The Annals of Applied Probability

Belief propagation, robust reconstruction and optimal recovery of block models

Elchanan Mossel, Joe Neeman, and Allan Sly

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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

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

Received: September 2014
Revised: April 2015
First available in Project Euclid: 1 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
Secondary: 91D30: Social networks

Stochastic block model unsupervised learning belief propagation robust reconstruction


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.

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