Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Demystifying Oversmoothing in Attention-Based Graph Neural Networks

Xinyi Wu, Amir Ajorlou, Zihui Wu, Ali Jadbabaie

TL;DR

This work resolves the open question of whether graph attention can prevent oversmoothing by modeling attention-based GNNs as nonlinear time-varying dynamical systems and analyzing long-horizon dynamics via products of inhomogeneous matrices and the joint spectral radius. It proves that oversmoothing occurs exponentially for a broad class of attention-based architectures, including GATs and graph transformers, under general nonlinear activations such as ReLU, LeakyReLU, GELU, and SiLU. The contributions include establishing a common connectivity structure across layers, proving ergodicity of matrix products, and deriving explicit exponential convergence rates for node representations. The findings have significant implications for designing deeper GNNs and motivate connectivity-altering methods to mitigate oversmoothing in practical applications.

Abstract

Oversmoothing in Graph Neural Networks (GNNs) refers to the phenomenon where increasing network depth leads to homogeneous node representations. While previous work has established that Graph Convolutional Networks (GCNs) exponentially lose expressive power, it remains controversial whether the graph attention mechanism can mitigate oversmoothing. In this work, we provide a definitive answer to this question through a rigorous mathematical analysis, by viewing attention-based GNNs as nonlinear time-varying dynamical systems and incorporating tools and techniques from the theory of products of inhomogeneous matrices and the joint spectral radius. We establish that, contrary to popular belief, the graph attention mechanism cannot prevent oversmoothing and loses expressive power exponentially. The proposed framework extends the existing results on oversmoothing for symmetric GCNs to a significantly broader class of GNN models, including random walk GCNs, Graph Attention Networks (GATs) and (graph) transformers. In particular, our analysis accounts for asymmetric, state-dependent and time-varying aggregation operators and a wide range of common nonlinear activation functions, such as ReLU, LeakyReLU, GELU and SiLU.

Demystifying Oversmoothing in Attention-Based Graph Neural Networks

TL;DR

This work resolves the open question of whether graph attention can prevent oversmoothing by modeling attention-based GNNs as nonlinear time-varying dynamical systems and analyzing long-horizon dynamics via products of inhomogeneous matrices and the joint spectral radius. It proves that oversmoothing occurs exponentially for a broad class of attention-based architectures, including GATs and graph transformers, under general nonlinear activations such as ReLU, LeakyReLU, GELU, and SiLU. The contributions include establishing a common connectivity structure across layers, proving ergodicity of matrix products, and deriving explicit exponential convergence rates for node representations. The findings have significant implications for designing deeper GNNs and motivate connectivity-altering methods to mitigate oversmoothing in practical applications.

Abstract

Oversmoothing in Graph Neural Networks (GNNs) refers to the phenomenon where increasing network depth leads to homogeneous node representations. While previous work has established that Graph Convolutional Networks (GCNs) exponentially lose expressive power, it remains controversial whether the graph attention mechanism can mitigate oversmoothing. In this work, we provide a definitive answer to this question through a rigorous mathematical analysis, by viewing attention-based GNNs as nonlinear time-varying dynamical systems and incorporating tools and techniques from the theory of products of inhomogeneous matrices and the joint spectral radius. We establish that, contrary to popular belief, the graph attention mechanism cannot prevent oversmoothing and loses expressive power exponentially. The proposed framework extends the existing results on oversmoothing for symmetric GCNs to a significantly broader class of GNN models, including random walk GCNs, Graph Attention Networks (GATs) and (graph) transformers. In particular, our analysis accounts for asymmetric, state-dependent and time-varying aggregation operators and a wide range of common nonlinear activation functions, such as ReLU, LeakyReLU, GELU and SiLU.
Paper Structure (54 sections, 17 theorems, 90 equations, 2 figures, 1 table)

This paper contains 54 sections, 17 theorems, 90 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Oversmoothing problem in GNNs
  4. Theoretical analysis of attention-based GNNs
  5. Problem Setup
  6. Notations
  7. Graph attention mechanism
  8. Attention-based GNNs
  9. Measure of oversmoothing
  10. Assumptions
  11. Main Results
  12. Attention-based GNNs as nonlinear time-varying dynamical systems
  13. Common connectivity structure among aggregation operators across different layers
  14. Ergodicity of infinite products of matrices
  15. Remark
  16. ...and 39 more sections

Key Result

Proposition 1

$\|X - \mathbf{1}\gamma_X \|_F$ is a node similarity measure.

Figures (2)

  • Figure 1: Evolution of $\mu(X^{(t)})$ (in log-log scale) on the largest connected component of each dataset (top 2 rows: homophilic graphs; bottom 2 rows: heterophilic graphs). Oversmoothing happens exponentially in both GCNs and GATs with the rates varying depending on the choice of activation function. Notably, GCNs demonstrate faster rates of oversmoothing compared to GATs.
  • Figure 2: Evolution of $\mu(X^{(t)})$ (in log-log scale) on the largest connected component of the large-scale benchmark dataset: Flickr.

Theorems & Definitions (31)

  • Definition 1
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Definition 2
  • Definition 3: Ergodicity
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Definition 4: Joint Spectral Radius
  • ...and 21 more