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 $d\ge2$ and $α\in(0,1]$ 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.

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