## The Annals of Statistics

### Consistency of spectral clustering in stochastic block models

#### Abstract

We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\log n$, with $n$ the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical $k$-median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.

#### Article information

Source
Ann. Statist., Volume 43, Number 1 (2015), 215-237.

Dates
First available in Project Euclid: 9 December 2014

https://projecteuclid.org/euclid.aos/1418135620

Digital Object Identifier
doi:10.1214/14-AOS1274

Mathematical Reviews number (MathSciNet)
MR3285605

Zentralblatt MATH identifier
1308.62041

Subjects
Primary: 62F12: Asymptotic properties of estimators

#### Citation

Lei, Jing; Rinaldo, Alessandro. Consistency of spectral clustering in stochastic block models. Ann. Statist. 43 (2015), no. 1, 215--237. doi:10.1214/14-AOS1274. https://projecteuclid.org/euclid.aos/1418135620

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