Limit theorems for eigenvectors of the normalized Laplacian for random graphs
arXiv:1607.08601
Abstract
We prove a central limit theorem for the components of the eigenvectors corresponding to the 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.
arXiv comment52 pages, 4 figures
References in corpus (8)
- Stochastic blockmodels and community structure in networks
- Consistency of spectral clustering
- Graph limits and exchangeable random graphs
- Stochastic blockmodels with growing number of classes
- Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel
- Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
- Stochastic Block Models and Reconstruction
- A nonparametric two-sample hypothesis testing problem for random dot product graphs