Open Access
Translator Disclaimer
December 2015 Rate-optimal graphon estimation
Chao Gao, Yu Lu, Harrison H. Zhou
Ann. Statist. 43(6): 2624-2652 (December 2015). DOI: 10.1214/15-AOS1354


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.


Download Citation

Chao Gao. Yu Lu. Harrison H. Zhou. "Rate-optimal graphon estimation." Ann. Statist. 43 (6) 2624 - 2652, December 2015.


Received: 1 October 2014; Revised: 1 June 2015; Published: December 2015
First available in Project Euclid: 7 October 2015

zbMATH: 1332.60050
MathSciNet: MR3405606
Digital Object Identifier: 10.1214/15-AOS1354

Primary: 60G05

Keywords: graphon , Minimax rate , network , Nonparametric regression , Stochastic block model

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 6 • December 2015
Back to Top