On Concentration Inequality of the Laplacian Matrix of Erd\H{o}s-R\'enyi Graphs

TL;DR

This paper investigates the spectral norm concentration of the Laplacian matrix in Erdős-Rényi graphs, providing optimal bounds and new inequalities that enhance understanding of spectral properties in random graph models.

Contribution

It establishes the optimal concentration bounds for the regularized Laplacian and introduces a uniform concentration inequality for the Laplacian's spectral norm in homogeneous Erdős-Rényi graphs.

Findings

Optimal bound for regularized Laplacian spectral norm

Uniform concentration inequality for Laplacian in homogeneous case

Eigenvector normalization leads to spectral norm concentration around 1

Abstract

This paper focuses on the concentration properties of the spectral norm of the normalized Laplacian matrix for Erd\H{o}s-R\'enyi random graphs. First, We achieve the optimal bound that can be attained in the further question posed by Le et al. [24] for the regularized Laplacian matrix. Beyond that, we also establish a uniform concentration inequality for the spectral norm of the Laplacian matrix in the homogeneous case, relying on a key tool: the uniform concentration property of degrees, which may be of independent interest. Additionally, we prove that after normalizing the eigenvector corresponding to the largest eigenvalue, the spectral norm of the Laplacian matrix concentrates around 1, which may be useful in special cases.