Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Bertini, L., Giacomin, G., Pakdaman, K.: Dynamical aspects of mean field plane rotators and the Kuramoto model. J. Stat. Phys. 138, 270–290 (2010)
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)
Bollobás, B.: Random Graphs. Cambridge studies in advanced mathematics, vol. 73, 2nd edn (2001)
Britton, T., Deijfen, M., Martin-Löf, A.: Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124, 1377–1397 (2006)
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)
Collet, F., Dai Pra, P.: The role of disorder in the dynamics of critical fluctuations of mean field models. Electron. J. Probab. 17 (2012)
Dahms, D.: Time behavior of a spherical mean field model. PhD Thesis, Technische Universität Berlin (2002)
Dai Pra, P., den Hollander, F.: McKean-Vlasov limit for interacting random processes in random media. J. Stat. Phys. 84, 735–772 (1996)
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)
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)
Dembo, A., Montanari, A.: Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24, 137–211 (2010)
Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74 (2004)
Gärtner, J.: On McKean-Vlasov limit for interacting diffusions. Mathematische Nachrichten 137, 197–248 (1988)
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)
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)
Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, New York (1984)
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)
Lancellotti, C.: On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transp. Theory Stat. Phys. 34, 523–535 (2005)
Luçon, E.: Quenched limits and fluctuations of the empirical measure for plane rotators in random media. Electron. J. Probab. 16, 792–829 (2011)
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)
Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford University Press, Oxford (2009)
McKay, B.D., Wormald, N.C.: Uniform generation of random regular graphs of moderate degree. J Algorithms 11(1), 52–67 (1990)
Olivieri, E., Vares, M.E.: Large Deviations and Metastability, Encyclopedia of Mathematics and its Applications, 100. Cambridge University Press, Cambridge (2005)
Olmi, S., Navas, A., Boccaletti, S., Torcini, A.: Hysteretic transitions in the Kuramoto model with inertia. Phys. Rev. E 90, 042905 (2014)
Shinomoto, S., Kuramoto, Y.: Phase transitions in active rotator systems. Prog. Theory Phys. 75, 1105–1110 (1986)
Silver, H., Frankel, N.E., Ninham, B.W.: A class of mean field models. J. Math. Phys. 13, 468–474 (1972)
Sznitman, A.-S.: Topics in Propagation of Chaos. Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, Berlin (1991)
Vlasov, V., Macau, E.E.N., Pikovsky, A.: Synchronization of oscillators in a Kuramoto-type model with generic coupling. Chaos 24, 023120 (2014)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Delattre, S., Giacomin, G. & Luçon, E. A Note on Dynamical Models on Random Graphs and Fokker–Planck Equations. J Stat Phys 165, 785–798 (2016). https://doi.org/10.1007/s10955-016-1652-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-016-1652-3