We study the problem of detecting or recovering a planted ranked subgraph from a directed graph, an analog for directed graphs of the well-studied planted dense subgraph model. We suppose that, among a set of $n$ items, there is a subset $S$ of $k$ items having a latent ranking in the form of a permutation $\pi$ of $S$, and that we observe a fraction $p$ of pairwise orderings between elements of $\{1, \dots, n\}$ which agree with $\pi$ with probability $\frac{1}{2} + q$ between elements of $S$ and otherwise are uniformly random. Unlike in the planted dense subgraph and planted clique problems where the community $S$ is distinguished by its unusual density of edges, here the community is only distinguished by the unusual consistency of its pairwise orderings. We establish computational and statistical thresholds for both detecting and recovering such a ranked community. In the log-density setting where $k$, $p$, and $q$ all scale as powers of $n$, we establish the exact thresholds in the associated exponents at which detection and recovery become statistically and computationally feasible. These regimes include a rich variety of behaviors, exhibiting both statistical-computational and detection-recovery gaps. We also give finer-grained results for two extreme cases: (1) $p = 1$, $k = n$, and $q$ small, where a full tournament is observed that is weakly correlated with a global ranking, and (2) $p = 1$, $q = \frac{1}{2}$, and $k$ small, where a small "ordered clique" (totally ordered directed subgraph) is planted in a random tournament.

翻译：我们研究从有向图中检测或恢复一个植入的排序子图的问题，这是对有向图中已得到充分研究的植入稠密子图模型的类比。我们假设，在n个项目集合中，存在一个大小为k的子集S，其具有一个以S的排列π形式存在的潜在排序，并且我们观察到{1, ..., n}中元素间的一部分（比例为p）的成对排序关系：对于S中的元素，这些关系以概率1/2+q与π一致；否则，这些关系是均匀随机的。与植入稠密子图和植入团问题中社区S因其异常的边密度而不同，这里的社区仅因其成对排序关系异常一致而得以区分。我们为检测和恢复此类排序社区建立了计算与统计阈值。在k、p和q均按n的幂次标度的对数密度设定下，我们确定了检测和恢复在统计和计算上变得可行的精确指数阈值。这些区域展现出丰富多样的行为，既有统计-计算间隙，也有检测-恢复间隙。我们还针对两种极端情况给出了更精细的结果：（1）p=1，k=n，且q较小，此时观测到一个与全局排序弱相关的完整锦标赛；（2）p=1，q=1/2，且k较小，此时在一个随机锦标赛中植入了一个小的"有序团"（全序有向子图）。