Stochastic block model entropy and broadcasting on trees with survey

TL;DR

This paper characterizes the entropy limit of the stochastic block model in the sparse, assortative case for most signal-to-noise ratios, using a novel approach involving local tree entropies and belief propagation fixed points.

Contribution

It extends the understanding of SBM entropy limits to a broader parameter range and introduces a new integral approach based on broadcasting on trees with side information.

Findings

Established entropy limits for the sparse, assortative SBM outside a specific SNR interval.

Provided an approximation to the entropy limit within the unresolved SNR window.

Proved the uniqueness of belief propagation fixed points for certain SNR ranges.

Abstract

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.