  We describe a network clustering framework, based on finite mixture models,
that can be applied to discrete-valued networks with hundreds of thousands of
nodes and billions of edge variables. Relative to other recent model-based
clustering work for networks, we introduce a more flexible modeling framework,
improve the variational-approximation estimation algorithm, discuss and
implement standard error estimation via a parametric bootstrap approach, and
apply these methods to much larger data sets than those seen elsewhere in the
literature. The more flexible framework is achieved through introducing novel
parameterizations of the model, giving varying degrees of parsimony, using
exponential family models whose structure may be exploited in various
theoretical and algorithmic ways. The algorithms are based on variational
generalized EM algorithms, where the E-steps are augmented by a
minorization-maximization (MM) idea. The bootstrapped standard error estimates
are based on an efficient Monte Carlo network simulation idea. Last, we
demonstrate the usefulness of the model-based clustering framework by applying
it to a discrete-valued network with more than 131,000 nodes and 17 billion
edge variables.
