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

• Published in: Examples and counterexamples Vol. 1; p. 100011
• Main Authors: Janson, Svante, Olhede, Sofia
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2021; Elsevier
• Subjects: Graphon; Non-parametric statistical network inference; Smooth graphon; 62G05; Non-parametric statistical network inference; Smooth graphon; Graphon; 05C80
• ISSN: 2666-657X; 2666-657X
• DOI: 10.1016/j.exco.2021.100011

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.