A Spectral Framework for Assessing the Geodesic Distance Between Graphs

ICLR 2025 Conference Submission12713 Authors

Keywords: Graph Theory, Graph Neural Network, Graph Laplacian, Riemannian Manifold, Geodesic, Graph Classification

Abstract: This paper presents a spectral framework for quantifying the differentiation between graph data samples by introducing a novel metric named Graph Geodesic Distance (GGD). 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 of different sizes, a resistance-based spectral graph coarsening scheme is introduced to reduce the size of the larger 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 graph classification and stability evaluation of GNNs, especially when only partial node features are available.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 12713