A graphon that is defined on $[0,1]^d$ and is H\"older$(\alpha)$ continuous for some $d\ge2$ and $\alpha\in(0,1]$ can be represented by a graphon on $[0,1]$ that is H\"older$(\alpha/d)$ continuous. We give examples that show that this reduction in smoothness to $\alpha/d$ is the best possible, for any $d$ and $\alpha$; for $\alpha=1$, 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.
翻译：以 $[0,1,1 =d$] 定义的图形,为 H\"older$"( ALpha), 以一些 $d\ ge2$ 和 $\ alpha\ in( 0, 1) 继续 美元为 H\ " older$( ALpha) 和 $\ " older$( ALpha), 以 $\ " older$( ALpha), 以 $ d2$ 和 $\ alpha\ in( 0. 0, 1) 为单位, 可以用 $ $ $, $ 0. 0, 1$ 的图形表示, $ $0, 1 $, $, $, $, $, ALphapha\\\\\\\\\ $( $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $, $,
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