This paper studies the problem of recovering the hidden vertex correspondence between two correlated random graphs. We propose the partially correlated Erd\H{o}s-R\'enyi graphs model, wherein a pair of induced subgraphs with a certain number are correlated. We investigate the information-theoretic thresholds for recovering the latent correlated subgraphs and the hidden vertex correspondence. We prove that there exists an optimal rate for partial recovery for the number of correlated nodes, above which one can correctly match a fraction of vertices and below which correctly matching any positive fraction is impossible, and we also derive an optimal rate for exact recovery. In the proof of possibility results, we propose correlated functional digraphs, which partition the edges of the intersection graph into two types of components, and bound the error probability by lower-order cumulant generating functions. The proof of impossibility results build upon the generalized Fano's inequality and the recovery thresholds settled in correlated Erd\H{o}s-R\'enyi graphs model.

翻译：本文研究了恢复两个相关随机图之间隐藏顶点对应关系的问题。我们提出了部分相关Erd\H{o}s-R\'enyi图模型，其中具有一定顶点数的诱导子图对是相关的。我们研究了恢复潜在相关子图与隐藏顶点对应关系的信息论阈值。我们证明了相关节点数量存在部分恢复的最优速率，高于该速率时可以正确匹配一定比例的顶点，而低于该速率时则不可能正确匹配任何正比例顶点；同时我们还推导出了精确恢复的最优速率。在可能性结果的证明中，我们提出了相关函数有向图，该结构将交图的边划分为两种类型的连通分量，并通过低阶累积量生成函数来界定错误概率。不可能性结果的证明建立在广义Fano不等式及相关Erd\H{o}s-R\'enyi图模型中已确立的恢复阈值基础之上。