On Concentration Inequality of the Laplacian Matrix of Erdős-Rényi Graphs
Yiming Chen, Xuanang Hu, Pengtao Li
TL;DR
This paper addresses concentration of the spectral norm of the normalized Laplacian for Erdős–Rényi graphs, particularly in sparse regimes. The authors develop an improved bound for the tau-regularized Laplacian by decomposing the deviation into a degree-driven component and a degree-normalization component, leveraging uniform degree concentration to achieve a rate of $||\mathcal{L}(A_tau) - \mathcal{L}(\mathbb{E}A_tau)|| \le C \frac{r^2}{\sqrt{\tau}} (1 + \frac{d}{\tau})^{1/2}$, which sharpens prior results. They also establish uniform concentration results over edge probabilities, including tau-independent bounds, and prove that the Laplacian norm concentrates around 1, aided by eigenvector delocalization and degree concentration arguments. These contributions enhance understanding of Laplacian concentration in sparse random graphs and have implications for spectral clustering and community detection in networks.
Abstract
This paper focuses on the concentration properties of the spectral norm of the normalized Laplacian matrix for Erdős-Rényi 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.