Electronic Communications in Probability

A large deviation principle for the Erdős–Rényi uniform random graph

Amir Dembo and Eyal Lubetzky

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Starting with the large deviation principle (LDP) for the Erdős–Rényi binomial random graph ${\mathcal G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph ${\mathcal G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in ${\mathcal G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random graphs, e.g., the edge/triangle model.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 79, 13 pp.

Received: 30 April 2018
Accepted: 9 October 2018
First available in Project Euclid: 24 October 2018

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60F10: Large deviations

Large deviations Erdős–Rényi graphs constrained random graphs

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Dembo, Amir; Lubetzky, Eyal. A large deviation principle for the Erdős–Rényi uniform random graph. Electron. Commun. Probab. 23 (2018), paper no. 79, 13 pp. doi:10.1214/18-ECP181. https://projecteuclid.org/euclid.ecp/1540346603

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  • [1] N. Alon and J. H. Spencer, The probabilistic method, fourth ed., Wiley Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2016.
  • [2] F. Augeri, Nonlinear large deviation bounds with applications to traces of Wigner matrices and cycles counts in Erdös-Renyi graphs, preprint, arXiv:1809.11148.
  • [3] B. B. Bhattacharya, S. Ganguly, E. Lubetzky, and Y. Zhao, Upper tails and independence polynomials in random graphs, Adv. Math. 319 (2017), 313–347.
  • [4] C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós, and K. Vesztergombi, Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing, Adv. Math. 219 (2008), no. 6, 1801–1851.
  • [5] S. Chatterjee and A. Dembo, Nonlinear large deviations, Adv. Math. 299 (2016), 396–450.
  • [6] S. Chatterjee and S. R. S. Varadhan, The large deviation principle for the Erdős-Rényi random graph, European J. Combin. 32 (2011), no. 7, 1000–1017.
  • [7] N. Cook and A. Dembo, Large deviations of subgraph counts for sparse Erdős–Rényi graphs, preprint, arXiv:1809.11148.
  • [8] A. Dembo and O. Zeitouni, Large deviations techniques and applications, Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010.
  • [9] F. den Hollander, M. Mandjes, A. Roccaverde, and N. J. Starreveld, Ensemble equivalence for dense graphs, Electron. J. Probab. 23 (2018), Paper No. 12, 26 pp.
  • [10] R. Eldan, Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations, Geom. Funct. Anal., to appear.
  • [11] W. Hoeffding, On the distribution of the number of successes in independent trials, Ann. Math. Statist. 27 (1956), 713–721.
  • [12] S. Janson, T. Łuczak, and A. Rucinski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000.
  • [13] R. Kenyon, C. Radin, K. Ren, and L. Sadun, Multipodal structure and phase transitions in large constrained graphs, J. Stat. Phys. 168 (2017), no. 2, 233–258.
  • [14] R. Kenyon, C. Radin, K. Ren, and L. Sadun, The phases of large networks with edge and triangle constraints, J. Phys. A 50 (2017), no. 43, 435001, 22.
  • [15] R. Kenyon, C. Radin, K. Ren, and L. Sadun, Bipodal structure in oversaturated random graphs, Int. Math. Res. Not. (IMRN) 2018 (2018), no. 4, 1009–1044.
  • [16] L. Lovász, Large networks and graph limits, American Mathematical Society Colloquium Publications, vol. 60, American Mathematical Society, Providence, RI, 2012.
  • [17] L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006), no. 6, 933–957.
  • [18] E. Lubetzky and Y. Zhao, On replica symmetry of large deviations in random graphs, Random Structures Algorithms 47 (2015), no. 1, 109–146.
  • [19] E. Lubetzky and Y. Zhao, On the variational problem for upper tails in sparse random graphs, Random Structures Algorithms 50 (2017), no. 3, 420–436.
  • [20] C. Radin and L. Sadun, Phase transitions in a complex network, J. Phys. A 46 (2013), no. 30, 305002, 12.
  • [21] Y. Zhao, On the lower tail variational problem for random graphs, Combin. Probab. Comput. 26 (2017), no. 2, 301–320.