IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v37y2024i2d10.1007_s10959-023-01271-8.html
   My bibliography  Save this article

Gradient Flows on Graphons: Existence, Convergence, Continuity Equations

Author

Listed:
  • Sewoong Oh

    (University of Washington)

  • Soumik Pal

    (University of Washington)

  • Raghav Somani

    (University of Washington)

  • Raghavendra Tripathi

    (University of Washington)

Abstract

Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential. However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable. Such large graphs are known to converge to continuum limits called graphons as their size grows to infinity. We show that the Euclidean gradient flow of a suitable function of the edge weights converges to a novel continuum limit given by a curve on the space of graphons that can be appropriately described as a gradient flow or, more technically, a curve of maximal slope. Several natural functions on graphons, such as homomorphism functions and the scalar entropy, are covered by our setup, and the examples have been worked out in detail.

Suggested Citation

  • Sewoong Oh & Soumik Pal & Raghav Somani & Raghavendra Tripathi, 2024. "Gradient Flows on Graphons: Existence, Convergence, Continuity Equations," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1469-1522, June.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01271-8
    DOI: 10.1007/s10959-023-01271-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-023-01271-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-023-01271-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kallenberg, Olav, 1989. "On the representation theorem for exchangeable arrays," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 137-154, July.
    2. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    3. Sirignano, Justin & Spiliopoulos, Konstantinos, 2020. "Mean field analysis of neural networks: A central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1820-1852.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Olav Kallenberg, 1999. "Multivariate Sampling and the Estimation Problem for Exchangeable Arrays," Journal of Theoretical Probability, Springer, vol. 12(3), pages 859-883, July.
    2. Chen, Kaicheng & Vogelsang, Timothy J., 2024. "Fixed-b asymptotics for panel models with two-way clustering," Journal of Econometrics, Elsevier, vol. 244(1).
    3. Daniel Gaigall & Stefan Weber, 2025. "Jointly Exchangeable Collective Risk Models: Interaction, Structure, and Limit Theorems," Papers 2504.06287, arXiv.org.
    4. Paolo Leonetti, 2018. "Finite Partially Exchangeable Laws Are Signed Mixtures of Product Laws," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 195-214, August.
    5. Kaicheng Chen, 2025. "Inference in High-Dimensional Panel Models: Two-Way Dependence and Unobserved Heterogeneity," Papers 2504.18772, arXiv.org.
    6. Eric Auerbach, 2019. "Identification and Estimation of a Partially Linear Regression Model using Network Data," Papers 1903.09679, arXiv.org, revised Jun 2021.
    7. Davezies, Laurent & D’Haultfœuille, Xavier & Guyonvarch, Yannick, 2022. "The Marcinkiewicz–Zygmund law of large numbers for exchangeable arrays," Statistics & Probability Letters, Elsevier, vol. 188(C).
    8. Ricardo Vélez & Tomás Prieto-Rumeau, 2015. "Random assignment processes: strong law of large numbers and De Finetti theorem," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(1), pages 136-165, March.
    9. Jochmans, Koen, 2024. "Nonparametric identification and estimation of stochastic block models from many small networks," Journal of Econometrics, Elsevier, vol. 242(2).
    10. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    11. Bikramjit Das & Tiandong Wang & Gengling Dai, 2022. "Asymptotic Behavior of Common Connections in Sparse Random Networks," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2071-2092, September.
    12. Dmitry Arkhangelsky & Guido Imbens, 2023. "Causal Models for Longitudinal and Panel Data: A Survey," Papers 2311.15458, arXiv.org, revised Jun 2024.
    13. Bryan S. Graham, 2020. "Sparse network asymptotics for logistic regression," Papers 2010.04703, arXiv.org.
    14. Bryan S. Graham, 2019. "Network Data," Papers 1912.06346, arXiv.org.
    15. François Caron & Emily B. Fox, 2017. "Sparse graphs using exchangeable random measures," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1295-1366, November.
    16. Graham, Bryan S., 2020. "Network data," Handbook of Econometrics,, Elsevier.
    17. Ruixiang Zhang & Ziyu Zhu & Meng Yuan & Yihan Guo & Jie Song & Xuanxuan Shi & Yu Wang & Yaojie Sun, 2023. "Regional Residential Short-Term Load-Interval Forecasting Based on SSA-LSTM and Load Consumption Consistency Analysis," Energies, MDPI, vol. 16(24), pages 1-17, December.
    18. Kameswarrao S. Casukhela, 1997. "Symmetric Distributions of Random Measures in Higher Dimensions," Journal of Theoretical Probability, Springer, vol. 10(3), pages 759-771, July.
    19. Graham, Bryan S. & Niu, Fengshi & Powell, James L., 2024. "Kernel density estimation for undirected dyadic data," Journal of Econometrics, Elsevier, vol. 240(2).
    20. Bryan S. Graham, 2019. "Dyadic Regression," Papers 1908.09029, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01271-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.