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

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 $\Theta(\log N)$ or as $\Theta(N)$ 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 $\mathcal{O}((\log N)^{-1/2})$.