A Graph-Theoretic Network Security Game

Consider a network vulnerable to viral infection, where the security software can guarantee safety only to a limited part of it. We model this practical network scenario as a non-cooperative multiplayer game on a graph, with two kinds of players, a set of attackers and a protector player, representing the viruses and the system security software, respectively. Each attacker player chooses a node of the graph via a probability distribution to infect. The protector player chooses either an edge or a simple path of the network and cleans this part from attackers. Each attacker wishes to maximize the probability of escaping its cleaning by the protector. In contrast, the protector aims at maximizing the expected number of extinguished attackers. We call the two games obtained the Path and the Edge model, respectively. We are interested in the associated Nash equilibria, where no network entity can unilaterally improve its local objective. We obtain the following results: For certain families of graphs, mixed Nash equilibria can be computed in polynomially time. These families include, among others, regular graphs, graphs with perfect matchings and trees. The corresponding Price of Anarchy for any mixed Nash equilibria of the game is upper and lower bounded by a linear function of the number of vertices of the graph. (We define the Price of Anarchy to reflect the utility of the protector.) The problem of existence of a pure Nash equilibrium for the Path model is NP-complete.