We consider the problem of finding the initial vertex (Adam) in a Barab\'asi--Albert tree process $(\mathcal{T}(n) : n \geq 1)$ at large times. More precisely, given $ \varepsilon>0$, one wants to output a subset $ \mathcal{P}_{ \varepsilon}(n)$ of vertices of $ \mathcal{T}(n)$ so that the initial vertex belongs to $ \mathcal{P}_ \varepsilon(n)$ with probability at least $1- \varepsilon$ when $n$ is large. It has been shown by Bubeck, Devroye & Lugosi, refined later by Banerjee & Huang, that one needs to output at least $ \varepsilon^{-1 + o(1)}$ and at most $\varepsilon^{-2 + o(1)}$ vertices to succeed. We prove that the exponent in the lower bound is sharp and the key idea is that Adam is either a ``large degree" vertex or is a neighbor of a ``large degree" vertex (Eve).

翻译：我们考虑在Barabási–Albert树过程$(\mathcal{T}(n) : n \geq 1)$中，在长时间尺度下寻找初始顶点（亚当）的问题。更精确地说，给定$\varepsilon>0$，需要输出$\mathcal{T}(n)$的一个顶点子集$\mathcal{P}_{\varepsilon}(n)$，使得当$n$较大时，初始顶点以至少$1-\varepsilon$的概率属于$\mathcal{P}_{\varepsilon}(n)$。Bubeck、Devroye和Lugosi已证明，随后由Banerjee和Huang加以改进，成功完成该任务至少需要输出$\varepsilon^{-1 + o(1)}$个顶点，至多需要输出$\varepsilon^{-2 + o(1)}$个顶点。我们证明下界中的指数是紧的，其关键思想在于：亚当要么是一个“大度”顶点，要么是一个“大度”顶点（夏娃）的邻居。