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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

TL;DR

This work provides a unified, mathematically grounded framework to analyze oversmoothing in diffusion-based GNNs using operator semigroup theory. It shows that oversmoothing arises from the ergodicity of the diffusion generator and that the fixed-point behavior is determined by an invariant measure, with the convergence rate tied to the spectral gap. By introducing an ergodicity-breaking term, the approach generalizes existing diffusion remedies and offers a principled design principle to mitigate oversmoothing, complemented by a probabilistic interpretation via Markov processes and a killing process. Empirical results on node classification across diverse datasets demonstrate reduced oversmoothing (higher Dirichlet energy) and improved performance, validating the theoretical claims and highlighting the method's practical impact.

Abstract

This paper presents an analytical study of the oversmoothing issue in diffusion-based Graph Neural Networks (GNNs). Generalizing beyond 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. Relying on semigroup method, we can quantitatively analyze the dynamic of graph diffusion and give a specific mathematical form of the smoothing feature by ergodicity and invariant measure of operator, which improves previous works only show existence of oversmoothing. 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 relieve 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.

Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theory

TL;DR

This work provides a unified, mathematically grounded framework to analyze oversmoothing in diffusion-based GNNs using operator semigroup theory. It shows that oversmoothing arises from the ergodicity of the diffusion generator and that the fixed-point behavior is determined by an invariant measure, with the convergence rate tied to the spectral gap. By introducing an ergodicity-breaking term, the approach generalizes existing diffusion remedies and offers a principled design principle to mitigate oversmoothing, complemented by a probabilistic interpretation via Markov processes and a killing process. Empirical results on node classification across diverse datasets demonstrate reduced oversmoothing (higher Dirichlet energy) and improved performance, validating the theoretical claims and highlighting the method's practical impact.

Abstract

This paper presents an analytical study of the oversmoothing issue in diffusion-based Graph Neural Networks (GNNs). Generalizing beyond 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. Relying on semigroup method, we can quantitatively analyze the dynamic of graph diffusion and give a specific mathematical form of the smoothing feature by ergodicity and invariant measure of operator, which improves previous works only show existence of oversmoothing. 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 relieve 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.
Paper Structure (37 sections, 26 equations, 2 figures, 4 tables)

This paper contains 37 sections, 26 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Organization
  4. Background
  5. Operator Semigroup and Generator of Graph Diffusion
  6. Operator View for Oversmoothing
  7. Operator Representation of Feature
  8. Ergodicity and Oversmoothing
  9. Ergodicity Breaking Term
  10. Probability View for Oversmoothing
  11. Markov Chain on Graph with Generator $\mathcal{A}$
  12. Limiting Distribution of Node Features
  13. Killing Process
  14. Connections and Extensions to Existing Works
  15. Confirmation of Oversmoothing
  16. ...and 22 more sections

Figures (2)

  • Figure 1: Paper Organization
  • Figure 2: Demonstration on Oversmoothing and Performance Metrics. The figure comprises three panels: \ref{['os-wisconsin']} showcases the increase in oversmoothing of various GNN models with increasing depth on the Wisconsin dataset, with models like GCN and GAT experiencing significant effects; \ref{['os-homo']} illustrates the impact of synthetic Cora dataset with varying homophily levels on oversmoothing; and \ref{['os-test-prerformance']} displays the corresponding decrease in test accuracy on Cora dataset as network depth increases, highlighting performance declines in models such as GCN and GAT at greater depths due to the oversmoothing effects noted in the other panels.