Mean Field Games on Weighted and Directed Graphs via Colored Digraphons

Multi-agent systems are in general hard to model and control due to their complex nature involving many individuals. Numerous approaches focus on empirical and algorithmic aspects of approximating outcomes and behavior in multi-agent systems and lack a rigorous theoretical foundation. Graphon mean field games (GMFGs) on the other hand provide a mathematically well-founded and numerically scalable framework for a large number of connected agents. In standard GMFGs, the connections between agents are undirected, unweighted and invariant over time. Our paper introduces colored digraphon mean field games (CDMFGs) which allow for weighted and directed links between agents that are also adaptive over time. Thus, CDMFGs are able to model more complex connections than standard GMFGs. Besides a rigorous theoretical analysis including both existence and convergence guarantees, we employ the online mirror descent algorithm to learn equilibria. To conclude, we illustrate our findings with an epidemics model and a model of the systemic risk in financial markets.