This paper presents a spectral framework for assessing the generalization and stability of Graph Neural Networks (GNNs) by introducing a Graph Geodesic Distance (GGD) metric. For two different graphs with the same number of nodes, our framework leverages a spectral graph matching procedure to find node correspondence so that the geodesic distance between them can be subsequently computed by solving a generalized eigenvalue problem associated with their Laplacian matrices. For graphs with different sizes, a resistance-based spectral graph coarsening scheme is introduced to reduce the size of the bigger graph while preserving the original spectral properties. We show that the proposed GGD metric can effectively quantify dissimilarities between two graphs by encapsulating their differences in key structural (spectral) properties, such as effective resistances between nodes, cuts, the mixing time of random walks, etc. Through extensive experiments comparing with the state-of-the-art metrics, such as the latest Tree-Mover's Distance (TMD) metric, the proposed GGD metric shows significantly improved performance for stability evaluation of GNNs especially when only partial node features are available.

通过引入图测地线距离（GGD）度量，本文提出了一种用于评估图神经网络（GNNs）泛化性和稳定性的光谱框架。通过引入光谱图匹配过程找到节点对应关系并计算其测地线距离，通过减小较大图的尺寸但保留原始光谱特性的阻抗型光谱图粗化方案降低了图的大小。实验证明，所提出的GGD度量能够有效量化两个图之间的差异，包括节点之间的有效电阻、切割、随机行走的混合时间等结构（光谱）属性，与最新的Tree-Mover's Distance（TMD）度量相比，提出的GGD度量在仅有部分节点特征可用时，对GNN的稳定性评估表现出显著改进。

图之间的测地距离：一种用于评估图神经网络稳定性的谱度量