  Most recent developments on the stochastic block model (SBM) 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
[AS15] for linear size communities. The results are three-fold: (i) in the
constant degree regime, an algorithm is developed that requires only a
lower-bound on the relative sizes of the communities and detects communities
with an optimal accuracy scaling for large degrees; (ii) in the regime where
degrees are scaled by $\omega(1)$ (diverging degrees), this is enhanced into a
fully agnostic algorithm that only takes the graph in question and
simultaneously learns the model parameters (including the number of
communities) and detects communities with accuracy $1-o(1)$, with an overall
quasi-linear complexity; (iii) in the logarithmic degree regime, an agnostic
algorithm is developed that learns the parameters and achieves the optimal
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 general SBM with linear size
communities.
