Role of subgraphs in epidemics over finite-size networks under the scaled SIS process

Abstract

In previous work, we developed the scaled SIS process, which models the dynamics of SIS epidemics over networks. We derived for the scaled SIS process a closed-form expression for the time-asymptotic probability distribution of the configurations of all the agents in the network, which explicitly exhibits the underlying network topology through its adjacency matrix. This is accomplished for networks that are of finite-size and of arbitrary topology. This paper determines which network configuration is the most probable. We prove that, for a range of epidemic parameters, this combinatorial inference problem leads to a submodular optimization problem, which can be solved in polynomial time. We relate the most-probable configuration to the network structure, and in particular to the existence of high-density subgraphs. Depending on the model parameters, subset of agents may be more likely to be infected than others; these more vulnerable agents form subgraphs that are denser than the overall network. We illustrate our results with a 193 node social network of drug users and with the 4941 node Western US power grid under different model parameters.