Section: New Results

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

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on n nodes, having i.i.d. weights ${\left({\phi }_u\right)}_{u=1}^n$ (possibly heavy-tailed), partitioned into $q\ge 2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b>0$ and the finite second moment of the weights ${\Phi }^{\left(2\right)}$. Vertices $u$ and $v$ are connected by an edge with probability $({\phi }_u{\phi }_v/n)a$ when they are in the same class and with probability $({\phi }_u{\phi }_v/n)b$ otherwise. We prove that it is information-theoretically impossible to estimate the clusters in a way positively correlated with the true community structure when $\left(a-b\right)2{\Phi }^{\left(2\right)}\le q(a+b)$. As by-products of our proof we obtain (1) a precise coupling result for local neighbourhoods in DC-SBM's, that we use in a follow up paper [Gulikers et al., 2017] to establish a law of large numbers for local-functionals and (2) that long-range interactions are weak in (power-law) DC-SBM's.