Multisection in the Stochastic Block Model using Semidefinite Programming

Identifying underlying structure in graphs is a primitive question for scientists: can existing communities be located in a large graph? Is it possible to partition the vertices of a graph into strongly connected clusters? Several of these questions have been shown to be hard to answer, even approximately, so instead of looking for worst-case guarantees attention has shifted towards average-case analyses. In order to study such questions, the usual approach is to consider a random [McS01] or a semi-random [FK01, MMV14] generative model of graphs, and use it as a benchmark to test existing algorithms or to develop new ones. With respect to identifying underlying community structure, the Stochastic Block Model (SBM) (or planted partition model) has, in recent times, been one of the most popular choices. Its growing popularity is largely due to the fact that its structure is simple to describe, but at the same time it has interesting and involved phase transition properties which have only recently been discovered ([DKMZ11, MNS12, MNS13, ABH14, CX14, MNS14b, HWX14, HWX15, AS15, Ban15]). In this paper we consider the SBM on k-communities defined as follows.