A GAME ON A DISTRIBUTED NETWORK

Consider a distributed information network with harmful procedures called attackers (e.g., viruses); each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is the system protector scanning and cleaning from attackers some part of the network (e.g., an edge or a simple path), which it chooses independently using another probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the system protector; towards a conflicting objective, the system protector aims at maximizing the expected number of cleaned attackers. In [8, 9], we model this network scenario as a non-cooperative strategic game on graphs. We focus on two basic cases for the protector; where it may choose a single edge or a simple path of the network. The two games obtained are called as the Path and the Edge model, respectively. For these games, we are interested in the associated Nash equilibria, where no network entity can unilaterally improve its local objective. For the Edge model we obtain the following results: No instance of the model possesses a pure Nash equilibrium. Every mixed Nash equilibrium enjoys a graph-theoretic structure, which enables a (typically exponential) algorithm to compute it. We coin a natural subclass of mixed Nash equilibria, which we call matching Nash equilibria, for this game on graphs. Matching Nash equilibria are defined using structural parameters of graphs - We derive a characterization of graphs possessing matching Nash equilibria. The characterization enables a linear time algorithm to compute a matching Nash equilibrium on any such graph. - Bipartite graphs and trees are shown to satisfy the characterization; we derive polynomial time algorithms that compute matching Nash equilibria on corresponding instances of the game. We proceed with other graph families. Utilizing graph-theoretic arguments and the characterization of mixed NE proved before, we compute, in polynomial time, mixed Nash equilibria on corresponding graph instances. The graph families considered are regular graphs, graphs with, polynomial time computable, r-regular factors and graphs with perfect matchings. We define the social cost of the game to be the expected number of attackers catch by the protector. We prove that the corresponding Price of Anarchy in any mixed Nash equilibria of the Edge model is upper and lower bounded by a linear function of the number of vertices of the graph. Finally, we consider the more generalized variation of the problem considered, captured by the Path model. We prove that the problem of existence of a pure Nash equilibrium is NP-complete for this model.