Linear-Quadratic Mean Field Games with Hybrid Local-Global Interactions on Manifolds

Abstract

This paper studies linear-quadratic mean field games on compact Riemannian manifolds with a hybrid interaction topology. The network structure is a superposition of a deterministic graph for local geometric connectivity and a stochastic directed graph for non-local interactions. The global graph is constructed via random sampling based on a continuous kernel $K$. The out-degree of each node scales as $Θ(\log N)$ or as $Θ(N)$ to represent a sparse or dense network, respectively. In the infinite-population limit, the continuum system is governed by a coupled system of forward-backward partial differential equations, where the dynamics of the expected state incorporate the integral operator corresponding to the non-local sampling. The existence of a Nash equilibrium is established for this limit system. Furthermore, the approximation error is analyzed using operator concentration inequalities and analytic semigroup theory. Non-asymptotic high-probability error bounds between the finite-population empirical state and the continuum limit are derived. The convergence rates differ depending on the two topological regimes. Under the dense regime, the tracking error exhibits a polynomial decay rate dependent on the manifold dimension and Sobolev regularity, while under the sparse regime, the error decays at a rate of $\mathcal{O}((\log N)^{-1/2})$.

Disclosure

“ogies. Acknowledgments The author would like to acknowledge the assistance of Prof. Peter E. Caines in providing valuable suggestions and insightful comments. Declarations During the preparation of this manuscript, the authors utilized ChatGPT to assist with language polishing and textual error correction. All mathematical results, computational experiments, and conclusions were verified by the authors. The authors assume responsibility for all content. A Operator Error De”

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2026-08-24T00:34:12.480Z

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