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

Stochastic block model entropy and broadcasting on trees with survey