# Comment on "A Note on Over-Smoothing for Graph Neural Networks"

arXiv: https://arxiv.org/abs/2509.04178 · HTML: https://mlsanity.com/paper/2509.04178/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2025.

## Abstract (arXiv)

We comment on Cai and Wang (2020, arXiv:2006.13318), who analyze over-smoothing in GNNs via Dirichlet energy. We show that under mild spectral conditions (including with Leaky-ReLU), the Dirichlet energy of node embeddings decreases exponentially with depth; we further extend the result to spectral polynomial filters and provide a short proof for the Leaky-ReLU case. Experiments on edge deletion and weight amplification illustrate when Dirichlet energy increases, hinting at practical ways to relieve over-smoothing.