Strategic geometric graphs through mean field games

TL;DR

This paper develops a mean field game framework for analyzing strategic dynamic geometric graphs on Riemannian manifolds, capturing intrinsic geometric features like Ollivier curvature in the limit of infinitely many nodes.

Contribution

It introduces a novel mean field game model set on manifolds, incorporating geometric properties and establishing existence and uniqueness of solutions for quadratic Hamiltonians.

Findings

Derived a mean field game system on Riemannian manifolds.

Proved existence and uniqueness of solutions under certain conditions.

Preserved geometric information such as Ollivier curvature in the limit.

Abstract

We exploit the structure of geometric graphs on Riemannian manifolds to analyze strategic dynamic graphs at the limit, when the number of nodes tends to infinity. This framework allows to preserve intrinsic geometrical information about the limiting graph structure, such as the Ollivier curvature. After introducing the setting, we derive a mean field game system, which models a strategic equilibrium between the nodes. It has the usual structure with the distinction of being set on a manifold. Finally, we establish existence and uniqueness of solutions to the system when the Hamiltonian is quadratic for a class of non-necessarily compact Riemannian manifolds, referred to as manifolds of bounded geometry.