Exact mean field dynamics for epidemic-like processes on heterogeneous networks

TL;DR

This paper derives exact mean field solutions for epidemic-like processes on heterogeneous networks, providing a unified approach for SIR, SI, rumor, and recommendation models, and validating them with simulations.

Contribution

It introduces a method to exactly solve mean field equations for various epidemic and spreading models on complex networks, including arbitrary degree distributions.

Findings

Exact solutions for SIR epidemic dynamics on arbitrary networks.

Validation of mean field theory through simulations on scale-free networks.

Extension of the method to rumor and recommendation spreading models.

Abstract

We show that the mean field equations for the SIR epidemic can be exactly solved for a network with arbitrary degree distribution. Our exact solution consists of reducing the dynamics to a lone first order differential equation, which has a solution in terms of an integral over functions dependent on the degree distribution of the network, and reconstructing all mean field functions of interest from this integral. Irreversibility of the SIR epidemic is crucial for the solution. We also find exact solutions to the sexually transmitted disease SI epidemic on bipartite graphs, to a simplified rumor spreading model, and to a new model for recommendation spreading, via similar techniques. Numerical simulations of these processes on scale free networks demonstrate the qualitative validity of mean field theory in most regimes.