Title: Role of Normalization in Spectral Clustering for Stochastic Blockmodels

Abstract: Spectral Clustering clusters elements using the top few eigenvectors of their (possibly normalized) similarity matrix. The quality of Spectral Clustering is closely tied to the convergence properties of these principal eigenvectors. This rate of convergence has been shown to be identical for both the normalized and unnormalized variants ([17]). However normalization for Spectral Clustering is the common practice ([16], [19]). Indeed, our experiments also show that normalization improves prediction accuracy. In this paper, for the popular Stochastic Blockmodel, we theoretically show that under spectral embedding, normalization shrinks the variance of points in a class by a constant fraction. As a byproduct of our work, we also obtain sharp deviation bounds of empirical principal eigenvalues of graphs generated from a Stochastic Blockmodel.