Topological Data Analysis on Graphs: Euler characteristics, Persistent Homology, or Spectrum?
Keywords: graph representation learning, topological data analysis, Euler characteristics, persistent homology, Laplacian spectrum, graph neural networks
TL;DR: We characterize the expressivity, stability, and computational cost of topological descriptors for graphs; and design and analyze new descriptors and filtration functions.
Abstract: Graph neural networks (GNNs) are limited by the Weisfeiler-Leman (WL) hierarchy and cannot compute graph properties such as cycles. Topological descriptors (TDs) such as the Euler characteristics (EC), persistent homology (PH), and Laplacian spectrums have thus been employed to enhance the GNNs. However, despite empirical successes, the theoretical underpinnings of these TDs remain largely underexplored. We bridge this gap with a rigorous characterization of TDs focusing on three key aspects: expressivity (representational power), stability (robustness to data perturbations), and computation (implementation cost). We evaluate the expressivity of different TDs, and design a novel scheme $\operatorname{RePHINE}^{Spec}$ that is strictly more expressive. We also propose new metrics to assess the stability of the state-of-the-art RePHINE method and the newly proposed $\operatorname{RePHINE}^{Spec}$ method. To address computational costs, we introduce and analyze weaker variants for several descriptors. TDs find significant applications in molecular contexts, so we also explore new filtration functions on the molecular graphs. Finally, we formalize the properties of filtration functions derived from graph products. Overall, this work lays the foundation for the principled design and analysis of new TDs that can be tailored to specific applications.
Primary Area: learning on graphs and other geometries & topologies
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Submission Number: 11241
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