April 2022 Rare event asymptotics for exploration processes for random graphs
Shankar Bhamidi, Amarjit Budhiraja, Paul Dupuis, Ruoyu Wu
Author Affiliations +
Ann. Appl. Probab. 32(2): 1112-1178 (April 2022). DOI: 10.1214/21-AAP1704


Large deviations for random graph models has been a topic of significant recent research activity. Much work in this area is focused on the class of dense random graph models (number of edges in the graph scale as n2, where n is the number of vertices) where the theory of graphons has emerged as a principal tool in the study of large deviation properties. These tools do not give a good approach to large deviation problems for random graph models in the sparse regime. The aim of this paper is to study an approach for large deviation problems in this regime by establishing large deviation principles (LDP) on suitable path spaces for certain exploration processes of the associated random graph sequence. Exploration processes are an important tool in the study of sparse random graph models and have been used to understand detailed asymptotics of many functionals of sparse random graphs, such as component sizes, surplus, deviations from trees, etc. In the context of rare event asymptotics of interest here, the point of view of exploration process transforms a large deviation analysis of a static random combinatorial structure to the study of a small noise LDP for certain stochastic dynamical systems with jumps.

Our work focuses on one particular class of random graph models, namely the configuration model; however, the general approach of using exploration processes for studying large deviation properties of sparse random graph models has broader applicability. The goal is to study asymptotics of probabilities of nontypical behavior in the large network limit. The first key step for this is to establish a LDP for an exploration process associated with the configuration model. A suitable exploration process here turns out to be an infinite-dimensional Markov process with transition probability rates that diminish to zero in certain parts of the state space. Large deviation properties of such Markovian models is challenging due to poor regularity behavior of the associated local rate functions. Our proof of the LDP relies on a representation of the exploration process in terms of a system of stochastic differential equations driven by Poisson random measures and variational formulas for moments of nonnegative functionals of Poisson random measures. Uniqueness results for certain controlled systems of deterministic equations play a key role in the analysis. Next, using the rate function in the LDP for the exploration process we formulate a calculus of variations problem associated with the asymptotics of component degree distributions. The second key ingredient in our study is a careful analysis of the infinite-dimensional Euler–Lagrange equations associated with this calculus of variations problem. Exact solutions of these systems of nonlinear differential equations are identified which then provide explicit formulas for decay rates of probabilities of nontypical component degree distributions and related quantities.

Funding Statement

The research of SB was supported in part by the NSF (DMS-1606839, DMS-1613072 and DMS-2113662) and the Army Research Office (W911NF-17-1-0010). The research of AB was supported in part by the NSF (DMS-1305120, DMS-1814894, DMS-1853968). The research of PD was supported in part by the NSF (DMS-1904992) and DARPA (W911NF-15-2-0122). The research of RW was supported in part by DARPA (W911NF-15-2-0122).


We would like to thank the two referees for a careful review of our work and for the many helpful suggestions.


Download Citation

Shankar Bhamidi. Amarjit Budhiraja. Paul Dupuis. Ruoyu Wu. "Rare event asymptotics for exploration processes for random graphs." Ann. Appl. Probab. 32 (2) 1112 - 1178, April 2022. https://doi.org/10.1214/21-AAP1704


Received: 1 January 2021; Revised: 1 April 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

MathSciNet: MR4414702
zbMATH: 1487.05239
Digital Object Identifier: 10.1214/21-AAP1704

Primary: 05C80 , 60C05 , 60F10
Secondary: 90B15

Keywords: branching processes , calculus of variations problems , configuration model , diminishing rates , Euler–Lagrange equations , Exploration process , Giant component , large deviation principle , Poisson random measures , Random graphs , singular dynamics , sparse regime , Variational representations

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 2 • April 2022
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