LQG Graphon Mean Field Games: Analysis via Graphon-Invariant Subspaces

This article studies approximate solutions to large-scale linear quadratic stochastic games with homogeneous nodal dynamics' parameters and heterogeneous network couplings within the graphon mean field game framework. A graphon time-varying dynamical system model is first formulated to study the finite and then limit problems of linear quadratic Gaussian graphon mean field games (LQG-GMFGs). The Nash equilibrium of the limit problem is then characterized by two coupled graphon time-varying dynamical systems. Sufficient conditions are established for the existence of a unique solution to the limit LQG-GMFG problem. For the computation of LQG-GMFG solutions, two methods are established and employed where one is based on fixed point analysis and the other on a decoupling operator Riccati equation; furthermore, two corresponding sets of solutions are established based on spectral decompositions. Finally, a set of numerical simulations on networks associated with different types of graphons are presented.