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 H & ouml;lder(a) continuous for some d 2 and a is an element of (0, 1] can be represented by a graphon on [0, 1] that is H & ouml;lder(a/d) continuous. We give examples that show that this reduction in smoothness to a/d is the best possible, for any d and a; for a = 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.