Defining an optimal protection strategy against viruses, spam propagation, or any other kind of contamination process is an important feature for designing new networks and architectures. In this paper, we consider decentralized optimal protection strategies when a virus is propagating over a network through an SIS epidemic process. We assume that each node in the network can fully protect itself from infection at a constant cost, or the node can use recovery software, once it is infected. We model our system using a game-theoretic framework and find pure, mixed equilibria, and the Price of Anarchy in several network topologies. Further, we propose a decentralized algorithm and an iterative procedure to compute a pure equilibrium in the general case of a multiple communities network. Finally, we evaluate the algorithms and give numerical illustrations of all our results.
Virus spread processes in networks have been studied in the past @cite_44 @cite_38 @cite_39 @cite_19 , usually considering the number of infected nodes @cite_44 over the time and in stationary regimes, the epidemic threshold @cite_38 or the relation with eigenvalues @cite_31 . One of the widely explored Susceptible Infected Susceptible (SIS) approximations is the N-intertwined mean-field approximation NIMFA @cite_44 @cite_45 .
@cite_38 @cite_39 @cite_44 @cite_19 @cite_45 @cite_31
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The influence of the network characteristics on the virus spread is analyzed in a new-the N -intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The N -intertwined model has been compared with the exact 2N-state Markov model and with previously proposed ldquohomogeneousrdquo or ldquolocalrdquo models. The sharp epidemic threshold tauc , which is a consequence of mean field theory, is rigorously shown to be equal to tauc = 1 (lambdamax(A)) , where lambdamax(A) is the largest eigenvalue-the spectral radius-of the adjacency matrix A . A continued fraction expansion of the steady-state infection probability at node j is presented as well as several upper bounds. Besides the epidemic threshold, the recently proposed viral conductance @j by [11] may be regarded as an additional characterizer of the viral robustness of a network, that measures the overall ease in which viruses can spread in a particular network. Motivated to explain observed features of the viral conductance @j in simulations [29], we have analysed this metric in depth using the N-intertwined SIS epidemic model, that upper bounds the real infection probability in any network and, hence, provides safe-side bounds on which network protection can be based. Our study here derives a few exact results for @j, a number of different lower and upper bounds for @j with variable accuracy. We also extend the theory of the N-intertwined SIS epidemic model, by deducing formal series expansions of the steady-state fraction of infected nodes for any graph and any effective infection rate, that result in a series for the viral conductance @j. Though approximate, we illustrate here that the N-intertwined SIS epidemic model is so far the only SIS model on networks that is analytically tractable, and valuable to provide first order estimates of the epidemic impact in networks. Finally, inspired by the analogy between virus spread and synchronization of coupled oscillators in a network, we propose the synchronizability as the analogue of the viral conductance. Given a contact network that changes over time (say, day vs night connectivity), and the SIS (susceptible infected susceptible, flu like) virus propagation model, what can we say about its epidemic threshold? That is, can we determine when a small infection will "take-off" and create an epidemic? Consequently then, which nodes should we immunize to prevent an epidemic? This is a very real problem, since, e.g. people have different connections during the day at work, and during the night at home. Static graphs have been studied for a long time, with numerous analytical results. Time-evolving networks are so hard to analyze, that most existing works are simulation studies [5]. Specifically, our contributions in this paper are: (a) we formulate the problem by approximating it by a Non-linear Dynamical system (NLDS), (b) we derive the first closed formula for the epidemic threshold of time-varying graphs under the SIS model, and finally (c) we show the usefulness of our threshold by presenting efficient heuristics and evaluate the effectiveness of our methods on synthetic and real data like the MIT reality mining graphs. The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent β > 2, we show that the PoA is bounded by O(T2(β -1))), where T = γ α is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations.