been investigated using different mean field approaches which predict different outcomes we performed extensive simulations in the quasistationary state for a comparison with these mean field theories we observed concomitant multiple transitions in individual networks presenting large gaps in the degree distribution and the obtained multiple epidemic thresholds are well described by different mean field theories we observed that the transitions involving thresholds which vanish at the thermodynamic limit involve localized states in which a vanishing fraction of the network effectively contributes to epidemic activity whereas an endemic state with a finite density of infected vertices occurs at a finite threshold the multiple transitions are related to the activations of distinct subdomains of the network which are not directly connected

phase transition of the susceptible infected susceptible dynamics on time varying configuration model networks we present a degree based theoretical framework to study the susceptible infected susceptible sis dynamics on time varying rewired configuration model networks using this framework on a given degree distribution we provide a detailed analysis of the stationary state using the rewiring rate to explore the whole range of the time variation of the structure relative to that of the sis process this analysis is suitable for the characterization of the phase transition and leads to three main contributions 1 we obtain a self consistent expression for the absorbing state threshold able to capture both collective and hub activation 2 we recover the predictions of a number of existing approaches as limiting cases of our analysis providing thereby a unifying point of view for the sis dynamics on random networks 3 we obtain bounds for the critical exponents of a number of quantities in the stationary state this allows us to reinterpret the concept of hub dominated phase transition within our framework it appears as a heterogeneous critical phenomenon observables for different degree classes have a different scaling with the infection rate this phenomenon is followed by the successive activation of the degree classes beyond the epidemic threshold

survival time of the susceptible infected susceptible infection process on a graph the survival time t is the longest time that a virus a meme or a failure can propagate in a network using the hitting time of the absorbing state in an uniformized embedded markov chain of the continuous time susceptible infected susceptible sis markov process we derive an exact expression for the average survival time e t of a virus in the complete graph kn and the star graph k1 n 1 by using the survival time instead of the average fraction of infected nodes we propose a new method to approximate the sis epidemic threshold c that at least for kn and k1 n 1 correctly scales with the number of nodes n and that is superior to the epidemic threshold 1 c 1 1 of the n intertwined mean field approximation where 1 is the spectral radius of the adjacency matrix of the graph g although this new approximation of the epidemic threshold offers a more intuitive understanding of the sis process it remains difficult to compare outbreaks in different graph types for example the survival in an arbitrary graph seems upper bounded by the complete graph and lower bounded by the star graph as a function of the normalized effective infection rate 1 c however when the average fraction of infected nodes is used as a basis for comparison the virus will survive in the star graph longer than in any other graph making the star graph the worst case graph instead of the complete graph finally in non markovian sis the distribution of the spreading attempts over the infectious period of a node influences the survival time even if the expected number of spreading attempts during an infectious period the non markovian equivalent of the effective infection rate is kept constant both early and late infection attempts lead to shorter survival times interestingly just as in markovian sis the survival times appear to be exponentially distributed regardless of the infection and curing time distributions

phase transitions with infinitely many absorbing states in complex networks we investigate the properties of the threshold contact process tcp a process showing an absorbing state phase transition with infinitely many absorbing states on random complex networks the finite size scaling exponents characterizing the transition are obtained in a heterogeneous mean field hmf approximation and compared with extensive simulations particularly in the case of heterogeneous scale free networks we observe that the tcp exhibits the same critical properties as the contact process which undergoes an absorbing state phase transition to a single absorbing state the accordance among the critical exponents of different models and networks leads to conjecture that the critical behavior of the contact process in a hmf theory is a universal feature of absorbing state phase transitions in complex networks depending only on the locality of the interactions and independent of the number of absorbing states the conditions for the applicability of the conjecture are discussed considering a parallel with the susceptible infected susceptible epidemic spreading model which in fact belongs to a different universality class in complex networks

slow dynamics and rare region effects in the contact process on weighted tree networks we show that generic slow dynamics can occur in the contact process on complex networks with a tree like structure and a superimposed weight pattern in the absence of additional nontopological sources of quenched disorder the slow dynamics is induced by rare region effects occurring on correlated subspaces of vertices connected by large weight edges and manifests in the form of a smeared phase transition we conjecture that more sophisticated network motifs could be able to induce griffiths phases as a consequence of purely topological disorder

universal finite size scaling behavior and universal dynamical scaling behavior of absorbing phase transitions with a conserved field we analyze numerically three different models exhibiting an absorbing phase transition we focus on the finite size scaling as well as the dynamical scaling behavior an accurate determination of several critical exponents allows one to validate certain hyperscaling relations using these hyperscaling relations it is possible to express the avalanche exponents of a self organized critical system in terms of the ordinary exponents of a continuous absorbing phase transition

traffic model with an absorbing state phase transition we consider a modified nagel schreckenberg ns model in which drivers do not decelerate if their speed is smaller than the headway number of empty sites to the car ahead in the original ns model such a reduction in speed occurs with probability p independent of the headway as long as the current speed is greater than zero in the modified model the free flow state with all vehicles traveling at the maximum speed v max is absorbing for densities smaller than a critical value c 1 v max 2 the phase diagram in the p plane is reentrant for densities in the range c c both small and large values of p favor free flow while for intermediate values a nonzero fraction of vehicles have speeds v max in addition to representing a more realistic description of driving behavior this change leads to a better understanding of the phase transition in the original model our results suggest an unexpected connection between traffic models and stochastic sandpiles

