Diffusion in Social Networks as SIS Epidemics: Beyond Full Mixing and Complete Graphs

Peer influence and interactions between agents in a population give rise to complex, nonlinear behaviors. This paper adopts the SIS (susceptible-infected-susceptible) framework from epidemiology to analytically study how network topology affects the diffusion of ideas/opinions/beliefs/innovations in social networks. We introduce the scaled SIS process, which models peer influence as neighbor-to-neighbor infections. We model the scaled SIS process as a continuous-time Markov process and derive for this process its closed form equilibrium distribution. The adjacency matrix that describes the underlying social network is explicitly reflected in this distribution. The paper shows that interesting population asymptotic behaviors occur for scenarios where the individual tendencies of each agent oppose peer influences. Specifically, we determine how the most probable configuration of agent states (i.e., the population configuration with maximum equilibrium distribution) depends on both model parameters and network topology. We show that, for certain regions of the parameter space, this and related issues reduce to standard graph questions like the maximum independent set problem.