Accuracy of the Graphon Mean Field Approximation for Interacting Particle Systems

We consider a system of N particles whose interactions are characterized by a (weighted) graph [Formula: see text]. Each particle is a node of the graph with an internal state. This state changes according to Markovian dynamics that depend on the states of neighboring particles. We study the limiting properties of the state dynamics, focusing on the dense graph regime, in which the average degree of a node grows linearly with N. We show that, when [Formula: see text] converges to a piecewise Lipschitz 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 [Formula: see text] and G. We apply these results for two subcases: when [Formula: see text] is a discretization of the graph G with individually weighted edges; when [Formula: see text] is a random graph obtained by sampling edges with probabilities obtained from G. In the case of weighted interactions, we obtain a bound of order [Formula: see text]. In the random graph case, the error is of order [Formula: see text] 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. Funding: This work was supported by the ANR (Agence National de la Recherche), via the project REFINO (ANR-19-CE23-0015). Supplemental Material: The online appendix is available at https://doi.org/10.1287/stsy.2024.0070 .