Markov processes on quasi-random graphs

TL;DR

This paper investigates the accuracy of mean-field approximations for Markov processes on large graphs, establishing conditions under which these approximations are valid and providing explicit error bounds, especially for epidemic models.

Contribution

It proves that the homogeneous mean-field approximation is accurate for quasi-random graphs and provides explicit error bounds, linking graph properties to approximation validity.

Findings

HMFA is accurate on quasi-random graphs with explicit error bounds.

Error bounds are of order 1/√N plus graph discrepancy.

Diverging average degree is necessary for HMFA accuracy.

Abstract

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. Explicit error bound is given and being of order $\frac{1}{\sqrt{N}}$ plus the largest discrepancy of the graph. For Erd\H{o}s R\'{e}nyi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.