Spectral clustering and the high-dimensional stochastic blockmodel

TLDR: In this article, the authors studied spectral clustering under the stochastic block model and showed that the eigenvectors of the normalized graph Laplacian asymptotically converges to the eigens of a population normalized graph.

Abstract: Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social ne tworks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasi ble method to discover these communities. The Stochastic Block Model (Holland et al., 1983) is a social network model with well defined communities; each node is a member of one community. For a network generated from the Stochastic Block Model, we bound the number of nodes "misclus- tered" by spectral clustering. The asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the Stochastic Block Model, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a "population" normal- ized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original. AMS 2000 subject classifications: Primary 62H30, 62H25; secondary 60B20.