mlsanityData to 13 Oct 2026

Measuring Over-smoothing beyond Dirichlet energy

arXiv 2512.06782 ↗First posted 7 Dec 2025cs.LGcited by 1

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Abstract (arXiv)

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.

Key facts. Cited by 1 paper in the map (data through 13 Oct 2026). Builds on 24 of its 31 references that are in the map. First posted on arXiv 7 Dec 2025.

Builds on

24 of 31 references.

References · 24

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,318
  2. 2017Graph Attention Networkscited by 8,888
  3. 2019Simplifying Graph Convolutional Networkscited by 1,474
  4. 2016Revisiting Semi-Supervised Learning with Graph Embeddingscited by 973
  5. 2019Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological Viewcited by 545
  6. 2019PairNorm: Tackling Oversmoothing in GNNscited by 297
  7. 2021GRAND: Graph Neural Diffusioncited by 258
  8. 2020A Note on Over-Smoothing for Graph Neural Networkscited by 248
  9. 2023A Survey on Oversmoothing in Graph Neural Networkscited by 247
  10. 2022Graph-Coupled Oscillator Networkscited by 124
  11. 2019Continuous Graph Neural Networkscited by 114
  12. 2020Towards Deeper Graph Neural Networks with Differentiable Group Normalizationcited by 114
12 more
  1. 2022FoSR: First-order spectral rewiring for addressing oversquashing in GNNscited by 105
  2. 2022Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvaturecited by 97
  3. 2022On the Trade-off between Over-smoothing and Over-squashing in Deep Graph Neural Networkscited by 64
  4. 2021Dissecting the Diffusion Process in Linear Graph Convolutional Networkscited by 58
  5. 2022GREAD: Graph Neural Reaction-Diffusion Networkscited by 54
  6. 2022Gradient Gating for Deep Multi-Rate Learning on Graphscited by 49
  7. 2023Demystifying Oversmoothing in Attention-Based Graph Neural Networkscited by 48
  8. 2023A Fractional Graph Laplacian Approach to Oversmoothingcited by 45
  9. 2022ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networkscited by 31
  10. 2024Spectral Graph Pruning Against Over-Squashing and Over-Smoothingcited by 24
  11. 2022TIDE: Time Derivative Diffusion for Deep Learning on Graphscited by 17
  12. 2024Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FRONDcited by 16

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Cited by year

Citing papers per year: 2025: 1.

1 paper.

  1. 2025Solving Oversmoothing in GNNs via Nonlocal Message Passing: Algebraic Smoothing and Depth Scalabilitycited by 0