  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.
