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 asymptotically equal-sized clusters. The model parameters are two constants $a,b > 0$ and an i.i.d. sequence of weights $(\phi_u)_{u=1}^n$, with finite second moment $\Phi^{(2)}$. Vertices $u$ and $v$ are joined by an edge with probability $\frac{\phi_u \phi_v}{n}a$ when they are in the same class and with probability $\frac{\phi_u \phi_v}{n}b$ otherwise. We prove that it is information-theoretically impossible to estimate the spins in a way positively correlated with the true community structure when $(a-b)^2 \Phi^{(2)} \leq 2(a+b)$. 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.