Abstract
We study a security game over a network played between a defender and kattackers. 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. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. 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 further 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 Nash equilibrium. On the other hand, we prove that it is {mathtt {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 ; this is the worst equilibrium defense ratio over all graphs.
Highlights
With technology becoming a ubiquitous and integral part of our lives, we find ourselves using several different types of computer networks
In this work we depart from and significantly extend the line of work of Mavronicolas et al in their seminal paper [11] on defense games in graphs; we term the type of games we consider Connected Subgraph Defense (CSD) games
We study many questions related to CSD games
Summary
With technology becoming a ubiquitous and integral part of our lives, we find ourselves using several different types of computer networks. The study of network security has attracted a lot of attention over the years [18] Such breaches are often inevitable, since some parts of a large system are expected to have weaknesses that expose them to security attacks; history has shown several successful and highly-publicized such incidents [17]. The defender seeks to protect the network as much as possible, and on the other hand, every attacker seeks to increase the likelihood of not being caught. Since attacks and defenses over a large computer network are self-interested procedures that seek to maximize damage and protection, respectively, it is natural to model this network security scenario as a non-cooperative strategic game on graphs with two kinds of players: k ≥ 1 attackers, each playing a vertex of the graph, and a single defender playing a connected induced subgraph of the graph. We are interested in understanding and characterizing the networks that allow for a good defense ratio: given a strategy profile, i.e. a combination of strategies for the network entities (attackers and defender), the defense ratio of a network is the ratio of the total number of attackers over the defender’s expected payoff in that strategy profile
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