Mean-field limit of non-exchangeable multi-agent systems over hypergraphs with unbounded rank

TL;DR

This paper establishes a mean-field limit for multi-agent systems with higher-order, non-binary interactions modeled by dense hypergraphs of unbounded rank, leading to a Vlasov-type equation with complex interaction orders.

Contribution

It introduces the first rigorous derivation of a mean-field limit for hypergraph-based multi-agent systems with unbounded rank, extending classical models to include infinite interaction orders.

Findings

Mean-field limit described by a Vlasov-type equation.

Hypergraph limit encoded by UR-hypergraphon.

Force admits infinitely-many interaction orders.

Abstract

Interacting particle systems are known for their ability to generate large-scale self-organized structures from simple local interaction rules between each agent and its neighbors. In addition to studying their emergent behavior, a main focus of the mathematical community has been concentrated on deriving their large-population limit. In particular, the mean-field limit consists of describing the limit system by its population density in the product space of positions and labels. The strategy to derive such limits is often based on a careful combination of methods ranging from analysis of PDEs and stochastic analysis, to kinetic equations and graph theory. In this article, we focus on a generalization of multi-agent systems that includes higher-order interactions, which has largely captured the attention of the applied community in the last years. In such models, interactions between individuals are no longer assumed to be binary (i.e. between a pair of particles). Instead, individuals are allowed to interact by groups so that a full group jointly generates a non-linear force on any given individual. The underlying graph of connections is then replaced by a hypergraph, which we assume to be dense, but possibly non uniform and of unbounded rank. For the first time in the literature, we show that when the interaction kernels are regular enough, then the mean-field limit is determined by a limiting Vlasov-type equation, where the hypergraph limit is encoded by a so-called UR-hypergraphon (unbounded-rank hypergraphon), and where the resulting mean-field force admits infinitely-many orders of interactions.