An impossibility result for reconstruction in the degree-corrected stochastic block model

Abstract

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on $n$ nodes, having i.i.d. weights $(\phi_{u})_{u=1}^{n}$ (possibly heavy-tailed), partitioned into $q\geq2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b>0$ and the finite second moment of the weights $\Phi^{(2)}$. Vertices $u$ and $v$ are connected 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 clusters in a way positively correlated with the true community structure when $(a-b)^{2}\Phi^{(2)}\leq q(a+b)$. As by-products of our proof we obtain $(1)$ a precise coupling result for local neighbourhoods in DC-SBMs, that we use in Gulikers, Lelarge and Massoulie (2016) to establish a law of large numbers for local-functionals and $(2)$ that long-range interactions are weak in (power-law) DC-SBMs.