## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 21 (2016), paper no. 21, 24 pp.

### Consistency thresholds for the planted bisection model

Elchanan Mossel, Joe Neeman, and Allan Sly

#### Abstract

The planted bisection model is a random graph model in which the nodes are divided into two equal-sized communities and then edges are added randomly in a way that depends on the community membership. We establish necessary and sufficient conditions for the asymptotic recoverability of the planted bisection in this model. When the bisection is asymptotically recoverable, we give an efficient algorithm that successfully recovers it. We also show that the planted bisection is recoverable asymptotically if and only if with high probability every node belongs to the same community as the majority of its neighbors.

Our algorithm for finding the planted bisection runs in time almost linear in the number of edges. It has three stages: spectral clustering to compute an initial guess, a “replica” stage to get almost every vertex correct, and then some simple local moves to finish the job. An independent work by Abbe, Bandeira, and Hall establishes similar (slightly weaker) results but only in the case of logarithmic average degree.

#### Article information

**Source**

Electron. J. Probab., Volume 21 (2016), paper no. 21, 24 pp.

**Dates**

Received: 12 March 2015

Accepted: 27 January 2016

First available in Project Euclid: 11 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1457706457

**Digital Object Identifier**

doi:10.1214/16-EJP4185

**Mathematical Reviews number (MathSciNet)**

MR3485363

**Zentralblatt MATH identifier**

1336.05117

**Subjects**

Primary: 05C80: Random graphs [See also 60B20]

**Keywords**

stochastic block model planted partition model consistency threshold phase transition community detection random network

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Mossel, Elchanan; Neeman, Joe; Sly, Allan. Consistency thresholds for the planted bisection model. Electron. J. Probab. 21 (2016), paper no. 21, 24 pp. doi:10.1214/16-EJP4185. https://projecteuclid.org/euclid.ejp/1457706457