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.
Game theory has been used in several studies @cite_25 @cite_22 @cite_15 @cite_48 @cite_18 @cite_29 @cite_1 @cite_37 @cite_43 @cite_12 related to epidemic protection or curing, for example, in a generalized game settings @cite_18 without considering the infection state of the neighbors; by assigning nodal weights to reflect the security level @cite_48 etc. Omi ' c @cite_25 tune the strength of the nodal antivirus protection i.e. how big those (different) @math should be taken. Contrarily to @cite_25 , (i) we fix the curing and infection rates, which are not part of the game, and the decision consists of a player's choice to invest in an antivirus or not; (ii) we also consider mixed strategies Nash Equilibrium and (iii) propose a convergence algorithm to the equilibrium point. The goal of @cite_25 is in finding the optimal curing rates @math for each player @math , while this paper targets the optimal decision of taking an anti-virus that fully protects the host, because today's antivirus software packages provide accurate and up-to-date virus protection.
@cite_18 @cite_37 @cite_22 @cite_48 @cite_29 @cite_1 @cite_43 @cite_15 @cite_25 @cite_12
2398538542 2152769037 2025392770 2140356130
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. We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install anti-virus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. The goal of individual nodes is to minimize their individual expected cost. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that allowing selfish users to choose Nash equilibrium strategies is highly undesirable, because the price of anarchy is an unacceptable Θ(n) in the worst case. This shows in particular that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NP-hard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sum-of-squares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O(log2 n), giving the same approximation ratio for the inoculation game. We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install anti-virus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NP-hard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sum-of-squares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O(log1.5 n). Over the last years, game theory has provided great insights into the behavior of distributed systems by modeling the players as utility-maximizing agents. In particular, it has been shown that selfishness causes many systems to perform in a globally suboptimal fashion. Such systems are said to have a large Price of Anarchy. In this paper, we extend this active field of research by allowing some players to be malicious or Byzantine rather than selfish. We ask: What is the impact of Byzantine players on the system's efficiency compared to purely selfish environments or compared to the social optimum? In particular, we introduce the Price of Malice which captures this efficiency degradation. As an example, we analyze the Price of Malice of a game which models the containment of the spread of viruses. In this game, each node can choose whether or not to install anti-virus software. Then, a virus starts from a random node and iteratively infects all neighboring nodes which are not inoculated. We establish various results about this game. For instance, we quantify how much the presence of Byzantine players can deteriorate or---in case of highly risk-averse selfish players---improve the social welfare of the distributed system.