Hypergraphon mean field games

Cui, Kai; KhudaBukhsh, Wasiur R.; Koeppl, Heinz

Abstract

We propose an approach to modeling large-scale multi-agent dynamical systems allowing interactions among more than just pairs of agents using the theory of mean field games and the notion of hypergraphons, which are obtained as limits of large hypergraphs. To the best of our knowledge, ours is the first work on mean field games on hypergraphs. Together with an extension to a multi-layer setup, we obtain limiting descriptions for large systems of non-linear, weakly interacting dynamical agents. On the theoretical side, we prove the well-foundedness of the resulting hypergraphon mean field game, showing both existence and approximate Nash properties. On the applied side, we extend numerical and learning algorithms to compute the hypergraphon mean field equilibria. To verify our approach empirically, we consider a social rumor spreading model, where we give agents intrinsic motivation to spread rumors to unaware agents, and an epidemic control problem.

Recent developments in the field of complex systems have shown that real-world multi-agent systems are often not restricted to pairwise interactions, bringing to light the need for tractable models allowing higher-order interactions. At the same time, the complexity of analysis of large-scale multi-agent systems on graphs remains an issue even without considering higher-order interactions. An increasingly popular and tractable approach of analysis is the theory of mean field games. We combine mean field games with higher-order structure by means of hypergraphons, a limiting description of very large hypergraphs. To motivate our model, we build a theoretical foundation for the limiting system, showing that the limiting system has a solution and that it approximates finite, sufficiently large systems well. This allows us to analyze otherwise intractable, large hypergraph games with theoretical guarantees, which we verify using two examples of rumor spreading and epidemics control.

• Journal Article Type: Article
• Acceptance Date: Oct 24, 2022
• Online Publication Date: Nov 10, 2022
• Publication Date: 2022-11
• Deposit Date: Nov 11, 2022
• Publicly Available Date: Nov 11, 2022
• Journal: Chaos: An Interdisciplinary Journal of Nonlinear Science
• Print ISSN: 1054-1500
• Electronic ISSN: 1089-7682
• Publisher: AIP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 32
• Issue: 11
• Article Number: 113129
• DOI: https://doi.org/10.1063/5.0093758
• Keywords: Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics
• Public URL: https://nottingham-repository.worktribe.com/output/13463397
• Publisher URL: https://aip.scitation.org/doi/10.1063/5.0093758