Information-theoretic thresholds from the cavity method

TL;DR

This paper rigorously establishes an information-theoretic formula using the cavity method, revealing phase transitions in random graph models and solving several longstanding conjectures in statistical inference and graph theory.

Contribution

It provides the first rigorous proof of the cavity method's predictions for mutual information and phase transitions in various random graph models and inference problems.

Findings

Pinpointed the exact condensation phase transition in the Potts antiferromagnet.

Proved the conjecture on the condensation phase transition in random graph coloring for q≥3.

Confirmed the conjectured mutual information formula for Low-Density Generator Matrix codes.

Abstract

Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results [Contucci et al.: Communications in Mathematical Physics 2013]. Further, we prove the conjecture from [Krzakala et al.: PNAS 2007] about the condensation phase transition in the random graph coloring problem for any number $q\geq3$ of colors. Moreover, we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model [Decelle et al.: Phys. Rev. E 2011]. Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes [Montanari: IEEE Transactions on Information Theory 2005].