### Abstract

We study a random graph model called the “stochastic block model” in statistics and the “planted partition model” in theoretical computer science. In its simplest form, this is a random graph with two equal-sized classes of vertices, with a within-class edge probability of q and a between-class edge probability of q′. A striking conjecture of Decelle, Krzkala, Moore and Zdeborová [9], based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if q=a/n and q′=b/n, s=(a−b)/2 and d=(a+b)/2, then Decelle et al. conjectured that it is possible to efficiently cluster in a way correlated with the true partition if s^{2}>d and impossible if s^{2}<d. By comparison, until recently the best-known rigorous result showed that clustering is possible if s^{2}>Cdlnd for sufficiently large C. In a previous work, we proved that indeed it is information theoretically impossible to cluster if s^{2} ≤ d and moreover that it is information theoretically impossible to even estimate the model parameters from the graph when s^{2} < d. Here we prove the rest of the conjecture by providing an efficient algorithm for clustering in a way that is correlated with the true partition when s^{2}>d. A different independent proof of the same result was recently obtained by Massoulié [20].

Original language | English (US) |
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Pages (from-to) | 665-708 |

Number of pages | 44 |

Journal | Combinatorica |

Volume | 38 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2018 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*38*(3), 665-708. https://doi.org/10.1007/s00493-016-3238-8