Spectral clustering and the high-dimensional Stochastic Block Model

July, 2010
Report Number: 
791
Authors: 
Karl Rohe, Sourav Chatterjee, Bin Yu
Abstract: 

Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, 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 feasible 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 ``misclustered" by spectral clustering. The asymptotic results in this 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" normalized 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.

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