Probabilistic Analysis of Graphon Mean Field Control
Zhongyuan Cao, Mathieu Laurière
TL;DR
<3-5 sentence high-level summary> This paper develops a probabilistic framework for graphon mean field control to handle large populations with heterogeneous, non-exchangeable interactions. It establishes well-posed GMFC forward–backward SDEs, derives a graphon-specific Pontryagin maximum principle (necessary and sufficient conditions), and analyzes solvability, continuity, and stability of the optimal FBSDE system, including a continuation-method approach in the linear-quadratic setting. It further connects the GMFC limit to finite-N systems via propagation of chaos, proving approximate optimality of GMFC-derived controls for large networks and quantifying convergence rates. The results provide a rigorous bridge between continuum GMFC models and large but finite networks with nonuniform interaction structures.
Abstract
Motivated by recent interest in graphon mean field games and their applications, this paper provides a comprehensive probabilistic analysis of graphon mean field control (GMFC) problems, where the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous mean field interactions. We formulate the GMFC problem with general graphon mean field dependence and establish the existence and uniqueness of the associated graphon mean field forward-backward stochastic differential equations (FBSDEs). We then derive a version of the Pontryagin stochastic maximum principle tailored to GMFC problems. Furthermore, we analyze the solvability of the GMFC problem for linear dynamics and study the continuity and stability of the graphon mean field FBSDEs under the optimal control profile. Finally, we show that the solution to the GMFC problem provides an approximately optimal solution for large systems with heterogeneous mean field interactions, based on a propagation of chaos result.