  We study the stochastic block model which is often used to model community
structures and study community-detection algorithms. We consider the case of
two blocks in regard to its largest connected component and largest biconnected
component, respectively. We are especially interested in the distributions of
their sizes including the tails down to probabilities smaller than $10^{-800}$.
For this purpose we use sophisticated Markov chain Monte Carlo simulations to
sample graphs from the stochastic block model ensemble. We use this data to
study the large-deviation rate function and conjecture that the large-deviation
principle holds. Further we compare the distribution to the well known
Erd\H{o}s-R\'{e}nyi ensemble, where we notice subtle differences at and above
the percolation threshold.
