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.

翻译：本文对基于扩散的图神经网络（GNNs）中的过平滑问题进行了创新性研究。不同于现有基于随机游走分析或粒子系统的方法，我们通过算子半群理论来探讨这一问题。该理论框架使我们能够严格证明过平滑与扩散算子的遍历性存在内在关联。这一发现进一步提出了一个通用且温和的遍历性破缺条件，涵盖了以往提出的各种特定解决方案，从而为缓解基于扩散的GNNs中的过平滑问题提供了一种更通用且具有理论依据的方法。此外，我们还对理论进行了概率解释，建立了与先前研究的联系，拓展了理论视野。实验结果表明，该遍历性破缺项能有效缓解以狄利克雷能量度量的过平滑现象，同时提升节点分类任务的性能。