Belief propagation, robust reconstruction and optimal recovery of block models

TL;DR

This paper presents an optimal belief propagation-based algorithm for reconstructing sparse symmetric block models, achieving the best possible accuracy under certain conditions, and extends results on robust Ising model reconstruction.

Contribution

Introduces a belief propagation variant that is proven to be optimal for block model reconstruction, surpassing previous methods in accuracy guarantees.

Findings

Algorithm maximizes correct node labeling when $(a-b)^2>C(a+b)$

Proves optimality of the reconstruction algorithm

Provides new results on robust Ising model reconstruction

Abstract

We consider the problem of reconstructing sparse symmetric block models with two blocks and connection probabilities $a/n$ and $b/n$ for inter- and intra-block edge probabilities, respectively. It was recently shown that one can do better than a random guess if and only if $(a-b)^2>2(a+b)$. Using a variant of belief propagation, we give a reconstruction algorithm that is optimal in the sense that if $(a-b)^2>C(a+b)$ for some constant $C$ then our algorithm maximizes the fraction of the nodes labeled correctly. Ours is the only algorithm proven to achieve the optimal fraction of nodes labeled correctly. Along the way, we prove some results of independent interest regarding robust reconstruction for the Ising model on regular and Poisson trees.