A semidefinite program for unbalanced multisection in the stochastic block model

We analyze semidefinite programming (SDP) algorithms that exactly recover community structure in graphs generated from the stochastic block model. In this model, a graph is randomly generated on a vertex set that is partitioned into multiple communities of potentially different sizes, where edges are more probable within communities than between communities. We achieve exact recovery of the community structure, up to the information-theoretic limits determined by Abbe and Sandon. By virtue of a semidefinite approach, our algorithms succeed against a semirandom form of the stochastic block model, guaranteeing generalization to scenarios with radically different noise structure.