## A network game with attackers and a defender

##### Date

2008##### Source

Algorithmica (New York)##### Volume

51##### Issue

3##### Pages

315-341Google Scholar check

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Consider an information network with threats called attackers each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is a protector entity called defender the defender scans and cleans from attacks some part of the network (in particular, a link), which it chooses independently using its own probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the defender towards a conflicting objective, the defender aims at maximizing the expected number of attackers it catches. We model this network security scenario as a non-cooperative strategic game on graphs. We are interested in its associated Nash equilibria, where no network entity can unilaterally increase its local objective. We obtain the following results: • We obtain an algebraic characterization of (mixed) Nash equilibria. • No (non-trivial) instance of the graph-theoretic game has a pure Nash equilibrium. This is an immediate consequence of some covering properties we prove for the supports of the players in all (mixed) Nash equilibria. • We coin a natural subclass of mixed Nash equilibria, which we call Matching Nash equilibria, for this graph-theoretic game. Matching Nash equilibria are defined by enriching the necessary covering properties we proved with some additional conditions involving other structural parameters of graphs, such as Independent Sets. - We derive a characterization of graphs admitting Matching Nash equilibria. All such graphs have an Expanding Independent Set. The characterization enables a non-deterministic, polynomial time algorithm to compute a Matching Nash equilibrium for any such graph. - Bipartite graphs are shown to satisfy the characterization. So, using a polynomial time algorithm to compute a Maximum Matching for a bipartite graph, we obtain, as our main result, a deterministic, polynomial time algorithm to compute a Matching Nash equilibrium for any instance of the game with a bipartite graph. © 2007 Springer Science+Business Media, LLC.