Weakly interacting particle systems on inhomogeneous random graphs

TL;DR

This paper studies large systems of weakly interacting diffusions on dynamic inhomogeneous random graphs, establishing laws of large numbers, propagation of chaos, and a central limit theorem for such systems.

Contribution

It introduces a framework for analyzing weakly interacting diffusions on time-varying inhomogeneous random graphs, including new LLN, propagation of chaos, and CLT results.

Findings

Law of large numbers for multi-type populations

Propagation of chaos in dynamic inhomogeneous graphs

Central limit theorem under specific conditions

Abstract

We consider weakly interacting diffusions on time varying random graphs. The system consists of a large number of nodes in which the state of each node is governed by a diffusion process that is influenced by the neighboring nodes. The collection of neighbors of a given node changes dynamically over time and is determined through a time evolving random graph process. A law of large numbers and a propagation of chaos result is established for a multi-type population setting where at each instant the interaction between nodes is given by an inhomogeneous random graph which may change over time. This result covers the setting in which the edge probabilities between any two nodes is allowed to decay to $0$ as the size of the system grows. A central limit theorem is established for the single-type population case under stronger conditions on the edge probability function.