simulated network with no replenishment, then with random replenishment, then with scalefree replenishment. In the vanilla case the attack takes two rounds to disconnect the network; with random replenishment it takes three, and with scale-free replenishment it takes four.

![Naive defences against vertex-order decapitation attack[]{data-label="fig:BA00"}](replenish-graph.eps "fig:")\
![Naive defences against vertex-order decapitation attack[]{data-label="fig:BA00"}](replenish-key.eps "fig:")

It seems that, to defend against these kinds of decapitation attacks on networks, we will need smarter defence strategies. But how should these be evolved, and what sort of framework should we use to evaluate them?

A Model from Evolutionary Game Theory
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Previous researchers considered disruptive attacks on networks to be a single-round game. Such a model is suitable for applications such as a conventional war, in which the attacker has to expend a certain amount of effort to destroy the defender’s command, control and communications, and one wishes to estimate how much; or a single epidemic in which a certain amount of resource must be spent to bring the disease under control.

However, there are many applications in which attack and defense evolve through multiple rounds: terrorism and music-sharing are only two examples. We now develop a framework for considering this more general case. We 