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Activity Number: 330 - Advances in Inference of Networks
Type: Topic Contributed
Date/Time: Tuesday, August 1, 2017 : 10:30 AM to 12:20 PM
Sponsor: Section on Nonparametric Statistics
Abstract #323346 View Presentation
Title: Limit Theorems for Eigenvectors of the Normalized Laplacian for Random Graphs
Author(s): Minh Tang* and Carey E Priebe
Companies: Johns Hopkins University and Johns Hopkins University
Keywords: stochastic blockmodels ; spectral clustering ; Chernoff information

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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