An Impossibility Result for Reconstruction in a Degree-Corrected Planted-Partition Model

We consider a Degree-Corrected Planted-Partition model: a random graph on n nodes with two equal-sized clusters. The model parameters are two constants a, b > 0 and an i.i.d. sequence of weights (φu) n u=1, with finite second moment Φ . Vertices u and v are joined by an edge with probability φuφv n a when they are in the same class and with probability φuφv n b otherwise. We prove that it is information-theoretically impossible to estimate the spins in a way positively correlated with the true community structure when (a − b)Φ ≤ 2(a+ b). A by-product of our proof is a precise coupling-result for local-neighbourhoods in Degree-Corrected Planted-Partition models, which could be of independent interest.