This work studies fundamental limits for recovering the underlying correspondence among multiple correlated random graphs. We identify a necessary condition for any algorithm to correctly match all nodes across all graphs, and propose two algorithms for which the same condition is also sufficient. The first algorithm employs global information to simultaneously match all the graphs, whereas the second algorithm first partially matches the graphs pairwise and then combines the partial matchings by transitivity. Both algorithms work down to the information theoretic threshold. Our analysis reveals a scenario where exact matching between two graphs alone is impossible, but leveraging more than two graphs allows exact matching among all the graphs. Along the way, we derive independent results about the k-core of Erdos-Renyi graphs.

翻译：本文研究了多重相关随机图之间恢复潜在对应关系的基本极限。我们识别出任何算法正确匹配所有节点（跨所有图）的必要条件，并提出了两种算法，使得同一条件也成为充分条件。第一种算法利用全局信息同时匹配所有图，而第二种算法先以成对方式部分匹配图，再通过传递性合并部分匹配结果。两种算法均能达到信息论阈值。我们的分析揭示了一种场景：仅靠两张图无法实现精确匹配，但利用两张以上图时，所有图之间可以实现精确匹配。在此过程中，我们推导了关于Erdos-Renyi图的k-核的独立结论。