Embedded Vertexon-Graphons and Embedded GMFG Systems

An (embedded) vertexon in a connected compact set M in R
^m
is defined to be the vertex set of a graph together with an asymptotically dense partition hierarchy of M. It is shown that sequences of vertexons have subsequential vertexon limits measures in M independent of the particular partition hierarchy employed within a large class of hierarchies. Consequently, the differentiation of functions on vertexon limits with open support is well defined. Further, along these sequences the associated sequence of graphs have subsequential graph edge limits termed embedded graphons; the resulting embedded vertexon-graphon limit (measure) pair permits the definition of critical nodes and other features involving differentiation with respect to node location. This is significant for the analysis of Nash value functions for Embedded Graphon Mean Field Game (EGMFG) systems, that is to say, GMFG systems with nodes lying in a limiting vertexon in M ⊂ R
^m
with open support and associated embedded graphons defined on M
²
. A further feature of vertexon limits is that they provide a form of node population density which extends the local agent population aspect of embedded GMFG systems.