## The Annals of Statistics

- Ann. Statist.
- Volume 46, Number 1 (2018), 149-179.

### On semidefinite relaxations for the block model

Arash A. Amini and Elizaveta Levina

#### Abstract

The stochastic block model (SBM) is a popular tool for community detection in networks, but fitting it by maximum likelihood (MLE) involves a computationally infeasible optimization problem. We propose a new semidefinite programming (SDP) solution to the problem of fitting the SBM, derived as a relaxation of the MLE. We put ours and previously proposed SDPs in a unified framework, as relaxations of the MLE over various subclasses of the SBM, which also reveals a connection to the well-known problem of sparse PCA. Our main relaxation, which we call SDP-1, is tighter than other recently proposed SDP relaxations, and thus previously established theoretical guarantees carry over. However, we show that SDP-1 exactly recovers true communities over a wider class of SBMs than those covered by current results. In particular, the assumption of strong assortativity of the SBM, implicit in consistency conditions for previously proposed SDPs, can be relaxed to weak assortativity for our approach, thus significantly broadening the class of SBMs covered by the consistency results. We also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed SDP approaches and not an artifact of the proofs. Our analysis of SDPs is based on primal-dual witness constructions, which provides some insight into the nature of the solutions of various SDPs. In particular, we show how to combine features from SDP-1 and already available SDPs to achieve the most flexibility in terms of both assortativity and block-size constraints, as our relaxation has the tendency to produce communities of similar sizes. This tendency makes it the ideal tool for fitting network histograms, a method gaining popularity in the graphon estimation literature, as we illustrate on an example of a social networks of dolphins. We also provide empirical evidence that SDPs outperform spectral methods for fitting SBMs with a large number of blocks.

#### Article information

**Source**

Ann. Statist., Volume 46, Number 1 (2018), 149-179.

**Dates**

Received: January 2016

Revised: November 2016

First available in Project Euclid: 22 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1519268427

**Digital Object Identifier**

doi:10.1214/17-AOS1545

**Mathematical Reviews number (MathSciNet)**

MR3766949

**Zentralblatt MATH identifier**

06865108

**Subjects**

Primary: 62G20: Asymptotic properties 90C22: Semidefinite programming 62H99: None of the above, but in this section

**Keywords**

Community detection network semidefinite programming stochastic block model

#### Citation

Amini, Arash A.; Levina, Elizaveta. On semidefinite relaxations for the block model. Ann. Statist. 46 (2018), no. 1, 149--179. doi:10.1214/17-AOS1545. https://projecteuclid.org/euclid.aos/1519268427

#### Supplemental materials

- Supplement to “On semidefinite relaxations for the block model”. This supplement contains proofs of some of the results.Digital Object Identifier: doi:10.1214/17-AOS1545SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.