Limit theorems for eigenvectors of the normalized Laplacian for random graphs
This provides theoretical guarantees for spectral embedding methods in network analysis, aiding statisticians and data scientists working with graph data, though it is incremental as it builds on prior results for adjacency matrices.
The paper proves a central limit theorem for eigenvectors of the normalized Laplacian in random dot product graphs, showing that spectral embedding rows converge to multivariate normals with block-dependent parameters in stochastic blockmodels, and compares this to adjacency matrix embeddings via Chernoff information, finding neither method dominates for block recovery.
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.