Consider a network vulnerable to viral infection. The system security software can guarantee safety only to a limited part
of the network. We model this practical network scenario as a non-cooperative multi-player 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 (or a set of them, via a probability distribution) to infect. The protector player chooses independently, in a basic
case of the problem, a simple path or an edge of the graph (or a set of them, via a probability distribution) and cleans this
part of the network 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 cleaned attackers. We call the two games obtained from
the two basic cases considered, as the Path and the Edge model, respectively.
We are interested in the associated Nash equilibria on them, where no network entity can unilaterally improve its local objective. We obtain the following results:
– The problem of existence of a pure Nash equilibrium is
NP\cal NP-complete for the Path model. This opposed to that, no instance of the Edge model possesses a pure Nash equilibrium, proved
in [4].
– We compute, in polynomial time, mixed Nash equilibria on corresponding graph instances. These graph families include, regular
graphs, graphs that can be decomposed, in polynomially time, into vertex disjoint r-regular subgraphs, graphs with perfect matchings and trees.
– We utilize the notion of social cost [3] for measuring system performance on such scenario; here is defined to be the utility of the protector. We prove that
the corresponding Price of Anarchy in any mixed Nash equilibria of the game is upper and lower bounded by a linear function of the number of vertices of the
graph.
This work was partially supported by the IST Programs of the European Union under contract numbers IST-2001-33116 (FLAGS) and IST-2004-001907 (DELIS).