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

Abstract A graphon that is defined on [ 0 , 1 ] d and is Holder ( α ) continuous for some d ⩾ 2 and α ∈ ( 0 , 1 ] can be represented by a graphon on [ 0 , 1 ] that is Holder ( 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.