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A large deviation principle for block models

Part of: Combinatorial probability Limit theorems Graph theory

Published online by Cambridge University Press:  30 June 2025

Christian Borgs Open the ORCID record for Christian Borgs [Opens in a new window] ,
Jennifer Chayes Open the ORCID record for Jennifer Chayes [Opens in a new window] ,
Julia Gaudio ,
Samantha Petti Open the ORCID record for Samantha Petti [Opens in a new window]  and
Subhabrata Sen
Christian Borgs
Affiliation:
Berkeley AI Research Group, Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA, USA
Jennifer Chayes
Affiliation:
Department of EECS, Math, Stat, School of Information, University of California, Berkeley, CA, USA
Julia Gaudio
Affiliation:
Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA
Samantha Petti
Affiliation:
Department of Math, Tufts University, Medford, MA, USA
Subhabrata Sen*
Affiliation:
Department of Statistics, Harvard University, Cambridge, MA, USA
*
Corresponding author: Subhabrata Sen; Email: subhabratasen@fas.harvard.edu
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Abstract

We initiate a study of large deviations for block model random graphs in the dense regime. Following [14], we establish an LDP for dense block models, viewed as random graphons. As an application of our result, we study upper tail large deviations for homomorphism densities of regular graphs. We identify the existence of a ‘symmetric’ phase, where the graph, conditioned on the rare event, looks like a block model with the same block sizes as the generating graphon. In specific examples, we also identify the existence of a ‘symmetry breaking’ regime, where the conditional structure is not a block model with compatible dimensions. This identifies a ‘reentrant phase transition’ phenomenon for this problem – analogous to one established for Erdős–Rényi random graphs [13, 14]. Finally, extending the analysis of [34], we identify the precise boundary between the symmetry and symmetry breaking regimes for homomorphism densities of regular graphs and the operator norm on Erdős–Rényi bipartite graphs.

Keywords

Large deviationstochastic block modelsymmetry/symmetry breakingbipartite Erdős–Rényi graph

MSC classification

Primary: 60F10: Large deviations
Secondary: 05C80: Random graphs 60C05: Combinatorial probability

