Journal of Statistical Physics

, Volume 165, Issue 4, pp 785–798 | Cite as

A Note on Dynamical Models on Random Graphs and Fokker–Planck Equations

  • Sylvain Delattre
  • Giambattista Giacomin
  • Eric Luçon


We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e., a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erdős–Rényi graphs with edge probability \(p_n\), n is the number of vertices, such that \(\lim _{n \rightarrow \infty }p_n n= \infty \). The purpose of this note is twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker–Planck PDE (or, equivalently, by a nonlinear diffusion process) in the \(n=\infty \) limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with n large but finite, for example the values of n that can be reached in simulations or that correspond to the typical number of interacting units in a biological system.


Interacting diffusions on graphs Mean field Nonlinear diffusion Fokker–Planck PDE Kuramoto models 

Mathematics Subject Classification

82C20 60K35 



G.G. is grateful to Bastien Fernandez, Roberto Livi and Justin Salez for very helpful discussions. We would like to thank the referees for their careful reading and useful remarks on the paper.


  1. 1.
    Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005)ADSCrossRefGoogle Scholar
  2. 2.
    Bertini, L., Giacomin, G., Pakdaman, K.: Dynamical aspects of mean field plane rotators and the Kuramoto model. J. Stat. Phys. 138, 270–290 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertini, L., Giacomin, G., Poquet, C.: Synchronization and random long time dynamics for mean-field plane rotators. Probab. Theory Relat. Fields 160, 593–653 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bollobás, B.: Random Graphs. Cambridge studies in advanced mathematics, vol. 73, 2nd edn (2001)Google Scholar
  5. 5.
    Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124, 1377–1397 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cattiaux, P., Guillin, A., Malrieu, F.: Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields 140, 19–40 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Collet, F., Dai Pra, P.: The role of disorder in the dynamics of critical fluctuations of mean field models. Electron. J. Probab. 17 (2012)Google Scholar
  8. 8.
    Dahms, D.: Time behavior of a spherical mean field model. PhD Thesis, Technische Universität Berlin (2002)Google Scholar
  9. 9.
    Dai Pra, P., den Hollander, F.: McKean-Vlasov limit for interacting random processes in random media. J. Stat. Phys. 84, 735–772 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Del Genio, C.I., Kim, H., Toroczkai, Z., Bassler, K.E.: Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PLoS ONE 5(4), 1–7 (2010)Google Scholar
  11. 11.
    De Masi, A., Presutti, E., Vares, M.E.: Escape from the unstable equilibrium in a random process with infinitely many interacting particles. J. Stat. Phys. 44, 645–696 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dembo, A., Montanari, A.: Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24, 137–211 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74 (2004)Google Scholar
  14. 14.
    Gärtner, J.: On McKean-Vlasov limit for interacting diffusions. Mathematische Nachrichten 137, 197–248 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Giacomin, G., Luçon, E., Poquet, C.: Coherence stability and effect of random natural frequencies in populations of coupled oscillators. J. Dyn. Differ. Equ. 26, 333–367 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Giacomin, G., Pakdaman, K., Pellegrin, X.: Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators. Nonlinearity 25, 1247–1273 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kuramoto, Y., Shinomoto, S., Sakaguchi, H.: Active rotator model for large populations of oscillatory and excitable elements. In: Mathematical Topics in Population Biology, Morphogenesis and Neurosciences (Kyoto, 1985), Lecture Notes in Biomathematics, vol. 71, pp. 329–337. Springer (1987)Google Scholar
  19. 19.
    Lancellotti, C.: On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transp. Theory Stat. Phys. 34, 523–535 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Luçon, E.: Quenched limits and fluctuations of the empirical measure for plane rotators in random media. Electron. J. Probab. 16, 792–829 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Luçon, E., Poquet, C.: Long Time Ddynamics and Disorder-Induced Traveling Waves in the Stochastic Kuramoto Model. arXiv:1505.00497, Ann. IHP(B) (to appear)
  22. 22.
    Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford University Press, Oxford (2009)CrossRefzbMATHGoogle Scholar
  23. 23.
    McKay, B.D., Wormald, N.C.: Uniform generation of random regular graphs of moderate degree. J Algorithms 11(1), 52–67 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Olivieri, E., Vares, M.E.: Large Deviations and Metastability, Encyclopedia of Mathematics and its Applications, 100. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  25. 25.
    Olmi, S., Navas, A., Boccaletti, S., Torcini, A.: Hysteretic transitions in the Kuramoto model with inertia. Phys. Rev. E 90, 042905 (2014)Google Scholar
  26. 26.
    Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Prog. Theory Phys. 75, 1105–1110 (1986)ADSCrossRefGoogle Scholar
  27. 27.
    Silver, H., Frankel, N.E., Ninham, B.W.: A class of mean field models. J. Math. Phys. 13, 468–474 (1972)ADSCrossRefGoogle Scholar
  28. 28.
    Sznitman, A.-S.: Topics in Propagation of Chaos. Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, Berlin (1991)Google Scholar
  29. 29.
    Vlasov, V., Macau, E.E.N., Pikovsky, A.: Synchronization of oscillators in a Kuramoto-type model with generic coupling. Chaos 24, 023120 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sylvain Delattre
    • 1
  • Giambattista Giacomin
    • 1
  • Eric Luçon
    • 2
  1. 1.Laboratoire de Probabilités et Modèles Aléatoires, Sorbonne Paris Cité, UMR 7599Université Paris DiderotParisFrance
  2. 2.Laboratoire MAP5, Sorbonne Paris Cité, UMR 8145Université Paris DescartesParisFrance

Personalised recommendations