### Abstract

We consider the problem of reconstructing sparse symmetric block models with two blocks and connection probabilities a/n and b/n for inter- A nd 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.

Original language | English (US) |
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Pages (from-to) | 2211-2256 |

Number of pages | 46 |

Journal | Annals of Applied Probability |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2016 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Applied Probability*,

*26*(4), 2211-2256. https://doi.org/10.1214/15-AAP1145