Random Laplacian matrices and convex relaxations

TL;DR

This paper demonstrates that for many random Laplacian matrices, the largest eigenvalue closely matches the largest diagonal entry, providing insights into the spectral properties and optimality of convex relaxation algorithms in graph problems.

Contribution

It establishes the tightness of eigenvalue bounds for random Laplacian matrices and confirms the optimality of semidefinite relaxations for graph synchronization and community detection.

Findings

Largest eigenvalue often equals the largest diagonal entry in random Laplacian matrices

Semidefinite relaxations are optimal for Z2 synchronization and stochastic block model recovery

Connects spectral properties to phase transitions in Erdős-Rényi graphs

Abstract

The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms, often the size of the largest diagonal entry. Besides being a simple tool to obtain precise estimates on the largest eigenvalue of a large class of random Laplacian matrices, our main result settles a number of open problems related to the tightness of certain convex relaxation-based algorithms. It easily implies the optimality of the semidefinite relaxation approaches to problems such as $\mathbb{Z}_2$ Synchronization and Stochastic Block Model recovery. Interestingly, this result readily implies the connectivity threshold for Erd\H{o}s-R\'{e}nyi graphs and suggests that these three phenomena are manifestations of the same underlying principle. The main tool is a recent estimate on the spectral norm of matrices with independent entries by van Handel and the author.