在有噪声的树中找到‘亚当’
Finding Adam in noisy trees
AI总结：
研究在未标记树与埃尔德什 - 雷尼随机图并集情况下找随机均匀附着树根顶点的问题，核心方法是按顶点在高次顶点子图最大连通分量中的乔丹中心性排序，证实猜想并表明该方法在其他噪声模型也适用。
AI中文摘要：
我们考虑这样一个问题：当观察到未标记树与一个埃尔德什 - 雷尼随机图\(\mathbb{G}(n,p)\)的并集时，寻找随机均匀附着树的根顶点。我们证明，只要\(p = o(\log n / n)\)，对于任意\(\varepsilon > 0\)，可以构造一个仅依赖于\(\varepsilon\)而不依赖于\(n\)的大小为\(K(\varepsilon)\)的顶点置信集，使得它包含根的概率至少为\(1 - \varepsilon\)。这证实了克兰和徐（2021年）的一个猜想。我们的方法通过顶点在由高次顶点所张成子图的最大连通分量中的乔丹中心性对顶点进行排序。我们表明相同方法在其他噪声模型中也有效。
英文摘要：
We consider the problem of finding the root vertex of a random uniform attachment tree, when the union of the unlabeled tree and an Erdős-Rényi random graph $\mathbb{G}(n,p)$ is observed. We prove that, as long as $p=o(\log n /n)$, for any $\varepsilon>0$, one can construct a confidence set of vertices of size $K(\varepsilon)$ that depends only on $\varepsilon$ and not on $n$, such that it contains the root with probability at least $1-\varepsilon$. This affirms a conjecture of Crane and Xu (2021). Our approach ranks vertices by their Jordan centrality in the largest component of the subgraph spanned by high-degree vertices. We show that the same approach works in other noise models as well.
