  Our problem of interest is to cluster vertices of a graph by identifying
underlying community structure. Among various vertex clustering approaches,
spectral clustering is one of the most popular methods because it is easy to
implement while often outperforming more traditional clustering algorithms.
However, there are two inherent model selection problems in spectral
clustering, namely estimating both the embedding dimension and number of
clusters. This paper attempts to address the issue by establishing a novel
model selection framework specifically for vertex clustering on graphs under a
stochastic block model. The first contribution is a probabilistic model which
approximates the distribution of the extended spectral embedding of a graph.
The model is constructed based on a theoretical result of asymptotic normality
of the informative part of the embedding, and on a simulation result providing
a conjecture for the limiting behavior of the redundant part of the embedding.
The second contribution is a simultaneous model selection framework. In
contrast with the traditional approaches, our model selection procedure
estimates embedding dimension and number of clusters simultaneously. Based on
our conjectured distributional model, a theorem on the consistency of the
estimates of model parameters is presented, providing support for the validity
of our method. Algorithms for our simultaneous model selection for vertex
clustering are proposed, demonstrating superior performance in simulation
experiments. We illustrate our method via application to a collection of brain
graphs.
