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

In previous work, we developed the scaled SIS process, which models the dynamics of SIS epidemics over networks. With the scaled SIS process, we can consider networks that are finite-sized and of arbitrary topology (i.e., we are not restricted to specific classes of networks). We derived for the scaled SIS process a closed-form expression for the time-asymptotic probability distribution of the states of all the agents in the network. This closed-form solution of the equilibrium distribution explicitly exhibits the underlying network topology through its adjacency matrix. This paper determines which network configuration is the most probable. We prove that, for a range of epidemics parameters, this combinatorial problem leads to a submodular optimization problem, which is exactly solvable in polynomial time. We relate the most-probable configuration to the network structure, in particular, to the existence of high density subgraphs. Depending on the epidemics 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 and the 4941 node Western US power grid under different epidemics parameters.