Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective
  of Operator Semigroup Theory
Weichen Zhao Chenguang Wang Xinyan Wang Congying Han Tiande Guo Tianshu Yu
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.