Spectral clustering and the high-dimensional stochastic blockmodel

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 blockmodel [Social Networks 5 (1983) 109--137] is a social network model with well-defined communities; each node is a member of one community. For a network generated from the Stochastic Blockmodel, 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 blockmodel, 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.Comment: Published in at http://dx.doi.org/10.1214/11-AOS887 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random matrix theory in statistics: A review

“…Moreover, RMT has been utilized to understand the analytic properties of compressed sensing (Cai and Jiang, 2011;Vershynin, 2012). Effective uses of concepts and tools from RMT have also been made in the analysis of spectral clustering methods with a growing number of clusters (Rohe et al, 2011), in community detection problems in large random networks (Nadakuditi and Newman, 2012), and in kernel PCA methods for dimension reduction when the observations are high-dimensional and the argument of the kernel function depends on the inner products of pairs of observations (El Karoui, 2010b,a).…”

Impact of regularization on spectral clustering

“…In particular, for a graph with n nodes, previous theoretical analyses for spectral clustering, under the SBM and its extensions, [23], [7], [25], [10] assumed that the minimum degree of the graph scales at least by a polynomial power of log n. Even when this assumption is satisfied, the dependence on the minimum degree is highly restrictive when it comes to making inferences about cluster recovery. Our analysis provides cluster recovery results that potentially do not depend on the above mentioned constraint on the minimum degree.…”

“…Denoting τ as the regularization parameter, previous theoretical analyses of regularization ( [7], [23]) provided high-probability bounds on this spectral norm. These bounds have a 1/ √ τ dependence on τ , for large τ .…”

Hierarchical Intuitionistic Fuzzy Possibilistic C Means Kernel Clustering Algorithm for Distributed Networks

“…This is related to tight clustering condition [13] and less stringent than earlier results which assume that within-and-between-cluster similarities are constant and bounded in expectation [14]. The condition enforces that within-and-between-cluster similarities concentrate away from each other.…”

“…The condition enforces that within-and-between-cluster similarities concentrate away from each other. This condition is satisfied if similarities are constant in expectation, perturbed with any subgaussian noise [14], [15].…”