Uniform linear embeddings of graphons

In a random graph with a spatial embedding, the probability of linking to a particular vertex $v$ decreases with distance, but the rate of decrease may depend on the particular vertex $v$, and on the direction in which the distance increases. In this article, we consider the question when the embedding can be chosen to be uniform, so the probability of a link between two vertices depends only on the distance between them. We give necessary and sufficient conditions for the existence of a uniform linear embedding (embedding into a one-dimensional space) for spatial random graphs where the link probability can attain only a finite number of values.