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

Examples and Counterexamples

Sofia Olhede¹  Svante Janson² 

[1] Corresponding author.;Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden;

关键词: Graphon;    Smooth graphon;    Non-parametric statistical network inference;   

DOI  :  

来源: DOAJ

【 摘 要 】

A graphon that is defined on [0,1]dand is Hölder(α)continuous for some d⩾2and 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/dis 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.

【 授权许可】

Unknown