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

TL;DR

This paper investigates how smooth graphons defined on multi-dimensional spaces can be represented on a one-dimensional domain, establishing the optimal reduction in smoothness and its implications for statistical network analysis.

Contribution

It proves that a graphon on $[0,1]^d$ with Hölder continuity can be represented on $[0,1]$ with reduced smoothness $rac{ ext{original smoothness}}{d}$, and shows this reduction is optimal.

Findings

The smoothness reduction from $ ext{Hölder}(\alpha)$ on $[0,1]^d$ to $ ext{Hölder}(rac{\alpha}{d})$ on $[0,1]$ is optimal.

Examples include a dot product graphon demonstrating the sharpness of the smoothness reduction.

Smoothness assumptions in different dimensions are not equivalent, affecting non-parametric network analysis.

Abstract

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.