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

non markovian infection spread dramatically alters the susceptible infected susceptible epidemic threshold in networks most studies on susceptible infected susceptible epidemics in networks implicitly assume markovian behavior the time to infect a direct neighbor is exponentially distributed much effort so far has been devoted to characterize and precisely compute the epidemic threshold in susceptible infected susceptible markovian epidemics on networks here we report the rather dramatic effect of a nonexponential infection time while still assuming an exponential curing time on the epidemic threshold by considering weibullean infection times with the same mean but different power exponent for three basic classes of graphs the erd s renyi random graph scale free graphs and lattices the average steady state fraction of infected nodes is simulated from which the epidemic threshold is deduced for all graph classes the epidemic threshold significantly increases with the power exponents hence real epidemics that violate the exponential or markovian assumption can behave seriously differently than anticipated based on markov theory

accuracy criterion for the mean field approximation in susceptible infected susceptible epidemics on networks mean field approximations mfas are frequently used in physics when a process such as an epidemic or a synchronization on a network is approximated by mfa a major hurdle is the determination of those graphs for which mfa is reasonably accurate here we present an accuracy criterion for markovian susceptible infected susceptible sis epidemics on any network based on the spectrum of the adjacency and sis covariance matrix we evaluate the mfa criterion for the complete and star graphs analytically and numerically for connected erd s renyi random graphs for small size n 14 the accuracy of mfa increases with average degree and with n precise simulations up to network sizes n 100 of the mfa accuracy criterion versus n for the complete graph star square lattice and path graphs lead us to conjecture that the worst mfa accuracy decreases for large n proportionally to the inverse of the spectral radius of the adjacency matrix of the graph

collective versus hub activation of epidemic phases on networks we consider a general criterion to discern the nature of the threshold in epidemic models on scale free sf networks comparing the epidemic lifespan of the nodes with largest degrees with the infection time between them we propose a general dual scenario in which the epidemic transition is either ruled by a hub activation process leading to a null threshold in the thermodynamic limit or given by a collective activation process corresponding to a standard phase transition with a finite threshold we validate the proposed criterion applying it to different epidemic models with waning immunity or heterogeneous infection rates in both synthetic and real sf networks in particular a waning immunity irrespective of its strength leads to collective activation with finite threshold in scale free networks with large degree exponent at odds with canonical theoretical approaches

perturbed brownian motions we study perturbed brownian motions that can be loosely speaking described as follows they behave exactly as linear brownian motion except when they hit their past maximum or and maximum where they get an extra push we define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain natural class of processes in the case where both perturbations at the maximum and at the minimum are self repelling we show that in fact more is true such a process can almost surely be constructed from brownian paths by a one to one measurable transformation this generalizes some results of carmona petit yor and davis we also derive some fine properties of perturbed brownian motions hausdorff dimension of points of monotonicity for example

susceptible infected susceptible model a comparison of n intertwined and heterogeneous mean field approximations we introduce the susceptible infected susceptible sis spreading model which is taken as a benchmark for the comparison between the n intertwined approximation and the pastor satorras and vespignani heterogeneous mean field hmf approximation of the sis model the n intertwined approximation the hmf approximation and the sis spreading model are compared for different graph types we focus on the epidemic threshold and the steady state fraction of infected nodes in networks with different degree distributions overall the n intertwined approximation is superior to the hmf approximation the n intertwined approximation is exactly the same as the hmf approximation in regular graphs however for some special graph types such as the square lattice graph and the path graph the two mean field approximations are both very different from the sis spreading model

sandpiles with height restrictions we study stochastic sandpile models with a height restriction in one and two dimensions a site can topple if it has a height of two as in manna s model but in contrast to previously studied sandpiles here the height or number of particles per site cannot exceed two this yields a considerable simplification over the unrestricted case in which the number of states per site is unbounded two toppling rules are considered in one the particles are redistributed independently while the other involves some cooperativity we study the fixedenergy system no input or loss of particles using cluster approximations and extensive simulations and find that it exhibits a continuous phase transition to an absorbing state at a critical value z c of the particle density the critical exponents agree with those of the unrestricted manna sandpile

a stochastic sis epidemic model with heterogeneous contacts abstract a stochastic model for the spread of an sis epidemic among a population consisting of n individuals each having heterogeneous infectiousness and or susceptibility is considered and its behavior is analyzed under the practically relevant situation when n is small the model is formulated as a finite time homogeneous continuous time markov chain x based on an appropriate labeling of states we first construct its infinitesimal rate matrix by using an iterative argument and we then present an algorithmic procedure for computing steady state measures such as the number of infected individuals the length of an outbreak the maximum number of infectives and the number of infections suffered by a marked individual during an outbreak the time till the epidemic extinction is characterized as a phase type random variable when there is no external source of infection and its laplace stieltjes transform and moments are derived in terms of a forward elimination backward substitution solution the inverse iteration method is applied to the quasi stationary distribution of x which provides a good approximation of the process x at a certain time conditional on non extinction after a suitable waiting time the basic reproduction number r 0 is defined here as a random variable rather than an expected value