LIMIT THEOREMS FOR EIGENVECTORS OF THE NORMALIZED LAPLACIAN FOR RANDOM GRAPHS

• 成果类型: Article
• 署名作者: Tang, Minh; Priebe, Carey E.
• 署名单位: Johns Hopkins University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1623
• 发表日期: 2018
• 页码: 2360-2415
• 关键词: stochastic blockmodel graphs; hypothesis-testing problem; Consistency; models

摘要

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.