Sparse Network Mean Field Games: Ring Structures and Related Topologies

For sequences of networks embedded in the unit cube $[0,1]^{m}$, (weak) measure limits of sequences of empirical measures of vertex densities (vertexon functions) exist, and the associated (weak) measure limits of sequences of empirical measures of edge densities (graphexon functions) in $[0,1]^{2 m}$ exist, regardless of the sparsity or density of the limit graphs. This paper presents an extension of Graphon Mean Field Game (GMFG) theory to the vertexon-graphexon MFG setup (denoted GXMFG). Specific second order dynamics are introduced for the inter-node influence mediated by the singular part of a network graphexon measure; this is analyzed in the particular cases of a network limit ring topology and a limit rectangular lattice topology. Existence and uniqueness results are presented for the corresponding GXMFG equations.