  The limit of the entropy in the stochastic block model (SBM) has been
characterized in the sparse regime for the special case of disassortative
communities [COKPZ17] and for the classical case of assortative communities but
in the dense regime [DAM16]. The problem has not been closed in the classical
sparse and assortative case. This paper establishes the result in this case for
any SNR besides for the interval (1, 3.513). It further gives an approximation
to the limit in this window.
  The result is obtained by expressing the global SBM entropy as an integral of
local tree entropies in a broadcasting on tree model with erasure
side-information. The main technical advancement then relies on showing the
irrelevance of the boundary in such a model, also studied with variants in
[KMS16], [MNS16] and [MX15]. In particular, we establish the uniqueness of the
BP fixed point in the survey model for any SNR above 3.513 or below 1. This
only leaves a narrow region in the plane between SNR and survey strength where
the uniqueness of BP conjectured in these papers remains unproved.
