The Annals of Applied Probability

An impossibility result for reconstruction in the degree-corrected stochastic block model

Lennart Gulikers, Marc Lelarge, and Laurent Massoulié

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91D30: Social networks 05C80: Random graphs [See also 60B20]
Secondary: 68W40: Analysis of algorithms [See also 68Q25] 91C20: Clustering [See also 62H30]

Social and information networks random graphs degree-corrected stochastic block model spectral algorithm machine learning


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.

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