Defense strategies for infrastructures with multiple systems of components

In several critical infrastructures correlations between the constituent systems represent certain vulnerabilities: disruptions to one may propagate to others and possibly to the entire infrastructure. The correlations between the systems are characterized in two ways in this paper: (i) the aggregate failure correlation function specifies the conditional failure probability of the infrastructure given the failure of an individual system, and (ii) the pairwise correlation function between two systems specifies the failure probability of one system given the failure of the other. The survival probabilities of individual systems satisfy first-order differential conditions that generalize the contest success functions and statistical independence conditions. We formulate a problem of ensuring the resilience of an infrastructure as a game between the provider and attacker; their utility functions are sums of infrastructure survival probability terms and cost terms, both expressed in terms of the numbers of system components attacked and reinforced. We derive Nash Equilibrium conditions and sensitivity functions that highlight the dependence of infrastructure resilience on the cost terms, correlation functions, and individual system survival probabilities. We apply these results to models of distributed cloud computing and energy grid infrastructures.