Mean Field Games on Dense and Sparse Networks: The Graphexon MFG Equations

For sequences of networks embedded in the unit cube
$[0,1]^{m}$
in
$\mathbb{R}^{m}$
(more generally compact sets in Riemannian manifolds), a notion related to that of graphons was introduced in [Caines, CDC 2022] in terms of (weak) measure limits of (sub-) sequences of empirical measures of vertex densities (vertexons) on
$[0,1]^{m}$
and the associated (weak) measure limits of (sub-) sequences of empirical measures of edge densities (graphexons) on
$\lceil 0,1\rceil^{2n}$
, both of which exist regardless of sparsity or density of the limit graphs. This paper presents an extension of the Graphon Mean Field Game (GMFG) theory of [Caines-Huang, SICON, 2021] to the vertexon-graphexon MFG set-up (here denoted GXMFG). In particular, for sparse limit graphexons, an LQG GXMFG example is presented where the influence between agent populations on neighbouring nodes is modeled via a first order PDE.