Measuring Over-smoothing beyond Dirichlet energy
This work addresses a fundamental issue in graph neural networks for researchers and practitioners by providing more comprehensive metrics for over-smoothing, though it is incremental as it builds on existing Dirichlet energy concepts.
The authors tackled the limitation of Dirichlet energy in measuring over-smoothing by proposing a generalized family of node similarity measures based on higher-order feature derivatives, and they demonstrated that attention-based Graph Neural Networks suffer from over-smoothing under these metrics.
While Dirichlet energy serves as a prevalent metric for quantifying over-smoothing, it is inherently restricted to capturing first-order feature derivatives. To address this limitation, we propose a generalized family of node similarity measures based on the energy of higher-order feature derivatives. Through a rigorous theoretical analysis of the relationships among these measures, we establish the decay rates of Dirichlet energy under both continuous heat diffusion and discrete aggregation operators. Furthermore, our analysis reveals an intrinsic connection between the over-smoothing decay rate and the spectral gap of the graph Laplacian. Finally, empirical results demonstrate that attention-based Graph Neural Networks (GNNs) suffer from over-smoothing when evaluated under these proposed metrics.