  Consider a random graph $G$ of size $N$ constructed according to a
\textit{graphon} $w \, : \, [0,1]^{2} \mapsto [0,1]$ as follows. First embed
$N$ vertices $V = \{v_1, v_2, \ldots, v_N\}$ into the interval $[0,1]$, then
for each $i < j$ add an edge between $v_{i}, v_{j}$ with probability $w(v_{i},
v_{j})$. Given only the adjacency matrix of the graph, we might expect to be
able to approximately reconstruct the permutation $\sigma$ for which
$v_{\sigma(1)} < \ldots < v_{\sigma(N)}$ if $w$ satisfies the following
\textit{linear embedding} property introduced in [Janssen 2019]: for each $x$,
$w(x,y)$ decreases as $y$ moves away from $x$. For a large and non-parametric
family of graphons, we show that (i) the popular spectral seriation algorithm
[Atkins 1998] provides a consistent estimator $\hat{\sigma}$ of $\sigma$, and
(ii) a small amount of post-processing results in an estimate $\tilde{\sigma}$
that converges to $\sigma$ at a nearly-optimal rate, both as $N \rightarrow
\infty$.
