Exact Recovery in the Stochastic Block Model

The stochastic block model with two communities, or equivalently the planted bisection model, is a popular model of random graph exhibiting a cluster behavior. In the symmetric case, the graph has two equally sized clusters and vertices connect with probability p within clusters and q across clusters. In the past two decades, a large body of literature in statistics and computer science has focused on providing lower bounds on the scaling of vertical bar p - q vertical bar to ensure exact recovery. In this paper, we identify a sharp threshold phenomenon for exact recovery: if alpha = pn/log(n) and beta = qn/log(n) are constant (with alpha > beta), recovering the communities with high probability is possible if (alpha + beta/2)- root alpha beta> 1 and is impossible if (alpha + beta/2)- root alpha beta < 1. In particular, this improves the existing bounds. This also sets a new line of sight for efficient clustering algorithms. While maximum likelihood (ML) achieves the optimal threshold (by definition), it is in the worst case NP-hard. This paper proposes an efficient algorithm based on a semidefinite programming relaxation of ML, which is proved to succeed in recovering the communities close to the threshold, while numerical experiments suggest that it may achieve the threshold. An efficient algorithm that succeeds all the way down to the threshold is also obtained using a partial recovery algorithm combined with a local improvement procedure.