Graphexon Mean Field Games on Large Network Limits

This paper analyzes graphexon mean field games (GXMFG) on large network limits. By considering edge distributions of a sequence of finite graphs, we define the empirical graphexon measures and study their subsequential weak limits in a graphexon space consisting of all symmetric probability measures on $[0,1{]}^{2d}$. This concept of network limits extends graphon theory to general, not necessarily dense, networks. To formulate the game using the graphexon limit, we exploit a useful property that a probability measure on a product space is the product of a probability kernel and a marginal distribution. Based on such graph limits and the probability kernel of the associated graphexon measure, we derive the GXMFG equations, and present a detailed analysis for the linear-quadratic case. For linear-quadratic models with finite populations residing over large networks, we use the GXMFG equations to construct decentralized strategies, for which an $\epsilon$-Nash equilibrium property is established.