Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-27T20:16:11.473Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly interacting oscillators on dense random graphs

Part of: Special processes Graph theory Equilibrium statistical mechanics Equations of mathematical physics and other areas of application Stochastic analysis

Published online by Cambridge University Press:  30 June 2023

Gianmarco Bet Open the ORCID record for Gianmarco Bet [Opens in a new window] ,
Fabio Coppini  and
Francesca Romana Nardi
Gianmarco Bet*
Affiliation:
Università degli Studi di Firenze
Fabio Coppini*
Affiliation:
Università degli Studi di Firenze
Francesca Romana Nardi*
Affiliation:
Università degli Studi di Firenze and Eindhoven University of Technology
*
*Postal address: Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università degli Studi di Firenze, Firenze, Italy.
*Postal address: Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università degli Studi di Firenze, Firenze, Italy.
*Postal address: Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università degli Studi di Firenze, Firenze, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit.

Keywords

Interacting oscillatorsrandom graphonsmean-field systemsFokker–Planck equationexchangeable graphsMcKean–Vlasov

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 05C80: Random graphs
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60H20: Stochastic integral equations 35Q84: Fokker-Planck equations
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 255 - 278
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alon, N. and Naor, A. (2006). Approximating the Cut-Norm via Grothendieck’s inequality. SIAM J. Comput. 35, 787–803.CrossRefGoogle Scholar
Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y. and Zhou, C. (2008). Synchronization in complex networks. Phys. Rep. 469, 93–153.CrossRefGoogle Scholar
Basak, A., Bhamidi, S., Chakraborty, S. and Nobel, A. (2016). Large subgraphs in pseudo-random graphs. Available at arXiv:1610.03762.Google Scholar
Bayraktar, E., Chakraborty, S. and Wu, R. (2020). Graphon mean field systems. Available at arXiv:2003.13180.Google Scholar
Bertini, L., Giacomin, G. and Pakdaman, K. (2010). Dynamical aspects of mean field plane rotators and the Kuramoto model. J. Statist. Phys. 138, 270–290.CrossRefGoogle Scholar
Bet, G., van der Hofstad, R. and van Leeuwaarden, J. S. (2020). Big jobs arrive early: from critical queues to random graphs. Stoch. Syst. 10, 310–334.CrossRefGoogle Scholar
Bhamidi, S., Budhiraja, A. and Wu, R. (2019). Weakly interacting particle systems on inhomogeneous random graphs. Stoch. Process. Appl. 129, 2174–2206.CrossRefGoogle Scholar
Bogerd, K., Castro, R. M. and van der Hofstad, R. (2020). Cliques in rank-1 random graphs: the role of inhomogeneity. Bernoulli 26, 253–285.CrossRefGoogle Scholar
Caines, P. E. and Huang, M. (2018). Graphon mean field games and the GMFG equations. In 2018 IEEE Conference on Decision and Control (CDC), pp. 4129–4134.CrossRefGoogle Scholar
Carmona, R., Cooney, D. B., Graves, C. V. and Lauriere, M. (2022). Stochastic graphon games I: The static case. Math. Operat. Res. 47, 750–778.CrossRefGoogle Scholar
Chiba, H. and Medvedev, G. S. (2019). The mean field analysis of the Kuramoto model on graphs I: The mean field equation and transition point formulas. Discrete Contin. Dyn. Syst. A 39, 131.CrossRefGoogle Scholar
Chung, F. R. K., Graham, R. L. and Wilson, R. M. (1989). Quasi-random graphs. Combinatorica 9, 345–362.CrossRefGoogle Scholar
Coppini, F. (2020). Weakly interacting diffusions on graphs. Doctoral thesis, Université de Paris.Google Scholar
Coppini, F. (2022). Long time dynamics for interacting oscillators on graphs. Ann. Appl. Prob. 32, 360–391.CrossRefGoogle Scholar
Coppini, F., Dietert, H. and Giacomin, G. (2020). A law of large numbers and large deviations for interacting diffusions on Erdős–Rényi graphs. Stoch. Dyn. 20, 2050010.CrossRefGoogle Scholar
Delattre, S., Giacomin, G. and Luçon, E. (2016). A note on dynamical models on random graphs and Fokker–Planck equations. J. Statist. Phys. 165, 785–798.CrossRefGoogle Scholar
Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. 28, 33–61.Google Scholar
Katznelson, Y. (2004). An Introduction to Harmonic Analysis, 3rd edn (Cambridge Mathematical Library). Cambridge University Press.CrossRefGoogle Scholar
Lovász, L. (2012). Large Networks and Graph Limits (Colloquium Publications 60). American Mathematical Society.Google Scholar
Luçon, E. (2020). Quenched asymptotics for interacting diffusions on inhomogeneous random graphs. Stoch. Process. Appl. 130, 6783–6842.CrossRefGoogle Scholar
Luçon, E. and Stannat, W. (2014). Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Prob. 24, 1946–1993.CrossRefGoogle Scholar
Medvedev, G. S. (2019). The continuum limit of the Kuramoto model on sparse random graphs. Commun. Math. Sci. 17, 883–898.CrossRefGoogle Scholar
Oelschläger, K. (1984). A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Prob. 12, 458–479.CrossRefGoogle Scholar
Oliveira, R. I. and Reis, G. H. (2019). Interacting diffusions on random graphs with diverging average degrees: hydrodynamics and large deviations. J. Statist. Phys. 176, 1057–1087.CrossRefGoogle Scholar
Rodrigues, F. A., Peron, T. K. D., Ji, P. and Kurths, J. (2016). The Kuramoto model in complex networks. Phys. Rep. 610, 1–98.CrossRefGoogle Scholar
Sznitman, A.-S. (1991). Topics in propagation of chaos. In Ecole d’Eté de Probabilités de Saint-Flour XIX - 1989 (Lecture Notes Math. 1464), ed. P.-L. Hennequin, pp. 165–251. Springer, Berlin and Heidelberg.Google Scholar
Van Enter, A. C. D. and Ruszel, W. M. (2009). Gibbsianness versus non-Gibbsianness of time-evolved planar rotor models. Stoch. Process. Appl. 119, 1866–1888.CrossRefGoogle Scholar