## The Annals of Applied Probability

### 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 Massoulié (2016) to establish a law of large numbers for local-functionals and $(2)$ that long-range interactions are weak in (power-law) DC-SBMs.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 3002-3027.

Dates
Received: September 2017
First available in Project Euclid: 28 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1535443240

Digital Object Identifier
doi:10.1214/18-AAP1381

Mathematical Reviews number (MathSciNet)
MR3847979

Zentralblatt MATH identifier
06974771

#### Citation

Gulikers, Lennart; Lelarge, Marc; Massoulié, Laurent. An impossibility result for reconstruction in the degree-corrected stochastic block model. Ann. Appl. Probab. 28 (2018), no. 5, 3002--3027. doi:10.1214/18-AAP1381. https://projecteuclid.org/euclid.aoap/1535443240

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