Community detection in sparse networks via Grothendieck's inequality

We present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs. The method is based on Grothendieck's inequality. Unlike the previous uses of this inequality that lead to constant relative accuracy, we achieve arbitrary relative accuracy by leveraging randomness. We illustrate the method with the problem of community detection in sparse networks. Despite much progress in the recent years, almost no rigorous results have been known for totally sparse networks -- those with bounded average degrees. We demonstrate that even in this regime, various natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices. The method is general; it can be applied to a variety of stochastic models of networks and semidefinite programs.