Large deviations for the greedy exploration process on configuration models

TL;DR

This paper establishes a large deviation principle for the greedy exploration process on configuration models, providing explicit formulas and insights into the probability of rare events in graph exploration.

Contribution

It introduces a novel large deviation framework for the greedy exploration of configuration models, including explicit rate functions and analysis for regular graphs.

Findings

Derived a closed-form rate function for large deviations

Provided explicit large deviation trajectories via optimization

Analyzed large deviations for the size of independent sets in graphs

Abstract

We prove a large deviation principle for the greedy exploration of configuration models, building on a time-discretized version of the method proposed by Bermolen et al. and Brightwell et al. for jointly constructing a random graph from a given degree sequence and its exploration. The proof of this result follows the general strategy to study large deviations of processes proposed by Feng and Kurtz, based on the convergence of non-linear semigroups. We provide an intuitive interpretation of the LD cost function using Cramer's theorem for the average of random variables with appropriate distribution, depending on the degree distribution of explored nodes. The rate function can be expressed in a closed-form formula, and the large deviations trajectories can be obtained through explicit associated optimization problems. We then deduce large deviations results for the size of the independent set constructed by the algorithm. As a particular case, we analyze these results for d-regular graphs.