Vlasov Equations on Directed Hypergraph Measures

TL;DR

This paper develops a measure-theoretic framework for analyzing the mean field limit of particle systems on directed hypergraphs, demonstrating well-posedness of the Vlasov equation and its approximation by large network distributions.

Contribution

It introduces directed hypergraph measures (DHGMs) as a new way to study hypergraph limits and extends existing methods to higher dimensions for analyzing Vlasov equations.

Findings

Vlasov equations on DHGMs are well-posed.

Solutions can be approximated by empirical distributions of large networks.

Applicable to physics, epidemic, and ecological networks with higher-order interactions.

Abstract

In this paper we propose a framework to investigate the mean field limit (MFL) of interacting particle systems on directed hypergraphs. We provide a non-trivial measure-theoretic viewpoint and make extensions of directed hypergraphs as directed hypergraph measures (DHGMs), which are measure-valued functions on a compact metric space. These DHGMs can be regarded as hypergraph limits which include limits of a sequence of hypergraphs that are sparse, dense, or of intermediate densities. Our main results show that the Vlasov equation on DHGMs are well-posed and its solution can be approximated by empirical distributions of large networks of higher-order interactions. The results are applied to a Kuramoto network in physics, an epidemic network, and an ecological network, all of which include higher-order interactions. To prove the main results on the approximation and well-posedness of the Vlasov equation on DHGMs, we robustly generalize the method of [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349] to higher-dimensions.