Analysis of Dirichlet Energies as Over-smoothing Measures
This work addresses ambiguities in monitoring over-smoothing dynamics for researchers and practitioners in graph neural networks, but it is incremental as it clarifies existing measures rather than introducing new ones.
The paper tackled the problem of distinguishing between two Dirichlet energy functionals used as over-smoothing measures in graph neural networks, demonstrating that the normalized graph Laplacian version fails to meet axiomatic node-similarity criteria and formalizing spectral properties to guide metric selection for compatibility with GNN architectures.
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.