We investigate contextual graph matching in the Gaussian setting, where both edge weights and node features are correlated across two networks. We derive precise information-theoretic thresholds for exact recovery, and identify conditions under which almost exact recovery is possible or impossible, in terms of graph and feature correlation strengths, the number of nodes, and feature dimension. Interestingly, whereas an all-or-nothing phase transition is observed in the standard graph-matching scenario, the additional contextual information introduces a richer structure: thresholds for exact and almost exact recovery no longer coincide. Our results provide the first rigorous characterization of how structural and contextual information interact in graph matching, and establish a benchmark for designing efficient algorithms.

翻译：我们研究了高斯设定下的上下文图匹配问题，其中两个网络的边权重和节点特征均存在相关性。我们推导了精确恢复的精确信息论阈值，并根据图与特征的相关强度、节点数量及特征维度，识别出可实现或无法实现近似精确恢复的条件。有趣的是，尽管标准图匹配场景中观察到“全有或全无”的相变现象，但额外上下文信息的引入带来了更丰富的结构：精确恢复与近似精确恢复的阈值不再一致。我们的结果首次严格刻画了图匹配中结构化信息与上下文信息如何相互作用，并为设计高效算法建立了基准。