🤖 AI Summary
This work addresses the challenge of quantifying structural discrepancies among graphs. We propose the Graph Geodesic Distance (GGD), a spectral metric grounded in the generalized eigenvalue problem of Laplacian matrices. GGD models inter-graph dissimilarities in key spectral properties—including effective resistance, cut size, and random-walk mixing time—thereby enabling principled assessment of GNN generalization and stability. A novel resistance-preserving spectral coarsening scheme is introduced, enabling, for the first time, stable and comparable distance computation across graphs of differing scales (i.e., unequal node counts). Experiments demonstrate that, in the absence of node features, GGD significantly outperforms state-of-the-art methods—including Tree-Mover’s Distance—in both discriminative power and robustness for evaluating GNN stability.
📝 Abstract
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.