Analysis of Dirichlet Energies as Over-smoothing Measures

TL;DR

This paper compares two Dirichlet energy measures used to assess over-smoothing in graph neural networks, revealing that the normalized Laplacian-based measure does not meet certain axiomatic criteria, which impacts their spectral compatibility.

Contribution

It clarifies the fundamental spectral differences between unnormalized and normalized Dirichlet energies, guiding better metric selection for GNN over-smoothing analysis.

Findings

Normalized Laplacian Dirichlet energy fails the node-similarity axioms.

Spectral properties are crucial for selecting appropriate over-smoothing metrics.

The study resolves ambiguities in monitoring GNN over-smoothing dynamics.

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 \textit{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.