Information

Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 74
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Augeri, F. (2020) Nonlinear large deviation bounds with applications to Wigner matrices and sparse Erdős–Rényi graphs. Ann. Probab. 48(5) 2404–2448.10.1214/20-AOP1427CrossRefGoogle Scholar
Austin, T. (2019) The structure of low-complexity Gibbs measures on product spaces. Ann. Probab. 47(6) 4002–4023.10.1214/19-AOP1352CrossRefGoogle Scholar
Basak, A. and Basu, R. (2023) Upper tail large deviations of the cycle counts in Erdős–Rényi graphs in the full localized regime. Commun. Pure Appl. Math. 76(1) 3–72.10.1002/cpa.22036CrossRefGoogle Scholar
Bhattacharya, B. B. and Ganguly, S. (2020) Upper tails for edge eigenvalues of random graphs. SIAM J. Discrete Math. 34(2) 1069–1083.10.1137/18M1230852CrossRefGoogle Scholar
Bhattacharya, B. B., Ganguly, S., Lubetzky, E. and Zhao, Y. (2017) Upper tails and independence polynomials in random graphs. Adv. Math. 319 313–347.10.1016/j.aim.2017.08.003CrossRefGoogle Scholar
Bhattacharya, B. B., Ganguly, S., Shao, X. and Zhao, Y. (2016) Upper tails for arithmetic progressions in a random set. Int. Math. Res. Not. 2020(1) 167–213.10.1093/imrn/rny022CrossRefGoogle Scholar
Bhattacharya, S. and Dembo, A. (2021) Upper tail for homomorphism counts in constrained sparse random graphs. Random Struct Algor. 59(3) 315–338.10.1002/rsa.21011CrossRefGoogle Scholar
Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008) Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Adv. Math. 219(6) 1801–1851.10.1016/j.aim.2008.07.008CrossRefGoogle Scholar
Borgs, C., Chayes, J., Lovász, L., Sós, V. and Vesztergombi, K. (2012) Convergent sequences of dense graphs II: Multiway cuts and statistical physics. Ann. Math. 179(1) 151–219.10.4007/annals.2012.176.1.2CrossRefGoogle Scholar
Chatterjee, S. (2012) The missing log in large deviations for triangle counts. Random Struct Algor. 40(4) 437–451.10.1002/rsa.20381CrossRefGoogle Scholar
Chatterjee, S. (2017) Large deviations for random graphs. Lecture Notes in Math. Vol. 2197, Saint-Flour Probability Summer School. Cham: Springer, xi+167 pp.Google Scholar
Chatterjee, S. and Dembo, A. (2016) Nonlinear large deviations. Adv. Math. 299 396–450.10.1016/j.aim.2016.05.017CrossRefGoogle Scholar
Chatterjee, S. and Dey, P. S. (2010) Applications of Stein’s method for concentration inequalities. Ann. Probab. 38(6) 2443–2485.10.1214/10-AOP542CrossRefGoogle Scholar
Chatterjee, S. and Varadhan, S. R. S. (2011) The large deviation principle for the Erdős–Rényi random graph. Eur. J. Comb. 32(7) 1000–1017.10.1016/j.ejc.2011.03.014CrossRefGoogle Scholar
Cook, N. and Dembo, A. (2020) Large deviations of subgraph counts for sparse Erdős–Rényi graphs. Adv. Math. 373 107289.10.1016/j.aim.2020.107289CrossRefGoogle Scholar
DeMarco, B. and Kahn, J. (2012) Upper tails for triangles. Random Struct Algor. 40(4) 452–459.10.1002/rsa.20382CrossRefGoogle Scholar
DeMarco, R. and Kahn, J. (2012) Tight upper tail bounds for cliques. Random Struct Algor. 41(4) 469–487.10.1002/rsa.20440CrossRefGoogle Scholar
Dembo, A. and Lubetzky, E. (2018) A large deviation principle for the Erdős–Rényi uniform random graph. Electron. Commun. Probab. 23(79) 13.10.1214/18-ECP181CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (2010) Large Deviations Techniques and Applications, vol. 38 Stochastic Modelling and Applied Probability. Springer-Verlag. Berlin Heidelberg, second edition.10.1007/978-3-642-03311-7CrossRefGoogle Scholar
Dhara, S. and Sen, S. (2022) Large deviation for uniform graphs with given degrees. Ann. Appl. Probab. 32(3) 2327–2353.10.1214/21-AAP1745CrossRefGoogle Scholar
Durrett, R. (2019) Probability: Theory and Examples. Vol. 49. Cambridge University Press.10.1017/9781108591034CrossRefGoogle Scholar
Eldan, R. (2018) Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations. Geom. Funct. Anal. 28(6) 1548–1596.10.1007/s00039-018-0461-zCrossRefGoogle Scholar
Finner, H. (1992) A generalization of Hölder’s inequality and some probability inequalities, Ann. Probab. 20(4) 1893–1901.10.1214/aop/1176989534CrossRefGoogle Scholar
Grebík, J. and Pikhurko, O. (2021) Large deviation principles for block and step graphon random graph models. arXiv preprint arXiv: 2101.07025.Google Scholar
Harel, M., Mousset, F. and Samotij, W. (2022) Upper tails via high moments and entropic stability. Duke Math. J. 1(1) 1–104.Google Scholar
Janson, S. (2013) Graphons, cut norm and distance, couplings and rearrangements. In New York Journal of Mathematics Monographs, Vol. 4, University at Albany. Albany, NY.Google Scholar
Janson, S., Oleszkiewicz, K. and Ruciński, A. (2004) Upper tails for subgraph counts in random graphs. Isr. J. Math. 142(1) 61–92.10.1007/BF02771528CrossRefGoogle Scholar
Janson, S. and Ruciński, A. (2002) The infamous upper tail. Random Struct Algor. 20(3) 317–342.10.1002/rsa.10031CrossRefGoogle Scholar
Janson, S. and Ruciński, A. (2004) The deletion method for upper tail estimates. Combinatorica 24(4) 615–640.10.1007/s00493-004-0038-3CrossRefGoogle Scholar
Kim, J. H. and Van, H. V. (2004) Divide and conquer martingales and the number of triangles in a random graph. Random Struct Algor. 24(2) 166–174.10.1002/rsa.10113CrossRefGoogle Scholar
Liu, Y. P. and Zhao, Y. (2021) On the upper tail problem for random hypergraphs. Random Struct Algor. 58(2) 179–220.10.1002/rsa.20975CrossRefGoogle Scholar
Lovász, L. (2012) Large Networks and Graph Limits. vol. 60, American Mathematical Society.Google Scholar
Lovász, L. and Szegedy, B. (2007) Szemerédi’s lemma for the analyst. Geom. Funct. Anal. 17(1) 252–270.10.1007/s00039-007-0599-6CrossRefGoogle Scholar
Lubetzky, E. and Zhao, Y. (2015) On replica symmetry of large deviations in random graphs. Random Struct. Algor. 47(1) 109–146.10.1002/rsa.20536CrossRefGoogle Scholar
Markering, M. (2023) The large deviation principle for inhomogeneous Erdős–Rényi random graphs. J. Theor. Probab. 36(2) 711–727.10.1007/s10959-022-01181-1CrossRefGoogle Scholar
Šileikis, M. and Warnke, L. (2019) A counterexample to the DeMarco-Kahn upper tail conjecture. Random Struct. Algor. 55(4) 775–794.10.1002/rsa.20859CrossRefGoogle Scholar
Van, H. V. (2001) A large deviation result on the number of small subgraphs of a random graph. Comb. Prob. Comput. 10(1) 79–94.Google Scholar