  In the present paper we study a sparse stochastic network enabled with a
block structure. The popular Stochastic Block Model (SBM) and the Degree
Corrected Block Model (DCBM) address sparsity by placing an upper bound on the
maximum probability of connections between any pair of nodes. As a result,
sparsity describes only the behavior of network as a whole, without
distinguishing between the block-dependent sparsity patterns. To the best of
our knowledge, the recently introduced Popularity Adjusted Block Model (PABM)
is the only block model that allows to introduce a {\it structural sparsity}
where some probabilities of connections are identically equal to zero while the
rest of them remain above a certain threshold. The latter presents a more
nuanced view of the network.
