Spectral clustering and the high-dimensional stochastic blockmodel

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.

💡 Research Summary

The paper investigates the theoretical performance of spectral clustering on networks generated by the stochastic blockmodel (SBM) when the number of communities grows with the number of vertices—a setting the authors refer to as “high‑dimensional.” The analysis proceeds in two major steps.

First, the authors consider the more general latent space model (LSM), in which each node i possesses a latent vector z_i and the probability of an edge between i and j is a function of the inner product or distance of (z_i, z_j). The SBM is a special case of the LSM where each community is associated with a single latent vector, so every node belongs to exactly one community. Within this framework they study the normalized graph Laplacian

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