On resilience of cyber-physical infrastructures using discrete product-form games

In critical infrastructures consisting of discrete cyber and physical components, the correlations between them may be exploited to launch strategic component attacks that may degrade the entire system. We capture such correlations between cyber and physical sub-infrastructures using the conditional probabilities, and between cyber and physical components using first-order differential conditions. By using a resilience measure specified by the infrastructure's survival probability, we formulate a discrete game between the provider and attacker. Their disutility functions are products of the survival (or failure) probability and cost terms expressed in terms of the number of components attacked and reinforced by the attacker and provider, respectively. The Nash Equilibrium conditions of the game provide the sensitivity functions that clearly show the dependence of the infrastructure resilience on cost terms, correlation function and sub-infrastructure survival probabilities. These results for product-form disutility functions complement the sum-form results from previous works, and more closely represent the provider's objectives for a certain class of infrastructures. We apply these results to simple models of network testbed infrastructures and cyber infrastructures of smart energy grids.