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

01/19/2021

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by   Svante Janson, et al.

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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.