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.