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 $N \rightarrow \infty$.

翻译：随机图形 $G$, 大小為 $N美元, 以 clookit{ magon} 美元建造 : \, \, [0, 1,\\2} \ mpsto [0, 1] 美元。 首先在间隔( $V= v_ 1, v_ 2, v_ 2, v_ 2) 中嵌入 $美元, v_ N$, 然后对于每个美元 < j$ < j$ >, 加上 $@i}, v ⁇ j} 美元之间的边际。 仅根据图表的对齐度矩阵表, 我们也许能够大致重建 $\ gmad$ = = {v_ 1, v_ 2, v_, eldots, v_ 美元在间隔( $w$) 满足 [Janssen 2019] 中引入的以下文本{线性嵌入 : 每美元, $x, $x, 美元递减 美元, 美元, 美元, 在1998 美元中, a clasmal a cal- clasmax, a cal a cal a cal a cal a cal $.