Phase transitions for the cavity approach to the clique problem on random graphs

TL;DR

This paper rigorously proves two phase transitions in a probabilistic system inspired by spin glass theory, designed to identify large cliques in Erdős-Rényi random graphs, advancing understanding of phase behavior in combinatorial problems.

Contribution

It provides the first rigorous proof of phase transitions in a cavity-inspired probabilistic automaton for the clique problem on random graphs.

Findings

Identification of two distinct phase transitions in the system

Rigorous mathematical proof of phase transition phenomena

Insights into the behavior of cavity methods in combinatorial optimization

Abstract

We give a rigorous proof of two phase transitions for a disordered system designed to find large cliques inside Erdos random graphs. Such a system is associated with a conservative probabilistic cellular automaton inspired by the cavity method originally introduced in spin glass theory.