Mean Field Games and Control on Large Expander Graphs

Abstract

This paper investigates mean field games on sparse networks. In the case of large expander graphs, the limit topologies are analyzed using the graphexon framework, which characterizes sparse connections. We prove that the associated sequence of discrete averaging operators converges strongly to a continuous operator and this is illustrated in the development of infinite limits of Gabber-Galil-Margulis expander graphs \cite{gabber1981explicit}. These properties enable the formulation and existence proof of equilibria for linear-quadratic mean field games in which each agent is identified by a spatial network label $\alpha \in X$ and only interacts with the neighborhood average characterized by the operator $\mathcal{G}$, i.e., the average state of a limited number of connected neighbors in a large expander graph. Furthermore, algebraic conditions induced by the spectral gap of $\mathcal{G}$ for the global asymptotic stability of the closed-loop system are established, and parameter thresholds that give rise to a Turing-type topological instability are established.

arXiv commentThe short version of this manuscript has been accepted by the 65th IEEE Conference on Decision and Control (CDC) 2026