Large Deviation Principle for the Exploration Process of the Configuration Model

TL;DR

This paper establishes a large deviation principle for the exploration process of the configuration model, providing insights into the asymptotic behavior of large random graphs with specified degree distributions.

Contribution

It introduces a novel large deviation framework for the exploration process using stochastic differential equations and variational formulas, advancing understanding of the configuration model's component structure.

Findings

Large deviation principle for the exploration process established.

Representation of the process via stochastic differential equations.

Applications to asymptotic degree sequence analysis.

Abstract

The configuration model is a sequence of random graphs constructed such that in the large network limit the degree distribution converges to a pre-specified probability distribution. The component structure of such random graphs can be obtained from an infinite dimensional Markov chain referred to as the exploration process. We establish a large deviation principle for the exploration process associated with the configuration model. Proofs rely on a representation of the exploration process as 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. Applications of the large deviation results, for studying asymptotic behavior of the degree sequence in large components of the random graphs, are discussed.