Concentration and mean field approximation results for Markov processes on large networks

TL;DR

This paper establishes concentration bounds for Markov processes on large networks and provides improved error estimates for the N-Intertwined Mean Field Approximation, enhancing understanding of dynamics on complex hypergraph structures.

Contribution

It introduces new concentration bounds for Markov processes on hypergraphs and refines error estimates for mean field approximations on large networks.

Findings

Concentration bounds for vertex states under mild assumptions.

Complete characterization of concentration for averages in undirected weighted graphs.

Error bounds for NIMFA improve previous results, especially for symmetric unweighted graphs.

Abstract

We study Markov processes on weighted directed hypergraphs where the state of at most one vertex can change at a time. Our setting is general enough to include simplicial epidemic processes, processes on multilayered networks or even the dynamics of the edges of a graph. Our results are twofold. Firstly, we prove concentration bounds for the number of vertices in a certain state under mild assumptions. Our results imply that even the empirical averages of subpopulations of diverging but possibly sublinear size are well concentrated around their mean. In the case of undirected weighted graphs, we completely characterize when said averages concentrate around their expected value. Secondly, we prove (under assumptions which are tight in some significant cases) upper bounds for the error of the N-Intertwined Mean Field Approximation (NIMFA). In particular, for symmetric unweighted graphs, the error has the same order of magnitude as the reciprocal of the average degree, improving the previously known state of the art bound of the inverse square root of the average degree.