Probability Theory and Related Fields

, Volume 174, Issue 3–4, pp 1033–1090 | Cite as

Optimal graphon estimation in cut distance

  • Olga KloppEmail author
  • Nicolas Verzelen


Consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a W-random graph. We establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons. Surprisingly, our results imply that, from the minimax point of view, the raw data, that is, the adjacency matrix of the observed graph, is already optimal and more involved procedures cannot improve the convergence rates for this metric. This phenomenon contrasts with optimal rates of convergence with respect to other classical distances for graphons such as the \(l_1\) or \(l_2\) metrics.


Inhomogeneous random graph Graphon W-random graphs Networks Stochastic block model Cut distance 

Mathematics Subject Classification

Primary 62G05 Secondary 60C05 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.ESSEC Business SchoolCergyFrance
  2. 2.CRESTPalaiseauFrance
  3. 3.INRA, UMR 729 MISTEAMontpellierFrance

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