Partial mean field limits in heterogeneous networks

TL;DR

This paper develops a partial mean field framework for heterogeneous stochastic networks, providing laws of large numbers and large deviation results that extend classical homogeneous system analyses.

Contribution

It introduces a novel partial mean field approach that accounts for heterogeneity in network weights and asymptotic relevance, with explicit error bounds and broad applicability.

Findings

Law of large numbers with explicit error bounds

Large deviation principles established

Applicability to heterogeneous networks including preferential attachment models

Abstract

We investigate systems of interacting stochastic differential equations with two kinds of heterogeneity: one originating from different weights of the linkages, and one concerning their asymptotic relevance when the system becomes large. To capture these effects we define a partial mean field system, and prove a law of large numbers with explicit bounds on the mean squared error. Furthermore, a large deviation result is established under reasonable assumptions. The theory will be illustrated by several examples: on the one hand, we recover the classical results of chaos propagation for homogeneous systems, and on the other hand, we demonstrate the validity of our assumptions for quite general heterogeneous networks including those arising from preferential attachment random graph models.