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.

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Consider an 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 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. We model this network scenario as a non-cooperative strategic game on graphs. We focus on the special case where the protector chooses a single edge. We are interested in the associated Nash equilibria, where no network entity can unilaterally improve its local objective. We obtain the following results: • No instance of the game 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, such as independent sets and matchings.