Mean-Field Limit of Non-exchangeable Multi-agent Systems Over Hypergraphs with Unbounded Rank

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 nonlinear force on any given individual. The underlying graph of connections is then replaced by a hypergraph, which we assume to be dense, but possibly nonuniform 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.

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Publication Details

Journal
Journal of Nonlinear Science
Published
2026-10-23
DOI
https://doi.org/10.1007/s00332-026-10338-9
Citations
241
Primary Topic
Oceanographic and Atmospheric Processes
Type
article
Field-Weighted Citation Impact
42.41

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article

Mean-Field Limit of Non-exchangeable Multi-agent Systems Over Hypergraphs with Unbounded Rank

David Poyato, Nastassia Pouradier Duteil, Nathalie Ayi
241 citations
Journal of Nonlinear Science
Oceanographic and Atmospheric Processes
42.41
article

Mean-Field Limit of Non-exchangeable Multi-agent Systems Over Hypergraphs with Unbounded Rank

David Poyato, Nastassia Pouradier Duteil, Nathalie Ayi
article en
241 citations

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 nonlinear force on any given individual. The underlying graph of connections is then replaced by a hypergraph, which we assume to be dense, but possibly nonuniform 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.

Journal of Nonlinear ScienceVol. 36(6)
Centre National de la Recherche Scientifique (FR), Universidad de Granada (ES), Université Paris Cité (FR), Sorbonne Université (FR), IMT Mines Alès (FR), Laboratoire Jacques-Louis Lions (FR)
Consejería de Universidad, Investigación e Innovación, Junta de Andalucía, National Science Foundation, National Geographic Society, European Federation of Pharmaceutical Industries and Associations, Ministerio de Ciencia, Innovación y Universidades, Centres de Recerca de Catalunya, Cancer Research UK, Leverhulme Trust, European Commission, Boehringer Ingelheim Fonds, Agence Nationale de la Recherche, European Synchrotron Radiation Facility, Österreichischen Akademie der Wissenschaften, Austrian Science Fund, Generalitat de Catalunya, Universidad Complutense de Madrid, Ege Üniversitesi, Universität Wien, Eusko Jaurlaritza, Ministerstvo školstva, vedy, výskumu a športu Slovenskej republiky, Ministerio de Economía y Competitividad, Generalitat Valenciana, KU Leuven, Max-Planck-Gesellschaft, Centre National de la Recherche Scientifique, Ministerio de Ciencia e Innovación, Associazione Italiana per la Ricerca sul Cancro, Banco Bilbao Vizcaya Argentaria, Agentúra na Podporu Výskumu a Vývoja, Norges Forskningsråd, Universitat de Barcelona, Vedecká Grantová Agentúra MŠVVaŠ SR a SAV, Ministério da Ciência, Tecnologia e Ensino Superior, Université de Bordeaux, Universidad de Cantabria, Universidad de Granada, Universidad de Zaragoza, Fondazione Italiana per la Ricerca sul Cancro, Gobierno de Aragón, Grand Équipement National De Calcul Intensif, Ministerio de Asuntos Económicos y Transformación Digital, Gobierno de España, Naturalis Biodiversity Center, Foundation for Research in Rheumatology, Universidad de Salamanca, Xarxa de Bancs de Tumors de Catalunya, Universiti Sultan Zainal Abidin, Instituto de Telecomunicações, Sociedad para el Desarrollo de Cantabria, Brazilian Biodivesity Fund, Sorbonne Université, Fundación Bancaria Caixa d'Estalvis i Pensions de Barcelona, Nature, Horizon 2020 Framework Programme, HORIZON EUROPE Framework Programme, Japan Society for the Promotion of Science, Fundação para a Ciência e a Tecnologia, Instituto de Salud Carlos III, European Regional Development Fund, Agencia Estatal de Investigación, European Social Fund, H2020 European Research Council, HORIZON EUROPE Marie Sklodowska-Curie Actions, NextGenerationEU
Openalex Percentile: Top 0%
Oceanographic and Atmospheric Processes
42.41
DOIOpen Access PDFOpenAlex
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