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Nonlinear Graphon mean-field systems

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  • Coppini, Fabio
  • De Crescenzo, Anna
  • Pham, Huyên

Abstract

We address a system of weakly interacting particles where the heterogeneous connections among the particles are described by a graph sequence and the number of particles grows to infinity. Our results extend the existing law of large numbers and propagation of chaos results to the case where the interaction between one particle and its neighbours is expressed as a nonlinear function of the local empirical measure. In the limit of the number of particles which tends to infinity, if the graph sequence converges to a graphon, then we show that the limit system is described by an infinite collection of processes and can be seen as a process in a suitable L2 space constructed via a Fubini extension. The proof is built on decoupling techniques and careful estimates of the Wasserstein distance.

Suggested Citation

  • Coppini, Fabio & De Crescenzo, Anna & Pham, Huyên, 2025. "Nonlinear Graphon mean-field systems," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:spapps:v:190:y:2025:i:c:s0304414925001693
    DOI: 10.1016/j.spa.2025.104728
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    References listed on IDEAS

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    1. Francesca Parise & Asuman Ozdaglar, 2023. "Graphon Games: A Statistical Framework for Network Games and Interventions," Econometrica, Econometric Society, vol. 91(1), pages 191-225, January.
    2. Bayraktar, Erhan & Wu, Ruoyu, 2022. "Stationarity and uniform in time convergence for the graphon particle system," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 532-568.
    3. Keliger, Dániel & Horváth, Illés & Takács, Bálint, 2022. "Local-density dependent Markov processes on graphons with epidemiological applications," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 324-352.
    4. Sun, Yeneng, 2006. "The exact law of large numbers via Fubini extension and characterization of insurable risks," Journal of Economic Theory, Elsevier, vol. 126(1), pages 31-69, January.
    5. Daniel Lacker & Agathe Soret, 2023. "A Label-State Formulation of Stochastic Graphon Games and Approximate Equilibria on Large Networks," Mathematics of Operations Research, INFORMS, vol. 48(4), pages 1987-2018, November.
    6. Bhamidi, Shankar & Budhiraja, Amarjit & Wu, Ruoyu, 2019. "Weakly interacting particle systems on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 2174-2206.
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