A sparse stochastic block model with two unequal communities

We show posterior convergence for the community structure in the planted bi-section model, for several interesting priors. Examples include where the label on each vertex is iid Bernoulli distributed, with some parameter $r\in(0,1)$. The parameter $r$ may be fixed, or equipped with a beta distribution. We do not have constraints on the class sizes, which might be as small as zero, or include all vertices, and everything in between. This enables us to test between a uniform (Erd\"os-R\'enyi) random graph with no distinguishable community or the planted bi-section model. The exact bounds for posterior convergence enable us to convert credible sets into confidence sets. Symmetric testing with posterior odds is shown to be consistent.