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October 2018 Limit theorems for eigenvectors of the normalized Laplacian for random graphs
Minh Tang, Carey E. Priebe
Ann. Statist. 46(5): 2360-2415 (October 2018). DOI: 10.1214/17-AOS1623

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.

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Minh Tang. Carey E. Priebe. "Limit theorems for eigenvectors of the normalized Laplacian for random graphs." Ann. Statist. 46 (5) 2360 - 2415, October 2018. https://doi.org/10.1214/17-AOS1623

Information

Received: 1 August 2016; Revised: 1 June 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964336
MathSciNet: MR3845021
Digital Object Identifier: 10.1214/17-AOS1623

Subjects:
Primary: 62H12
Secondary: 62B10 , 62H30

Keywords: Chernoff information , convergence of eigenvectors , random dot product graph , spectral clustering , stochastic blockmodels

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 5 • October 2018
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