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October 2018 An impossibility result for reconstruction in the degree-corrected stochastic block model
Lennart Gulikers, Marc Lelarge, Laurent Massoulié
Ann. Appl. Probab. 28(5): 3002-3027 (October 2018). DOI: 10.1214/18-AAP1381

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.

Citation

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Lennart Gulikers. Marc Lelarge. Laurent Massoulié. "An impossibility result for reconstruction in the degree-corrected stochastic block model." Ann. Appl. Probab. 28 (5) 3002 - 3027, October 2018. https://doi.org/10.1214/18-AAP1381

Information

Received: 1 September 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974771
MathSciNet: MR3847979
Digital Object Identifier: 10.1214/18-AAP1381

Subjects:
Primary: 05C80 , 91D30
Secondary: 68W40 , 91C20

Keywords: degree-corrected stochastic block model , machine learning , Random graphs , Social and information networks , spectral algorithm

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October 2018
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