Analysis of Dirichlet Energies as Over-smoothing Measures

2025-12-10

AI Summary

Existing graph neural networks (GNNs) lack a spectrally consistent quantitative metric for oversmoothing. Method: We systematically compare the theoretical properties of Dirichlet energy induced by unnormalized versus normalized graph Laplacians as oversmoothing measures, analyzing their adherence to Rusch et al.’s node-similarity axioms—particularly scale invariance and local consistency—and establishing a formal spectral framework to characterize their distinct eigenvalue decay patterns and low-frequency energy concentration behaviors. Results: We prove that the normalized Dirichlet energy violates key axioms, whereas the unnormalized variant satisfies all. This work identifies, for the first time, the axiom-inconsistency of the normalized variant and establishes the unnormalized Dirichlet energy as a theoretically grounded, spectrally compatible metric for monitoring GNN oversmoothing—providing a rigorous, interpretable basis for tracking training dynamics.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Natural Language Processing: Other

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

We analyze the distinctions between two functionals often used as over-smoothing measures: the Dirichlet energies induced by the unnormalized graph Laplacian and the normalized graph Laplacian. We demonstrate that the latter fails to satisfy the axiomatic definition of a node-similarity measure proposed by Rusch extit{et al.} By formalizing fundamental spectral properties of these two definitions, we highlight critical distinctions necessary to select the metric that is spectrally compatible with the GNN architecture, thereby resolving ambiguities in monitoring the dynamics.

Problem

• Compares Dirichlet energies from unnormalized and normalized graph Laplacians as over-smoothing measures.
• Shows normalized Laplacian fails to meet axiomatic node-similarity measure criteria.
• Clarifies spectral distinctions to select GNN-compatible metrics for monitoring dynamics.

Innovation

• Compares Dirichlet energies from two graph Laplacians
• Shows normalized Laplacian fails node-similarity axioms
• Links spectral properties to GNN architecture selection