Optimal Network Topologies for Mitigating Security and Epidemic Risks

We consider networked environments under security and epidemic risks, where the probability of successful attack or infection at each vertex depends on the actions or states of its neighbors. In such settings, we consider the problem of designing an optimal network topology with a given number of vertices and edges in order to minimize the expected fraction of attacked or infected vertices. We show that such problems can be cast as minimizing the sum of a concave function of the vertex degrees, and generalize existing results on network design to obtain insights about the optimal network topologies. We first consider a class of interdependent security games where each vertex represents a user that invests in security to protect herself. The probability of successful attack at any given vertex is a function of the security investments in the neighborhood of that vertex. We introduce the notion of behavioral risk-attitudes, where each user perceives the security risks in a skewed manner (as prescribed by established models from the behavioral economics literature). We characterize an upper bound on the expected number of vertices that are successfully attacked under the Nash equilibrium security investments in such settings, and identify the network topologies that minimize this bound. We then consider the N-intertwined approximation of SIS epidemic dynamics, and characterize graphs that minimize (bounds on) the fraction of infected vertices in steady state.