A consistent adjacency spectral embedding for stochastic blockmodel graphs

TL;DR

This paper introduces a spectral embedding method based on the random dot product graph model for accurately identifying community structures in stochastic blockmodel graphs, applicable to large directed and undirected networks.

Contribution

The paper proposes a new consistent adjacency spectral embedding technique for stochastic blockmodels, extending to growing number of blocks and demonstrating computational efficiency.

Findings

Method is consistent for block assignment with negligible misclassification

Applicable to directed and undirected graphs

Performs well on simulated and real-world data

Abstract

We present a method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel. We use an embedding procedure motivated by the random dot product graph model, a particular example of the latent position model. The embedding associates each node with a vector; these vectors are clustered via minimization of a square error criterion. We prove that this method is consistent for assigning nodes to blocks, as only a negligible number of nodes will be mis-assigned. We prove consistency of the method for directed and undirected graphs. The consistent block assignment makes possible consistent parameter estimation for a stochastic blockmodel. We extend the result in the setting where the number of blocks grows slowly with the number of nodes. Our method is also computationally feasible even for very large graphs. We compare our method to Laplacian spectral clustering through analysis of simulated data and a graph derived from Wikipedia documents.