We study the problem of detecting and recovering a planted spanning tree $M_n^*$ hidden within a complete, randomly weighted graph $G_n$. Specifically, each edge $e$ has a non-negative weight drawn independently from $P_n$ if $e \in M_n^*$ and from $Q_n$ otherwise, where $P_n \equiv P$ is fixed and $Q_n$ scales with $n$ such that its density at the origin satisfies $\lim_{n\to\infty} n Q'_n(0)=1.$ We consider two representative cases: when $M_n^*$ is either a uniform spanning tree or a uniform Hamiltonian path. We analyze the recovery performance of the minimum spanning tree (MST) algorithm and derive a fixed-point equation that characterizes the asymptotic fraction of edges in $M_n^*$ successfully recovered by the MST as $n \to \infty.$ Furthermore, we establish the asymptotic mean weight of the MST, extending Frieze's $\zeta(3)$ result to the planted model. Leveraging this result, we design an efficient test based on the MST weight and show that it can distinguish the planted model from the unplanted model with vanishing testing error as $n \to \infty.$ Our analysis relies on an asymptotic characterization of the local structure of the planted model, employing the framework of local weak convergence.

翻译：我们研究在完全随机加权图$G_n$中检测并恢复隐藏的植入生成树$M_n^*$的问题。具体而言，每条边$e$的权重独立采样自$P_n$（若$e \in M_n^*$）或$Q_n$（否则），其中$P_n \equiv P$为固定分布，而$Q_n$随$n$变化且满足其在原点处的密度条件$\lim_{n\to\infty} n Q'_n(0)=1$。我们考虑两种典型情况：$M_n^*$为均匀生成树或均匀哈密顿路径。我们分析了最小生成树（MST）算法的恢复性能，推导出刻画MST算法在$n \to \infty$时成功恢复$M_n^*$中边比例的渐近固定点方程。此外，我们建立了MST渐近平均权重的表达式，将Frieze的$\zeta(3)$结果推广至植入模型。基于此结果，我们设计了一种基于MST权重的有效检验方法，并证明该方法能以趋于零的检验误差区分植入模型与无植入模型。我们的分析依赖于采用局部弱收敛框架对植入模型局部结构进行的渐近刻画。