Uniform linear embeddings of graphons

TL;DR

This paper investigates conditions under which a linear embedding of a graphon can be transformed into a uniform linear embedding by changing the underlying probability space, providing necessary and sufficient conditions for finite-valued graphons.

Contribution

It establishes necessary and sufficient conditions for the existence of uniform linear embeddings of finite-valued graphons, addressing a transformation problem in graphon theory.

Findings

Uniform linear embeddings exist under specific conditions for finite-valued graphons.

Most general graphons do not admit a uniform linear embedding.

The paper characterizes when such transformations are possible.

Abstract

Let $w:[0,1]^2\rightarrow [0,1]$ be a symmetric function, and consider the random process $G(n,w)$, where vertices are chosen from $[0,1]$ uniformly at random, and $w$ governs the edge formation probability. Such a random graph is said to have a linear embedding, if the probability of linking to a particular vertex $v$ decreases with distance. The rate of decrease, in general, depends on the particular vertex $v$. A linear embedding is called uniform if the probability of a link between two vertices depends only on the distance between them. In this article, we consider the question whether it is possible to "transform" a linear embedding to a uniform one, through replacing the uniform probability space $[0,1]$ with a suitable probability space on ${\mathbb R}$. We give necessary and sufficient conditions for the existence of a uniform linear embedding for random graphs where $w$ attains only a finite number of values. Our findings show that for a general $w$ the answer is negative in most cases.