Existence Theorems and Approximation Algorithms for Generalized Network Security Games

Aspnes et al introduced an innovative game for modeling the containment of the spread of viruses and worms (security breaches) in a network. In this model, nodes choose to install anti-virus software or not on an individual basis while the viruses or worms start from a node chosen uniformly at random and spread along paths consisting of insecure nodes. They showed the surprising result that a pure Nash Equilibrium always exists when all nodes have identical installation costs and identical infection costs. In this paper we present a substantial generalization of the model of that allows for arbitrary security and infection costs, and arbitrary distributions for the starting point of the attack. More significantly, our model GNS(d) incorporates a network locality parameter d which represents a hop-limit on the spread of infection as accounted for in the strategic decisions, due to either the intrinsic nature of the infection or the extent of neighborhood information that is available to a node. We determine that the network locality parameter plays a key role in the existence of pure Nash equilibria (NE): local (d = 1) and global games (d = ∞) have pure NE, while for GNS(d) games with 1 <; d <; ∞, pure NE may not exist, and in fact, it is NP-complete to determine whether a given instance has a pure NE. For local and global games, we also characterize the price of anarchy in terms of the maximum degree and vertex expansion of the contact network; these suggest natural heuristics to aid a network planner in enforcing efficient equilibria. We design a general LP-based framework for approximating the NP-complete problem of finding a socially optimal configuration in our game. Our framework yields a 2d-approximation for general GNS(d) games, and an O(log n)-approximation for the global model where n is the number of network nodes; the latter result improves on the approximation bound of O(log
^1.5
n) of achieved for a special case of our global model. We study the characteristics of NE and the quality of our approximations empirically in two distinct classes of graphs: random geometric graphs and power law graphs. We find that in local and global games on these real-world networks, best response dynamics converge in linear or sub-linear time and have costs comparable to the social optimum. Finally, we study the performance of our approximation algorithms, and find that the approximation guarantees with respect to social cost are much better in practice than our theoretical bounds.