Open Access
2022 Uniform estimation in stochastic block models is slow
Ismaël Castillo, Peter Orbanz
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Electron. J. Statist. 16(1): 2947-3000 (2022). DOI: 10.1214/22-EJS2014


We explicitly quantify the empirically observed phenomenon that estimation under a stochastic block model (SBM) is hard if the model contains classes that are similar. More precisely, we consider estimation of certain functionals of random graphs generated by a SBM. The SBM may or may not be sparse, and the number of classes may be fixed or grow with the number of vertices. Minimax lower and upper bounds of estimation along specific submodels are derived. The results are nonasymptotic and imply that uniform estimation of a single connectivity parameter is much slower than the expected asymptotic pointwise rate. Specifically, the uniform quadratic rate does not scale as the number of edges, but only as the number of vertices. The lower bounds are local around any possible SBM. An analogous result is derived for functionals of a class of smooth graphons.

Funding Statement

I. C.’s work is supported by ANR grant ANR-17-CE40-0001 (BASICS).


I. C. is very grateful for the hospitality of Columbia’s statistics department, where parts of this work where carried out.


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Ismaël Castillo. Peter Orbanz. "Uniform estimation in stochastic block models is slow." Electron. J. Statist. 16 (1) 2947 - 3000, 2022.


Received: 1 September 2021; Published: 2022
First available in Project Euclid: 4 May 2022

MathSciNet: MR4416678
zbMATH: 1493.62241
Digital Object Identifier: 10.1214/22-EJS2014

Primary: 62G20

Keywords: graphon model , Minimax rates , semiparametric estimation of functionals , spectral clustering , stochastic blockmodel

Vol.16 • No. 1 • 2022
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