  We study a security game over a network played between a $defender$ and $k$
$attackers$. Every attacker chooses, probabilistically, a node of the network
to damage. The defender chooses, probabilistically as well, a connected induced
subgraph of the network of $\lambda$ nodes to scan and clean.Each attacker
wishes to maximize the probability of escaping her cleaning by the defender. On
the other hand, the goal of the defender is to maximize the expected number of
attackers that she catches. This game is a generalization of the model from the
seminal paper of Mavronicolas et al. "The price of defense" (MFCS'06). We are
interested in Nash equilibria (NE) of this game, as well as in characterizing
$defense$-$optimal$ networks which allow for the best $equilibrium$ $defense$
$ratio$, termed Price of Defense; this is the ratio of $k$ over the expected
number of attackers that the defender catches in a NE. We provide
characterizations of the NEs of this game and defense-optimal networks. This
allows us to show that the NEs of the game coincide independently from the
coordination or not of the attackers. In addition, we give an algorithm for
computing NEs. Our algorithm requires exponential time in the worst case, but
it is polynomial-time for $\lambda$ constantly close to 1 or $n$. For the
special case of tree-networks, we refine our characterization which allows us
to derive a polynomial-time algorithm for deciding whether a tree is
defense-optimal and if this is the case it computes a defense-optimal NE. On
the other hand, we prove that it is NP-hard to find a best-defense strategy if
the tree is not defense-optimal. We complement this negative result with a
polynomial-time constant-approximation algorithm that computes solutions that
are close to optimal ones for general graphs. Finally, we provide
asymptotically (almost) tight bounds for the Price of Defense for any
$\lambda$.
