Network Security and Contagion

We develop a theoretical model of security investments in a network of interconnected agents. Network connections introduce the possibility of cascading failures, due to an exogenous or endogenous attack, depending on the profile of security investments by the agents. The general presumption in the literature, based on intuitive arguments or analysis of symmetric networks, is that because security investments create positive externalities on other agents, there will be under-investment in security. We show that this reasoning is incomplete because of a first-order economic force: security investments are also strategic substitutes. In a general (non-symmetric) network, this implies that under-investment by some agents will encourage over-investment by others. We demonstrate, by means of examples, there can be over-investment by some agents, and also that aggregate probabilities of infection can be lower in equilibrium compared to the social optimum. We then provide sufficient conditions for under-investment. This requires both sufficiently convex cost functions, (convexity alone is not enough), and networks that are either symmetric or locally tree-like. We also characterize the impact of network structure on equilibrium and optimal investments. Finally, we show that when the attack location is endogenized, (by assuming that the attacker chooses a probability distribution over the location of the attack in order to maximize damage), there is an additional incentive for over-investment: greater investment by an agent shifts the attack to other parts of the network