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

Abstract

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on n nodes, having i.i.d. weights (_u)_u=1^n (possibly heavy-tailed), partitioned into q 2 asymptotically equal-sized clusters. The model parameters are two constants a,b > 0 and the finite second moment of the weights ^(2). Vertices u and v are connected by an edge with probability _u _vna when they are in the same class and with probability _u _vnb 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 ^(2) 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.

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