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

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

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

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

epidemic spreading in correlated complex networks we study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node s connectivities for the case of markovian complex networks showing only correlations between pairs of nodes we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links which from a node with connectivity k go to nodes with connectivity k ensuremath numerical simulations on a correlated growing network model provide support for our conclusions

epidemic dynamics in finite size scale free networks many real networks present a bounded scale free behavior with a connectivity cutoff due to physical constraints or a finite network size we study epidemic dynamics in bounded scale free networks with soft and hard connectivity cutoffs the finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes the induced epidemic threshold is very small even at a relatively small cutoff showing that the neglection of connectivity fluctuations in bounded scale free networks leads to a strong overestimation of the epidemic threshold we provide the expression for the infection prevalence and discuss its finite size corrections the present paper shows that the highly heterogeneous nature of scale free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes

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

reduced representations of rooted trees abstract let k be a field and q a quiver the category of finite dimensional representations of q will be denoted by rep k q we can define the pointwise tensor product on the category rep k q i e for two representations n m the representation n m is defined at a vertex x q 0 as n m x n x m x and for an arrow q 1 we define n m n m this tensor product has been investigated in recent work of herschend and kinser 1 2 3 in 3 kinser investigated the representation ring of rooted trees a rooted tree q is a quiver q whose underlying graph is tree and which has a unique sink the representation ring of q is the free abelian group generated by the isomorphism classes of indecomposable objects in rep k q it is endowed with a multiplication induced by the pointwise tensor product see 3 for details for these rooted trees kinser constructed reduced representations which are defined inductively these reduced representations then lead to a set of orthogonal primitive idempotents in the representation ring of q on the other hand there is an inductive construction of so called radiation modules for a tree with arbitrary orientation in 5 in the special case of a rooted tree quiver this construction results in the reduced representations introduced by kinser the idea used in the construction of the radiation modules then allows us to prove a new characterization of reduced representations which is theorem let q be a rooted tree quiver an indecomposable representation v rep k q is reduced if and only if v is a direct summand of v v this theorem then allows us to prove further results for the special case where q is a rooted tree quiver whose underlying graph is of dynkin type in that case v being reduced is also equivalent to the condition that the vector space of v at the sink is one dimensional

how to cut a cake with a gram matrix abstract in this article we study a problem of fair division in particular we study a notion introduced by j barbanel that generalizes super envy free fair division we call this notion hyper envy free we give a new proof for the existence of such fair divisions our approach relies on classical linear algebra tools and allows us to give an explicit bound for this kind of fair division furthermore we also give a theoretical answer to an open problem posed by barbanel in 1996 roughly speaking this question is how can we decide if there exists a fair division satisfying some inequality constraints furthermore when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division

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

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

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

exceptional modules are tree modules abstract we consider representations of a quiver over an arbitrary field recall that an indecomposable representation m without self extensions is said to be exceptional we are going to show that exceptional representations can be exhibited using matrices involving as coefficients just 0 and 1 actually if d is the dimension of m there exists such a matrix presentation with precisely d 1 non zero coefficients the corresponding coefficient quiver is a tree

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

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

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

universal scaling of distances in complex networks universal scaling of distances between vertices of erd ifmmode mbox h o else h o fi s r enyi random graphs scale free barab asi albert models science collaboration networks biological networks internet autonomous systems and public transport networks are observed a mean distance between two nodes of degrees k i and k j equals to l ij a ensuremath b phantom rule 0 2em 0ex mathrm log k i k j the scaling is valid over several decades a simple theory for the appearance of this scaling is presented parameters a and b depend on the mean value of a node degree k nn calculated for the nearest neighbors and on network clustering coefficients

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

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