Properties of atypical graphs from negative complexities

TL;DR

This paper introduces a novel application of the one-step replica symmetry breaking cavity method to study large deviations in random graph ensembles, linking negative complexities with the probabilities of rare configurations.

Contribution

It establishes a new theoretical connection between negative complexities in spin-glass models and large deviation probabilities, validated through combinatorial calculations.

Findings

The cavity method effectively characterizes large deviations in random graphs.

Negative complexities correspond to probabilities of rare samples.

The approach is validated on multiple models with combinatorial comparisons.

Abstract

The one-step replica symmetry breaking cavity method is proposed as a new tool to investigate large deviations in random graph ensembles. The procedure hinges on a general connection between negative complexities and probabilities of rare samples in spin-glass like models. This relation between large deviations and replica theory is explicited on different models where it is confronted to direct combinatorial calculations.