  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.
