Susceptible-infected-susceptible process on Erdős-Rényi graphs: Determining the infected fraction

There are many methods to estimate the quasistationary infected fraction of the SIS process on (random) graphs. A challenge is to adequately incorporate correlations, which is especially important in sparse graphs. Methods typically are either significantly biased in sparse graphs, or computationally very demanding already for small network sizes. The former applies to heterogeneous mean field and to the N-intertwined mean field approximation, the latter to most higher order approximations. In this paper we introduce a method to determine the infected fraction in sparse graphs, which we test on Erdős-Rényi graphs. Our method is based on degree pairs, does take into account correlations, and gives accurate estimates. At the same time, computations are very feasible and can easily be done even for large networks.