Computational Complexity, Phase Transitions, and Message-Passing for Community Detection

TL;DR

This paper explores the intersection of computational complexity, phase transitions, and message-passing algorithms in community detection, highlighting theoretical bounds and phase transition phenomena in random graphs and related problems.

Contribution

It provides a comprehensive overview of phase transitions and message-passing techniques in community detection, connecting concepts from computer science and statistical physics.

Findings

Identification of critical thresholds for community detectability

Analysis of phase transition phenomena in random graph models

Application of belief propagation and non-backtracking matrices

Abstract

We take a whirlwind tour of problems and techniques at the boundary of computer science and statistical physics. We start with a brief description of P, NP, and NP-completeness. We then discuss random graphs, including the emergence of the giant component and the k-core, using techniques from branching processes and differential equations. Using these tools as well as the second moment method, we give upper and lower bounds on the critical clause density for random k-SAT. We end with community detection in networks, variational methods, the Bethe free energy, belief propagation, the detectability transition, and the non-backtracking matrix.