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We present a method to estimate block membership of nodes in a random graphgenerated by a stochastic blockmodel. We use an embedding procedure motivatedby the random dot product graph model, a particular example of the latentposition model. The embedding associates each node with a vector; these vectorsare clustered via minimization of a square error criterion. We prove that thismethod is consistent for assigning nodes to blocks, as only a negligible numberof nodes will be mis-assigned. We prove consistency of the method for directedand undirected graphs. The consistent block assignment makes possibleconsistent parameter estimation for a stochastic blockmodel. We extend theresult in the setting where the number of blocks grows slowly with the numberof nodes. Our method is also computationally feasible even for very largegraphs. We compare our method to Laplacian spectral clustering through analysisof simulated data and a graph derived from Wikipedia documents.

A consistent adjacency spectral embedding for stochastic blockmodel graphs