Analytical Framework for the Approximate Master Equation

TL;DR

This paper introduces an analytical framework to derive steady states of the approximate master equation on networks, improving understanding of dynamical processes like epidemic spread and evolutionary games.

Contribution

The authors develop a systematic approximation method that simplifies the steady state analysis of the AME, extending its applicability to various models.

Findings

Successfully derives steady states for SIS, voter, and evolutionary game models.

Reproduces pair approximation steady states and refines towards exact AME solutions.

Enables analytical derivation of time evolution in evolutionary games.

Abstract

The approximate master equation (AME) provides a highly accurate description of dynamical processes on networks, yet its steady states are generally analytically intractable. In this study, we develop an analytical framework to derive the steady states of the AME by introducing a controlled approximation that enables closure of the moment equations. This framework reproduces the steady state of the pair approximation by achieving closure with the minimum required order of moments, and can be systematically refined to approach the exact steady states of the AME. We apply this to the SIS model, the voter model, and evolutionary games, demonstrating that the steady states can be derived. In particular, for evolutionary games, we show that combining our framework with the singular perturbation method enables the analytical derivation of the time evolution.