  We study bipartite community detection in networks, or more generally the
network biclustering problem. We present a fast two-stage procedure based on
spectral initialization followed by the application of a pseudo-likelihood
classifier twice. Under mild regularity conditions, we establish the weak
consistency of the procedure (i.e., the convergence of the misclassification
rate to zero) under a general bipartite stochastic block model. We show that
the procedure is optimal in the sense that it achieves the optimal convergence
rate that is achievable by a biclustering oracle, adaptively over the whole
class, up to constants. This is further formalized by deriving a minimax lower
bound over a class of biclustering problems. The optimal rate we obtain
sharpens some of the existing results and generalizes others to a wide regime
of average degree growth, from sparse networks with average degrees growing
arbitrarily slowly to fairly dense networks with average degrees of order
$\sqrt{n}$. As a special case, we recover the known exact recovery threshold in
the $\log n$ regime of sparsity. To obtain the consistency result, as part of
the provable version of the algorithm, we introduce a sub-block partitioning
scheme that is also computationally attractive, allowing for distributed
implementation of the algorithm without sacrificing optimality. The provable
algorithm is derived from a general class of pseudo-likelihood biclustering
algorithms that employ simple EM type updates. We show the effectiveness of
this general class by numerical simulations.
