## The Annals of Applied Probability

### Belief propagation, robust reconstruction and optimal recovery of block models

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 4 (2016), 2211-2256.

Dates
Revised: April 2015
First available in Project Euclid: 1 September 2016

https://projecteuclid.org/euclid.aoap/1472745457

Digital Object Identifier
doi:10.1214/15-AAP1145

Mathematical Reviews number (MathSciNet)
MR3543895

Zentralblatt MATH identifier
1350.05154

#### Citation

Mossel, Elchanan; Neeman, Joe; Sly, Allan. Belief propagation, robust reconstruction and optimal recovery of block models. Ann. Appl. Probab. 26 (2016), no. 4, 2211--2256. doi:10.1214/15-AAP1145. https://projecteuclid.org/euclid.aoap/1472745457

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