Emmanuel Abbe, Colin Sandon
The stochastic block model (SBM) has recently gathered significant attention due to new threshold phenomena. However, most developments rely on the knowledge of the model parameters, or at least on the number of communities. This paper introduces efficient algorithms that do not require such knowledge and yet achieve the optimal information-theoretic tradeoffs identified in Abbe-Sandon FOCS15. In the constant degree regime, an algorithm is developed that requires only a lower-bound on the relative sizes of the communities and achieves the optimal accuracy scaling for large degrees. This lower-bound requirement is removed for the regime of arbitrarily slowly diverging degrees, and the model parameters are learned efficiently. For the logarithmic degree regime, this is further enhanced into a fully agnostic algorithm that achieves the CH-limit for exact recovery in quasi-linear time. These provide the first algorithms affording efficiency, universality and information-theoretic optimality for strong and weak consistency in the SBM.