Mean field interaction on random graphs with dynamically changing multi-color edges

TL;DR

This paper studies large systems of interacting jump processes on dynamic, multi-colored random graphs, establishing asymptotic behaviors like law of large numbers and propagation of chaos, with novel path-dependent limits.

Contribution

It introduces a new framework for analyzing mean field interactions on time-varying graphs with multi-color edges, revealing path-dependent limiting dynamics.

Findings

Law of large numbers for the system

Propagation of chaos established

Asymptotic behavior under accelerated edge dynamics analyzed

Abstract

We consider weakly interacting jump processes on time-varying random graphs with dynamically changing multi-color edges. The system consists of a large number of nodes in which the node dynamics depends on the joint empirical distribution of all the other nodes and the edges connected to it, while the edge dynamics depends only on the corresponding nodes it connects. Asymptotic results, including law of large numbers, propagation of chaos, and central limit theorems, are established. In contrast to the classic McKean-Vlasov limit, the limiting system exhibits a path-dependent feature in that the evolution of a given particle depends on its own conditional distribution given its past trajectory. We also analyze the asymptotic behavior of the system when the edge dynamics is accelerated. A law of large number and a propagation of chaos result is established, and the limiting system is given as independent McKean-Vlasov processes. Error between the two limiting systems, with and without acceleration in edge dynamics, is also analyzed.