  The stochastic block model (SBM) is a generative model revealing macroscopic
structures in graphs. Bayesian methods are used for (i) cluster assignment
inference and (ii) model selection for the number of clusters. In this paper,
we study the behavior of Bayesian inference in the SBM in the large sample
limit. Combining variational approximation and Laplace's method, a consistent
criterion of the fully marginalized log-likelihood is established. Based on
that, we derive a tractable algorithm that solves tasks (i) and (ii)
concurrently, obviating the need for an outer loop to check all model
candidates. Our empirical and theoretical results demonstrate that our method
is scalable in computation, accurate in approximation, and concise in model
selection.
