Graph alignment refers to the task of finding the vertex correspondence between two correlated graphs of $n$ vertices. Extensive study has been done on polynomial-time algorithms for the graph alignment problem under the Erd\H{o}s-R\'enyi graph pair model, where the two graphs are Erd\H{o}s-R\'enyi graphs with edge probability $q_\mathrm{u}$, correlated under certain vertex correspondence. To achieve exact recovery of the correspondence, all existing algorithms at least require the edge correlation coefficient $\rho_\mathrm{u}$ between the two graphs to be \emph{non-vanishing} as $n\rightarrow\infty$. Moreover, it is conjectured that no polynomial-time algorithm can achieve exact recovery under vanishing edge correlation $\rho_\mathrm{u}<1/\mathrm{polylog}(n)$. In this paper, we show that with a vanishing amount of additional \emph{attribute information}, exact recovery is polynomial-time feasible under \emph{vanishing} edge correlation $\rho_\mathrm{u} \ge n^{-\Theta(1)}$. We identify a \emph{local} tree structure, which incorporates one layer of user information and one layer of attribute information, and apply the subgraph counting technique to such structures. A polynomial-time algorithm is proposed that recovers the vertex correspondence for most of the vertices, and then refines the output to achieve exact recovery. The consideration of attribute information is motivated by real-world applications like LinkedIn and Twitter, where user attributes like birthplace and education background can aid alignment.

翻译：图对齐是指在两个具有$n$个顶点的相关图之间寻找顶点对应关系的任务。在Erdős–Rényi图对模型下，人们对图对齐问题的多项式时间算法进行了广泛研究。在该模型中，两个图均为Erdős–Rényi图，边概率为$q_\mathrm{u}$，并在特定顶点对应关系下相互关联。为了实现对应关系的精确恢复，所有现有算法至少要求两个图之间的边相关系数$\rho_\mathrm{u}$在$n\rightarrow\infty$时是**非消失**的。此外，据推测，在边相关消失（$\rho_\mathrm{u}<1/\mathrm{polylog}(n)$）的情况下，不存在多项式时间算法能够实现精确恢复。本文中，我们证明：在引入**消失量**的额外**属性信息**后，当边相关**消失**（$\rho_\mathrm{u} \ge n^{-\Theta(1)}$）时，精确恢复在多项式时间内是可行的。我们识别出一种**局部**树结构，该结构融合了一层用户信息和一层属性信息，并对此类结构应用子图计数技术。我们提出了一种多项式时间算法，该算法首先恢复大多数顶点的对应关系，然后优化输出以实现精确恢复。属性信息的考虑源于现实应用（如领英和推特），在这些应用中，用户的出生地、教育背景等属性有助于对齐。