Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]?

A graphon that is defined on [0,1]d and is Hölder(α) continuous for some d⩾2 and α∈(0,1] can be represented by a graphon on [0,1] that is Hölder(α/d) continuous. We give examples that show that this reduction in smoothness to α/d is the best possible, for any d and α; for α=1, the example is a dot p...

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Bibliographic Details
Published in:Examples and counterexamples 2021-11, Vol.1, p.100011, Article 100011
Main Authors: Janson, Svante, Olhede, Sofia
Format: Article
Language:English
Subjects:
Graphon
Non-parametric statistical network inference
Smooth graphon
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Summary:A graphon that is defined on [0,1]d and is Hölder(α) continuous for some d⩾2 and α∈(0,1] can be represented by a graphon on [0,1] that is Hölder(α/d) continuous. We give examples that show that this reduction in smoothness to α/d is the best possible, for any d and α; for α=1, the example is a dot product graphon and shows that the reduction is the best possible even for graphons that are polynomials. A motivation for studying the smoothness of graphon functions is that this represents a key assumption in non-parametric statistical network analysis. Our examples show that making a smoothness assumption in a particular dimension is not equivalent to making it in any other latent dimension.
ISSN:2666-657X
2666-657X
DOI:10.1016/j.exco.2021.100011