The Annals of Statistics

Rate-optimal graphon estimation

Chao Gao, Yu Lu, and Harrison H. Zhou

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Network analysis is becoming one of the most active research areas in statistics. Significant advances have been made recently on developing theories, methodologies and algorithms for analyzing networks. However, there has been little fundamental study on optimal estimation. In this paper, we establish optimal rate of convergence for graphon estimation. For the stochastic block model with $k$ clusters, we show that the optimal rate under the mean squared error is $n^{-1}\log k+k^{2}/n^{2}$. The minimax upper bound improves the existing results in literature through a technique of solving a quadratic equation. When $k\leq\sqrt{n\log n}$, as the number of the cluster $k$ grows, the minimax rate grows slowly with only a logarithmic order $n^{-1}\log k$. A key step to establish the lower bound is to construct a novel subset of the parameter space and then apply Fano’s lemma, from which we see a clear distinction of the nonparametric graphon estimation problem from classical nonparametric regression, due to the lack of identifiability of the order of nodes in exchangeable random graph models. As an immediate application, we consider nonparametric graphon estimation in a Hölder class with smoothness $\alpha$. When the smoothness $\alpha\geq1$, the optimal rate of convergence is $n^{-1}\log n$, independent of $\alpha$, while for $\alpha\in(0,1)$, the rate is $n^{-2\alpha/(\alpha+1)}$, which is, to our surprise, identical to the classical nonparametric rate.

Article information

Ann. Statist. Volume 43, Number 6 (2015), 2624-2652.

Received: October 2014
Revised: June 2015
First available in Project Euclid: 7 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G05: Foundations of stochastic processes

Network graphon stochastic block model nonparametric regression minimax rate


Gao, Chao; Lu, Yu; Zhou, Harrison H. Rate-optimal graphon estimation. Ann. Statist. 43 (2015), no. 6, 2624--2652. doi:10.1214/15-AOS1354.

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Supplemental materials

  • Supplement to “Rate-optimal graphon estimation”. In the supplement, we prove Theorem 2.4, Lemmas 2.1, 4.1, 4.2, 4.3, Proposition 4.2 and Theorem 3.6.