Graph matching is one of the most significant graph analytic tasks in practice, which aims to find the node correspondence across different graphs. Most existing approaches rely on adjacency matrices or node embeddings when matching graphs, whose performances are often sub-optimal because of not fully leveraging the multi-modal information hidden in graphs, such as node attributes, subgraph structures, etc. In this study, we propose a novel and effective graph matching method based on a differentiable hierarchical optimal transport (HOT) framework, called DHOT-GM. Essentially, our method represents each graph as a set of relational matrices corresponding to the information of different modalities. Given two graphs, we enumerate all relational matrix pairs and obtain their matching results, and accordingly, infer the node correspondence by the weighted averaging of the matching results. This method can be implemented as computing the HOT distance between the two graphs -- each matching result is an optimal transport plan associated with the Gromov-Wasserstein (GW) distance between two relational matrices, and the weights of all matching results are the elements of an upper-level optimal transport plan defined on the matrix sets. We propose a bi-level optimization algorithm to compute the HOT distance in a differentiable way, making the significance of the relational matrices adjustable. Experiments on various graph matching tasks demonstrate the superiority and robustness of our method compared to state-of-the-art approaches.

翻译：图匹配是实践中最重要的图分析任务之一，旨在寻找不同图之间的节点对应关系。现有方法大多依赖邻接矩阵或节点嵌入进行图匹配，由于未能充分利用图中隐藏的多模态信息（如节点属性、子图结构等），其性能往往次优。本研究提出一种基于可微分分层最优传输（HOT）框架的新型高效图匹配方法，称为DHOT-GM。本质上，该方法将每个图表示为对应不同模态信息的关系矩阵集合。给定两个图，我们枚举所有关系矩阵对并获取其匹配结果，进而通过匹配结果的加权平均推断节点对应关系。该方法可等价于计算两个图之间的HOT距离——每个匹配结果对应两个关系矩阵间Gromov-Wasserstein（GW）距离的最优传输方案，而所有匹配结果的权重则构成定义在矩阵集合上的上层最优传输方案的元素。我们提出一种双层优化算法，以可微分方式计算HOT距离，使关系矩阵的重要性可调节。在多种图匹配任务上的实验表明，与现有最优方法相比，本方法具有优越性和鲁棒性。