  Consider a random graph process where vertices are chosen from the interval
$[0,1]$, and edges are chosen independently at random, but so that, for a given
vertex $x$, the probability that there is an edge to a vertex $y$ decreases as
the distance between $x$ and $y$ increases. We call this a random graph with a
linear embedding. We define a new graph parameter $\Gamma^*$, which aims to
measure the similarity of the graph to an instance of a random graph with a
linear embedding. For a graph $G$, $\Gamma^*(G)=0$ if and only if $G$ is a unit
interval graph, and thus a deterministic example of a graph with a linear
embedding. We show that the behaviour of $\Gamma^*$ is consistent with the
notion of convergence as defined in the theory of dense graph limits. In this
theory, graph sequences converge to a symmetric, measurable function on
$[0,1]^2$. We define an operator $\Gamma$ which applies to graph limits, and
which assumes the value zero precisely for graph limits that have a linear
embedding. We show that, if a graph sequence $\{ G_n\}$ converges to a function
$w$, then $\{ \Gamma^*(G_n)\}$ converges as well. Moreover, there exists a
function $w^*$ arbitrarily close to $w$ under the box distance, so that
$\lim_{n\rightarrow \infty}\Gamma^*(G_n)$ is arbitrarily close to $\Gamma
(w^*)$.
