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 Erdős-Rényi 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. The epidemic threshold of a network distinguishes between the overall-healthy network regime and the effective infection regime where permanently a nonzero fraction of the nodes is infected. The epidemic threshold reflects the effectivity of an epidemic in a particular network and is a major indicator or tool to protect the nodes (people, computers, etc.) and to take preventive measures (governmental immunization strategies, antivirus software protection).Recently (see, e.g., Refs. [1][2][3][4][5][6][7]) much effort has been devoted to the precise computation of the epidemic threshold in the continuous-time susceptible-infected-susceptible (SIS) Markov model in networks. In that simple SIS model, the viral state of a node i at time t is specified by a Bernoulli random variable X i ðtÞ 2 f0; 1g: X i ðtÞ ¼ 0 for a healthy node and X i ðtÞ ¼ 1 for an infected node. A node i at time t can be in one of two states: infected, with probability v i ðtÞ ¼ Pr½X i ðtÞ ¼ 1 or healthy, with probability 1 À v i ðtÞ, but susceptible to the infection. The curing process per node i is a Poisson process with rate and the infection rate per link is a Poisson process with rate . Obviously, only when a node is infected, can it infect its direct neighbors that are still healthy. Both the curing and infection Poisson processes are independent. The network is represented by an adjacency matrix A, where a ij ¼ 1 if there is a link from node i to node j, otherwise a ij ¼ 0. A major complication in the SIS Markov model is the absorbing state to which the epidemic SIS process always converges after a sufficiently long time in any network G with a finite number N of nodes and L of links. Hence, the steady state is the overall-healthy (absorbing) state. Since the exact steady state is physically less meaningful, the epidemic threshold refers to the metastable or quasistationary state which is observed in practice. However, the metastable state needs to be defined (see, e.g., Refs. [4,8]).Since X i is a Bernoulli random ...

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 (x 1 ) i (x 1 ) j of the components of the eigenvector x 1 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.

We introduce the ε-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 ε-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 ε-SIS spreading model.

The classical, continuous-time susceptible-infected-susceptible (SIS) Markov epidemic model on an arbitrary network is extended to incorporate infection and curing or recovery times each characterized by a general distribution (rather than an exponential distribution as in Markov processes). This extension, called the generalized SIS (GSIS) model, is believed to have a much larger applicability to real-world epidemics (such as information spread in online social networks, real diseases, malware spread in computer networks, etc.) that likely do not feature exponential times. While the exact governing equations for the GSIS model are difficult to deduce due to their non-Markovian nature, accurate mean-field equations are derived that resemble our previous Nintertwined mean-field approximation (NIMFA) and so allow us to transfer the whole analytic machinery of the NIMFA to the GSIS model. In particular, we establish the criterion to compute the epidemic threshold in the GSIS model. Moreover, we show that the average number of infection attempts during a recovery time is the more natural key parameter, instead of the effective infection rate in the classical, continuous-time SIS Markov model. The relative simplicity of our mean-field results enables us to treat more general types of SIS epidemics, while offering an easier key parameter to measure the average activity of those general viral agents.

The interplay between disease dynamics on a network and the dynamics of the structure of that network characterizes many real-world systems of contacts. A continuous-time adaptive susceptible-infectious-susceptible (ASIS) model is introduced in order to investigate this interaction, where a susceptible node avoids infections by breaking its links to its infected neighbors while it enhances the connections with other susceptible nodes by creating links to them. When the initial topology of the network is a complete graph, an exact solution to the average metastable-state fraction of infected nodes is derived without resorting to any mean-field approximation. A linear scaling law of the epidemic threshold τ c as a function of the effective link-breaking rate ω is found. Furthermore, the bifurcation nature of the metastable fraction of infected nodes of the ASIS model is explained. The metastable-state topology shows high connectivity and low modularity in two regions of the τ,ω plane for any effective infection rate τ > τ c : (i) a "strongly adaptive" region with very high ω and (ii) a "weakly adaptive" region with very low ω. These two regions are separated from the other half-open elliptical-like regions of low connectivity and high modularity in a contour-line-like way. Our results indicate that the adaptation of the topology in response to disease dynamics suppresses the infection, while it promotes the network evolution towards a topology that exhibits assortative mixing, modularity, and a binomial-like degree distribution.

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-infectedsusceptible (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 Erdős-Rényi 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.

Abstract-For many networked games, such as the Defense of the Ancients and StarCraft series, the unofficial leagues created by players themselves greatly enhance user-experience, and extend the success of each game. Understanding the social structure that players of these games implicitly form helps to create innovative gaming services to the benefit of both players and game operators. But how to extract and analyse the implicit social structure? We address this question by first proposing a formalism consisting of various ways to map interaction to social structure, and apply this to real-world data collected from three different game genres. We analyse the implications of these mappings for in-game and gaming-related services, ranging from network and socially-aware matchmaking of players, to an investigation of social network robustness against player departure.

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 K N and the star graph K 1,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 τ c that, at least for K N and K 1,N−1 , correctly scales with the number of nodes N and that is superior to the epidemic threshold τ . 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.