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 work, we consider decentralized optimal protection strategies when a virus is propagating over a network through a 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 (PoA) in several network topologies. Further, we propose both 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.

We study the robustness of networks under node removal, considering random node failure, as well as targeted node attacks based on network centrality measures. Whilst both of these have been studied in the literature, existing approaches tend to study random failure in terms of average-case behavior, giving no idea of how badly network performance can degrade purely by chance. Instead of considering average network performance under random failure, we compute approximate network performance probability density functions as functions of the fraction of nodes removed. We find that targeted attacks based on centrality measures give a good indication of the worst-case behavior of a network. We show that many centrality measures produce similar targeted attacks and that a combination of degree centrality and eigenvector centrality may be enough to evaluate worst-case behavior of networks. Finally, we study the robustness envelope and targeted attack responses of networks that are rewired to have high-and lowdegree assortativities, discovering that moderate assortativity increases confer more robustness against targeted attacks whilst moderate decreases confer more robustness against random uniform attacks.

Abstract-Due to their importance to society, communication networks should be built and operated to withstand failures. However, cost considerations make network providers less inclined to take robustness measures against failures that are unlikely to manifest, like several failures coinciding simultaneously in different geographic regions of their network.Considering networks embedded in a two-dimensional plane, we study the problem of finding a critical region -a part of the network that can be enclosed by a given elementary figure of predetermined size -whose destruction would lead to the highest network disruption. We determine that only a polynomial, in the input, number of non-trivial positions for such a figure need to be considered and propose a corresponding polynomialtime algorithm. In addition, we consider region-aware network augmentation to decrease the impact of a regional failure. We subsequently address the region-disjoint paths problem, which asks for two paths with minimum total weight between a source (s) and a destination (d) that cannot both be cut by a single regional failure of diameter D (unless that failure includes s or d). We prove that deciding whether region-disjoint paths exist is NP-hard and propose a heuristic region-disjoint paths algorithm.

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.

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