Mean Field Games and Control on Large Expander Graphs

TL;DR

This paper studies mean field games on large expander graphs, analyzing their limits using graphexon framework, and establishes stability and instability conditions for the resulting systems.

Contribution

It introduces a novel analysis of mean field games on sparse networks using graphexon limits and characterizes stability thresholds for these systems.

Findings

Empirical graphexon measures converge weakly to a continuous limit.

Discrete averaging operators converge strongly to continuous operators.

Parameter thresholds for stability and Turing-type instabilities are identified.

Abstract

This paper investigates mean field games and control on sparse networks. In the case of large expander graphs, the limit topologies are analyzed using the graphexon framework, which characterizes both dense network limits and sparse connections. We prove that the sequence of empirical graphexon measures defined on finite graphs converges weakly to a limit graphexon measure on a continuous state space. Furthermore, the associated sequence of discrete averaging operators converges strongly to a continuous operator. These properties enable the formulation of a linear-quadratic mean field game in which each agent is identified by a spatial network label $\alpha \in X$ and only interacts with the neighborhood average defined by the operator $\mathcal{G}$ characterized by large expander graphs. In Section 5, algebraic conditions for the global asymptotic stability of the closed-loop system are established. The analysis identifies parameter thresholds that gives rise to a Turing-type topological instability, where the homogeneous mean state remains stable while the spatial deviation field diverges over the continuous spectrum of the limit operator.