Learning Mean Field Games on Sparse Graphs: A Hybrid Graphex Approach
Christian Fabian, Kai Cui, Heinz Koeppl
TL;DR
GXMFGs extend mean field games to sparse networks by leveraging graphexes, addressing scalability and realism in agent interactions. The authors propose a Hybrid graphex learning algorithm (HOMD) that decouples a highly connected core from a sparse periphery, and provide theoretical results showing convergence of the finite-system MF to the limiting GXMFG framework along with approximate optimality guarantees. They validate the approach on synthetic graphex networks and real-world graphs, reporting accurate mean-field predictions (typically within a few percent) and superior performance over LPGMFG baselines. The work advances scalable equilibrium learning for large, realistic networks and lays groundwork for extensions to continuous state/action spaces and time.
Abstract
Learning the behavior of large agent populations is an important task for numerous research areas. Although the field of multi-agent reinforcement learning (MARL) has made significant progress towards solving these systems, solutions for many agents often remain computationally infeasible and lack theoretical guarantees. Mean Field Games (MFGs) address both of these issues and can be extended to Graphon MFGs (GMFGs) to include network structures between agents. Despite their merits, the real world applicability of GMFGs is limited by the fact that graphons only capture dense graphs. Since most empirically observed networks show some degree of sparsity, such as power law graphs, the GMFG framework is insufficient for capturing these network topologies. Thus, we introduce the novel concept of Graphex MFGs (GXMFGs) which builds on the graph theoretical concept of graphexes. Graphexes are the limiting objects to sparse graph sequences that also have other desirable features such as the small world property. Learning equilibria in these games is challenging due to the rich and sparse structure of the underlying graphs. To tackle these challenges, we design a new learning algorithm tailored to the GXMFG setup. This hybrid graphex learning approach leverages that the system mainly consists of a highly connected core and a sparse periphery. After defining the system and providing a theoretical analysis, we state our learning approach and demonstrate its learning capabilities on both synthetic graphs and real-world networks. This comparison shows that our GXMFG learning algorithm successfully extends MFGs to a highly relevant class of hard, realistic learning problems that are not accurately addressed by current MARL and MFG methods.