From random point processes to hierarchical cavity master equations for stochastic dynamics of disordered systems in random graphs: Ising models and epidemics

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.

Dynamics of epidemics from cavity master equations: Susceptible-infectious-susceptible models

We apply the recently introduced cavity master equation (CME) to epidemic models and compare it to previously known approaches. We show that CME seems to be the formal way to derive (and correct) dynamic message passing (rDMP) equations that were previously introduced in an intuitive ad hoc manner. CME outperforms rDMP in all cases studied. Both approximations are nonbacktracking and this causes CME and rDMP to fail when the ecochamber mechanism is relevant, as in loopless topologies or scale free networks. However, we studied several random regular graphs and Erdős-Rényi graphs, where CME outperforms individual based mean field and a type of pair based mean field, although it is less precise than pair quenched mean field. We derive analytical results for endemic thresholds and compare them across different approximations.