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A large deviation principle for block models

Part of: Limit theorems Combinatorial probability 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 , Volume 34 , Issue 5 , September 2025 , pp. 680 - 753
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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