Uniform linear embeddings of graphons

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.