### Abstract

We consider the problem of identifying underlying community-like structures in graphs. Toward this end, we study the stochastic block model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal is to recover the initial “hidden” partition of [n]. We study semidefinite programming (SDP)-based algorithms in this context. In the regime (formula presented), we show that a certain natural SDP-based algorithm solves the problem of exact recovery in the k-community SBM, with high probability, whenever (formula presented), as long as k= o(log n). This threshold is known to be the information theoretically optimal. We also study the case when (formula presented). In this case however, we achieve recovery guarantees that no longer match the optimal condition (formula presented), thus leaving achieving optimality for this range an open question.

Original language | English (US) |
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Title of host publication | Applied and Numerical Harmonic Analysis |

Publisher | Springer International Publishing |

Pages | 125-162 |

Number of pages | 38 |

Edition | 9783319698014 |

DOIs | |

State | Published - Jan 1 2017 |

### Publication series

Name | Applied and Numerical Harmonic Analysis |
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Number | 9783319698014 |

ISSN (Print) | 2296-5009 |

ISSN (Electronic) | 2296-5017 |

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### Keywords

- Dual certificate
- Graph partitioning
- Random models
- Semidefinite programming
- Stochastic block model

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Applied and Numerical Harmonic Analysis*(9783319698014 ed., pp. 125-162). (Applied and Numerical Harmonic Analysis; No. 9783319698014). Springer International Publishing. https://doi.org/10.1007/978-3-319-69802-1_4