Optimal reconstruction of general sparse stochastic block models

This paper is motivated by the reconstruction problem on the sparse stochastic block model. Mossel, et. al. proved that a reconstruction algorithm that recovers an optimal fraction of the communities in the symmetric, 2-community case. The main contribution of their proof is to show that when the signal to noise ratio is sufficiently large, in particular $\lambda^2d > C$, the reconstruction accuracy for a broadcast process on a tree with or without noise on the leaves is asymptotically the same. This paper will generalize their results, including the main step, to a general class of the sparse stochastic block model with any number of communities that are not necessarily symmetric, proving that an algorithm closely related to Belief Propagation recovers an optimal fraction of community labels.