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The Large Deviation Principle for Interacting Dynamical Systems on Random Graphs


Using the weak convergence approach to large deviations, we formulate and prove the large deviation principle (LDP) for W-random graphs in the cut-norm topology. This generalizes the LDP for Erdős–Rényi random graphs by Chatterjee and Varadhan. Furthermore, we translate the LDP for random graphs to a class of interacting dynamical systems on such graphs. To this end, we demonstrate that the solutions of the dynamical models depend continuously on the underlying graphs with respect to the cut-norm and apply the contraction principle.

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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.


  1. Several proofs are available, including one that extends Sanov’s theorem. However, the most direct argument is to note that Mogulskii’s theorem asserts that if \(Y^n(x)=\int _0^x F^n(t)dt\), then \(\{Y^n\}\) satisfies an LDP in C([0, 1]) with the rate function \(I(\phi )\) equal to \(\int _0^1 L(\dot{\phi }(t))dt\) if \(\phi \) is absolutely continuous with \(\phi (0)=0\) and \(\infty \) otherwise. Using \(F^n(x)=\dot{Y}^n(x)\) a.s. in x, we can find the result stated for the sequence \(\{F^n\}\) using integration by parts.

  2. Here and below, C stands for a generic constant.


  1. Borgs, C., Chayes, J.T., Cohn, H., Zhao, Y.: An \(L^p\) theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions. Trans. Am. Math. Soc. 372(5), 3019–3062 (2019)

    Article  Google Scholar 

  2. Borgs, Christian., Chayes, Jennifer., Gaudio, Julia., Petti, Samantha., Sen, Subhabrata.: A large deviation principle for block models, arxiv:2007.14508, 2020

  3. Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the \(1/N\) limit of interacting classical particles. Commun. Math. Phys. 56(2), 101–113 (1977)

    ADS  MathSciNet  Article  Google Scholar 

  4. Budhiraja, Amarjit., Dupuis, Paul.: Analysis and approximation of rare events, Probability Theory and Stochastic Modelling, vol. 94, Springer, New York, 2019, Representations and weak convergence methods

  5. Chatterjee, S.: An introduction to large deviations for random graphs. Bull. Am. Math. Soc. (N.S.) 53(4), 617–642 (2016)

    MathSciNet  Article  Google Scholar 

  6. Chatterjee, S.: Large Deviations for Random Graphs. Lecture Notes in Mathematics, vol. 2197. Springer, Cham (2017)

    Book  Google Scholar 

  7. Chatterjee, S., Dembo, A.: Nonlinear large deviations. Adv. Math. 299, 396–450 (2016)

    MathSciNet  Article  Google Scholar 

  8. Chatterjee, S., Varadhan, S.R.S.: The large deviation principle for the Erdös–Rényi random graph. Eur. J. Combin. 32(7), 1000–1017 (2011)

    Article  Google Scholar 

  9. Coppini, Fabio., Dietert, Helge., Giacomin, Giambattista: A law of large numbers and large deviations for interacting diffusions on Erdos–Renyi graphs, Stoch. Dyn. 20 (2020), no. 2, 2050010, 19

  10. Dobrušin, R. L.: Vlasov equations, Funktsional. Anal. i Prilozhen. 13 (1979), no. 2, 48–58, 96

  11. Golse, François.: On the dynamics of large particle systems in the mean field limit, Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, Lect. Notes Appl. Math. Mech., vol. 3, Springer, [Cham], (2016), pp. 1–144

  12. Grebík, Jan., Pikhurko, Oleg.: Large deviation principles for block and step graphon random graph models, arXiv:2101.07025, (2021)

  13. Guédon, O., Vershynin, R.: Community detection in sparse networks via Grothendieck’s inequality. Probab. Theory Related Fields 165(3–4), 1025–1049 (2016)

    MathSciNet  Article  Google Scholar 

  14. Harel, Matan., Mousset, Frank., Samotij, Wojciech.: Upper tails via high moments and entropic stability, arXiv:1904.08212, (2021)

  15. Jabin, P.-E.: A review of the mean field limits for Vlasov equations. Kinet. Relat. Models 7(4), 661–711 (2014)

    MathSciNet  Article  Google Scholar 

  16. Kaliuzhnyi-Verbovetskyi, D., Medvedev, G.S.: Sparse Monte Carlo method for nonlocal diffusion problems, arXiv e-prints (2019), arXiv:1905.10844

  17. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984)

    Book  Google Scholar 

  18. Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg,: Translated from the French by P, p. 181. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band (1972)

  19. Lovász, L.: Large Networks and Graph Limits. AMS, Providence (2012)

    Book  Google Scholar 

  20. Lovász, L., Szegedy, B.: Szemerédi’s lemma for the analyst, GAFA, Geom. funct. anal. 17, 252–270

  21. Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B 96(6), 933–957 (2006)

    MathSciNet  Article  Google Scholar 

  22. Medvedev, G.S.: The nonlinear heat equation on dense graphs and graph limits. SIAM J. Math. Anal. 46(4), 2743–2766 (2014)

    MathSciNet  Article  Google Scholar 

  23. Medvedev, G.S.: The nonlinear heat equation on \(W\)-random graphs. Arch. Ration. Mech. Anal. 212(3), 781–803 (2014)

    MathSciNet  Article  Google Scholar 

  24. Medvedev, G.S.: The continuum limit of the Kuramoto model on sparse random graphs. Commun. Math. Sci. 17(4), 883–898 (2019)

    MathSciNet  Article  Google Scholar 

  25. Neunzert, H.: Mathematical investigations on particle - in - cell methods 9, 229–254 (1978)

  26. Oliveira, Roberto I., Reis, Guilherme H.: Interacting diffusions on random graphs with diverging average degrees: Hydrodynamics and large deviations, Journal of Statistical Physics (2019)

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Correspondence to Georgi S. Medvedev.

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Communicated by S. Chatterjee.

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P. Dupuis: research supported in part by the NSF (DMS-1904992). G. S. Medvedev: research supported in part by the NSF (DMS-2009233).

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Dupuis, P., Medvedev, G.S. The Large Deviation Principle for Interacting Dynamical Systems on Random Graphs. Commun. Math. Phys. 390, 545–575 (2022).

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