Multisection in the Stochastic Block Model Using Semidefinite Programming

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


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 \(p = \frac {\alpha \log (m)}{m}\) and \(q = \frac {\beta \log (m)}{m}\), we show that a certain natural SDP-based algorithm solves the problem of exact recovery in the k-community SBM, with high probability, whenever \(\sqrt {\alpha } - \sqrt {\beta } > \sqrt {1}\), as long as \(k=o(\log n)\). This threshold is known to be the information theoretically optimal. We also study the case when \(k=\theta (\log (n))\). In this case however, we achieve recovery guarantees that no longer match the optimal condition \(\sqrt {\alpha } - \sqrt {\beta } > \sqrt {1}\), thus leaving achieving optimality for this range an open question.


Graph partitioning Random models Stochastic block model Semidefinite programming Dual certificate 



Most of the work presented in this paper was conducted while ASB was at Princeton University and partly conducted while ASB was at the Massachusetts Institute of Technology. ASB acknowledges support from AFOSR Grant No. FA9550-12-1-0317, NSF DMS-1317308, NSF DMS-1712730, and NSF DMS-1719545.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Department of Mathematics, Courant Institute of Mathematical Sciences and Center for Data ScienceNew York UniversityNew YorkUSA
  3. 3.University of Illinois Urbana - ChampaignUrbanaUSA

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