This paper presents a novel study of the oversmoothing issue in diffusion-based Graph Neural Networks (GNNs). Diverging from extant approaches grounded in random walk analysis or particle systems, we approach this problem through operator semigroup theory. This theoretical framework allows us to rigorously prove that oversmoothing is intrinsically linked to the ergodicity of the diffusion operator. This finding further poses a general and mild ergodicity-breaking condition, encompassing the various specific solutions previously offered, thereby presenting a more universal and theoretically grounded approach to mitigating oversmoothing in diffusion-based GNNs. Additionally, we offer a probabilistic interpretation of our theory, forging a link with prior works and broadening the theoretical horizon. Our experimental results reveal that this ergodicity-breaking term effectively mitigates oversmoothing measured by Dirichlet energy, and simultaneously enhances performance in node classification tasks.

通过运算子半群理论，本论文探讨了扩散型图神经网络（GNNs）中的过平滑问题并证明了过平滑与扩散算子的遍历性相关。我们提出了一种普适且理论基础良好的方法来缓解扩散型GNNs中的过平滑问题，并对我们的理论进行了概率解释，从而与之前的研究建立关联并拓展了理论视野。实验结果显示，这种遍历性破坏项有效地缓解了通过Dirichlet能量度量的过平滑，并同时提升了节点分类任务的性能。

从算子半群理论的角度理解基于扩散的GNN中的过度平滑现象