Vlasov Equations on Directed Hypergraph Measures

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 measuretheoretic 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, arXiv:2107.08419, 2021] to higherdimensions. In particular, we successfully extend the arguments for the measure-valued functions f : X → M + (X) to those for f : X → M + (X k−1 ), where X is the vertex space of DHGMs and k ∈ N \ {1} is the cardinality of the DHGM.