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August 2021 Consistent nonparametric estimation for heavy-tailed sparse graphs
Christian Borgs, Jennifer T. Chayes, Henry Cohn, Shirshendu Ganguly
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Ann. Statist. 49(4): 1904-1930 (August 2021). DOI: 10.1214/20-AOS1985


We study graphons as a nonparametric generalization of stochastic block models, and show how to obtain compactly represented estimators for sparse networks in this framework. In contrast to previous work, we relax the usual boundedness assumption for the generating graphon and instead assume only integrability, so that we can handle networks that have long tails in their degree distributions. We also relax the usual assumption that the graphon is defined on the unit interval, to allow latent position graphs based on more general spaces.

We analyze three algorithms. The first is a least squares algorithm, which gives a consistent estimator for all square-integrable graphons, with errors expressed in terms of the best possible stochastic block model approximation. Next, we analyze an algorithm based on the cut norm, which works for all integrable graphons. Finally, we show that clustering based on degrees works whenever the underlying degree distribution is atomless.

Funding Statement

Shirshendu Ganguly was supported by an internship at Microsoft Research New England.


We thank David Choi, Sofia Olhede and Patrick Wolfe for initially introducing us to applications of graphons in machine learning of networks and, in particular, to the problem of graphon estimation. We are indebted to Sofia Olhede and Patrick Wolfe for numerous helpful discussions in the early stages of this work, to Alessandro Rinaldo for providing valuable feedback on our paper, and to the anonymous referees for their many helpful comments.


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Christian Borgs. Jennifer T. Chayes. Henry Cohn. Shirshendu Ganguly. "Consistent nonparametric estimation for heavy-tailed sparse graphs." Ann. Statist. 49 (4) 1904 - 1930, August 2021.


Received: 1 July 2017; Revised: 1 May 2020; Published: August 2021
First available in Project Euclid: 29 September 2021

Digital Object Identifier: 10.1214/20-AOS1985

Primary: 62G20
Secondary: 05C80 , 62H30

Keywords: estimation , graphons , Sparse networks

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 4 • August 2021
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