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 (possibly heavy-tailed), partitioned into asymptotically equal-sized clusters. The model parameters are two constants and the finite second moment of the weights . Vertices and are connected by an edge with probability when they are in the same class and with probability otherwise. We prove that it is information-theoretically impossible to estimate the clusters in a way positively correlated with the true community structure when . 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.