Multisection in the stochastic block model using semidefinite programming

Naman Agarwal, Afonso Bandeira, Konstantinos Koiliaris, Alexandra Kolla

Research output: Chapter in Book/Report/Conference proceedingChapter

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 languageEnglish (US)
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Pages125-162
Number of pages38
Edition9783319698014
DOIs
StatePublished - Jan 1 2017

Publication series

NameApplied and Numerical Harmonic Analysis
Number9783319698014
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

Fingerprint

Semidefinite Programming
Recovery
Model
Optimality
Partition
Graph in graph theory
Range of data
Community

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Agarwal, N., Bandeira, A., Koiliaris, K., & Kolla, A. (2017). Multisection in the stochastic block model using semidefinite programming. In 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

Multisection in the stochastic block model using semidefinite programming. / Agarwal, Naman; Bandeira, Afonso; Koiliaris, Konstantinos; Kolla, Alexandra.

Applied and Numerical Harmonic Analysis. 9783319698014. ed. Springer International Publishing, 2017. p. 125-162 (Applied and Numerical Harmonic Analysis; No. 9783319698014).

Research output: Chapter in Book/Report/Conference proceedingChapter

Agarwal, N, Bandeira, A, Koiliaris, K & Kolla, A 2017, Multisection in the stochastic block model using semidefinite programming. in Applied and Numerical Harmonic Analysis. 9783319698014 edn, Applied and Numerical Harmonic Analysis, no. 9783319698014, Springer International Publishing, pp. 125-162. https://doi.org/10.1007/978-3-319-69802-1_4
Agarwal N, Bandeira A, Koiliaris K, Kolla A. Multisection in the stochastic block model using semidefinite programming. In Applied and Numerical Harmonic Analysis. 9783319698014 ed. Springer International Publishing. 2017. p. 125-162. (Applied and Numerical Harmonic Analysis; 9783319698014). https://doi.org/10.1007/978-3-319-69802-1_4
Agarwal, Naman ; Bandeira, Afonso ; Koiliaris, Konstantinos ; Kolla, Alexandra. / Multisection in the stochastic block model using semidefinite programming. Applied and Numerical Harmonic Analysis. 9783319698014. ed. Springer International Publishing, 2017. pp. 125-162 (Applied and Numerical Harmonic Analysis; 9783319698014).
@inbook{4e8afb8f97144fcfad93807c72df04a6,
title = "Multisection in the stochastic block model using semidefinite programming",
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.",
keywords = "Dual certificate, Graph partitioning, Random models, Semidefinite programming, Stochastic block model",
author = "Naman Agarwal and Afonso Bandeira and Konstantinos Koiliaris and Alexandra Kolla",
year = "2017",
month = "1",
day = "1",
doi = "10.1007/978-3-319-69802-1_4",
language = "English (US)",
series = "Applied and Numerical Harmonic Analysis",
publisher = "Springer International Publishing",
number = "9783319698014",
pages = "125--162",
booktitle = "Applied and Numerical Harmonic Analysis",
edition = "9783319698014",

}

TY - CHAP

T1 - Multisection in the stochastic block model using semidefinite programming

AU - Agarwal, Naman

AU - Bandeira, Afonso

AU - Koiliaris, Konstantinos

AU - Kolla, Alexandra

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

AB - 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.

KW - Dual certificate

KW - Graph partitioning

KW - Random models

KW - Semidefinite programming

KW - Stochastic block model

UR - http://www.scopus.com/inward/record.url?scp=85042476724&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042476724&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-69802-1_4

DO - 10.1007/978-3-319-69802-1_4

M3 - Chapter

T3 - Applied and Numerical Harmonic Analysis

SP - 125

EP - 162

BT - Applied and Numerical Harmonic Analysis

PB - Springer International Publishing

ER -