Exact recovery in the binary stochastic block model with binary side information

We consider a generalization of the community detection problem, where for inference we have access to an additional observation that carries information (side information) about the label of each node. We study the effect of the quality of side information on the exact recovery phase transition of the binary symmetric stochastic block model by passing the true label through a binary symmetric channel (BSC) with crossover probability α. We show that when α is constant or approaching zero such that log(1-α/α) = o(log(n)), side information does not help exact recovery and the phase transition does not change. On the other hand, when log(1-α/α) = O(log(n)), we show that side information helps exact recovery. We provide necessary and sufficient conditions that are tight for different asymptotic regimes of α. An efficient algorithm that incorporates the effect of side information is proposed that uses a partial recovery algorithm combined with a local improvement procedure. Sufficient conditions are derived for exact recovery under this efficient algorithm.