Compressed Sensing and its Applications pp 125-162 | Cite as

# Multisection in the Stochastic Block Model Using Semidefinite Programming

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## 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 \(p = \frac {\alpha \log (m)}{m}\) 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.

## Keywords

Graph partitioning Random models Stochastic block model Semidefinite programming Dual certificate## Notes

### Acknowledgements

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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