  We present clustering methods for multivariate data exploiting the underlying
geometry of the graphical structure between variables. As opposed to standard
approaches that assume known graph structures, we first estimate the edge
structure of the unknown graph using Bayesian neighborhood selection
approaches, wherein we account for the uncertainty of graphical structure
learning through model-averaged estimates of the suitable parameters.
Subsequently, we develop a nonparametric graph clustering model on the lower
dimensional projections of the graph based on Laplacian embeddings using
Dirichlet process mixture models. In contrast to standard algorithmic
approaches, this fully probabilistic approach allows incorporation of
uncertainty in estimation and inference for both graph structure learning and
clustering. More importantly, we formalize the arguments for Laplacian
embeddings as suitable projections for graph clustering by providing
theoretical support for the consistency of the eigenspace of the estimated
graph Laplacians. We develop fast computational algorithms that allow our
methods to scale to large number of nodes. Through extensive simulations we
compare our clustering performance with standard clustering methods. We apply
our methods to a novel pan-cancer proteomic data set, and evaluate protein
networks and clusters across multiple different cancer types.
