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arXiv · 2207.03806

Vlasov Equations on Directed Hypergraph Measures

Christian Kuehn·Chuang Xu
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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.

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BibTeXRIS

Christian Kuehn, Chuang Xu. 2022-07-08. Vlasov Equations on Directed Hypergraph Measures. https://arxiv.org/abs/2207.03806

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