A Spectral Framework for Evaluating Geodesic Distances Between Graphs

TL;DR

This paper introduces a spectral framework and a novel Graph Geodesic Distance metric for quantifying dissimilarities between graphs, improving graph classification and analysis of graph neural networks.

Contribution

It proposes a new spectral-based metric, GGD, with a graph coarsening scheme, enhancing graph comparison tasks especially with partial node features.

Findings

GGD outperforms state-of-the-art metrics like TMD in graph classification.

The spectral framework effectively captures key structural differences between graphs.

GGD is versatile for applications beyond classification, including GNN stability and dataset comparison.

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, and the mixing time of random walks. Through extensive experiments comparing with state-of-the-art metrics, such as the latest Tree-Mover's Distance (TMD), the proposed GGD metric demonstrates significantly improved performance for graph classification, particularly when only partial node features are available. Furthermore, we extend the application of GGD beyond graph classification to stability analysis of GNNs and the quantification of distances between datasets, highlighting its versatility in broader machine learning contexts.