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Vlasov equations on directed hypergraph measures

Author

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  • Christian Kuehn

    (Technical University of Munich
    Technical University of Munich)

  • Chuang Xu

    (University of Hawai’i at Mānoa)

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.

Suggested Citation

  • Christian Kuehn & Chuang Xu, 2025. "Vlasov equations on directed hypergraph measures," Partial Differential Equations and Applications, Springer, vol. 6(1), pages 1-49, March.
    • Handle: RePEc:spr:pardea:v:6:y:2025:i:1:d:10.1007_s42985-025-00313-6
    DOI: 10.1007/s42985-025-00313-6
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    References listed on IDEAS

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    1. Keliger, Dániel & Horváth, Illés, 2023. "Accuracy criterion for mean field approximations of Markov processes on hypergraphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    2. Rainer Hegselmann & Ulrich Krause, 2002. "Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 5(3), pages 1-2.
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