Strategic geometric graphs through mean field games
Charles Bertucci, Matthias Rakotomalala
TL;DR
The paper develops mean-field games on Riemannian manifolds by letting interactions be driven by dynamic geometric graphs, preserving intrinsic geometry such as Ollivier curvature in the limit. It formulates a forward-backward MFG system on $\mathbb{R}^d \times M$ with diffusion given by the Laplace-Beltrami operator and derives explicit forms in special geometries (e.g., hyperbolic space). A curvature-aware, stationary example illustrates how curvature enters the mean-field costs, and a rigorous existence-uniqueness theory is established for second-order MFGs with quadratic Hamiltonians on manifolds of bounded geometry, employing Amann’s parabolic theory, a parabolic maximum principle, and a Schauder fixed-point framework via Cole-Hopf. The results bridge geometric graph limits and MFG analysis, providing tools to model strategic interaction structures that evolve with the agents’ positions on curved spaces and to handle non-compact settings with robust geometric control. This framework opens pathways for analyzing geometry-driven network formation and curvature-based incentives in large-scale agent systems.
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.