The Annals of Statistics

Limit theorems for eigenvectors of the normalized Laplacian for random graphs

Minh Tang and Carey E. Priebe

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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.

Article information

Source
Ann. Statist., Volume 46, Number 5 (2018), 2360-2415.

Dates
Received: August 2016
Revised: June 2017
First available in Project Euclid: 17 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1534492839

Digital Object Identifier
doi:10.1214/17-AOS1623

Mathematical Reviews number (MathSciNet)
MR3845021

Zentralblatt MATH identifier
06964336

Subjects
Primary: 62H12: Estimation
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62B10: Information-theoretic topics [See also 94A17]

Keywords
Spectral clustering random dot product graph stochastic blockmodels convergence of eigenvectors Chernoff information

Citation

Tang, Minh; Priebe, Carey E. Limit theorems for eigenvectors of the normalized Laplacian for random graphs. Ann. Statist. 46 (2018), no. 5, 2360--2415. doi:10.1214/17-AOS1623. https://projecteuclid.org/euclid.aos/1534492839


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