# Analysis of Dirichlet Energies as Over-smoothing Measures

arXiv: https://arxiv.org/abs/2512.09890 · HTML: https://mlsanity.com/paper/2512.09890/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 10 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 7 of 7)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learning](https://mlsanity.com/paper/2502.10818/) (2025)
- [A Spectral Interpretation of Redundancy in a Graph Reservoir](https://mlsanity.com/paper/2507.12963/) (2025)

## Built on by (most cited first, 1 of 1)

- [Not Just Oversmoothing: Detecting the Echo Chamber Effect in Graph Neural Networks](https://mlsanity.com/paper/2609.06521/) (2026)

## Abstract (arXiv)

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 textitet 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.