Accuracy of the Graphon Mean Field Approximation for Interacting Particle Systems

TL;DR

This paper analyzes how well the graphon mean field approximation models large interacting particle systems on dense graphs, providing convergence rates and demonstrating practical applications in load balancing and bike-sharing systems.

Contribution

It establishes convergence rates for the graphon mean field approximation in dense graph regimes, including weighted and random graphs, with explicit error bounds.

Findings

Weighted interactions achieve an $O(1/N)$ error bound.

Random graph interactions have an $O(\sqrt{rac{\log(N)}{N}})$ error bound with high probability.

Numerical examples demonstrate the approximation's efficiency and applicability.

Abstract

We consider a system of $N$ particles whose interactions are characterized by a (weighted) graph $G^N$. Each particle is a node of the graph with an internal state. The state changes according to Markovian dynamics that depend on the states and connection to other particles. We study the limiting properties, focusing on the dense graph regime, where the number of neighbors of a given node grows with $N$. We show that when $G^N$ converges to a graphon $G$, the behavior of the system converges to a deterministic limit, the graphon mean field approximation. We obtain convergence rates depending on the system size $N$ and cut-norm distance between $G^N$ and $G$. We apply the results for two subcases: When $G^N$ is a discretization of the graph $G$ with individually weighted edges; when $G^N$ is a random graph obtained through edge sampling from the graphon $G$. In the case of weighted interactions, we obtain a bound of order $O(1/N)$. In the random graph case, the error is of order $O(\sqrt{\log(N)/N})$ with high probability. We illustrate the applicability of our results and the numerical efficiency of the approximation through two examples: a graph-based load-balancing model and a heterogeneous bike-sharing system.