Multisection in the Stochastic Block Model using Semidefinite Programming

Naman Agarwal; Afonso S. Bandeira; Konstantinos Koiliaris; Alexandra Kolla

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arXiv · 1507.02323

Multisection in the Stochastic Block Model using Semidefinite Programming

Naman Agarwal·Afonso S. Bandeira·Konstantinos Koiliaris·Alexandra Kolla

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Abstract

We consider the problem of identifying underlying community-like structures in graphs. Towards 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 {\em 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.

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BibTeX RIS

Naman Agarwal, Afonso S. Bandeira, Konstantinos Koiliaris, Alexandra Kolla. 2015-07-08. Multisection in the Stochastic Block Model using Semidefinite Programming. https://arxiv.org/abs/1507.02323

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