Limit theorems for eigenvectors of the normalized Laplacian for random graphs

TL;DR

This paper establishes a central limit theorem for eigenvector components of the normalized Laplacian in random graphs, showing their convergence to normal distributions and analyzing implications for community detection.

Contribution

It introduces a CLT for eigenvectors of the normalized Laplacian in random graphs and compares spectral embedding methods for block recovery.

Findings

Eigenvector components follow a multivariate normal distribution asymptotically.

Spectral embeddings of the normalized Laplacian can recover block memberships.

No embedding method is universally superior for community detection.

Abstract

We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.