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A Dynamical Systems-Inspired Pruning Strategy for Addressing Oversmoothing in Graph Neural Networks

Biswadeep Chakraborty, Harshit Kumar, Saibal Mukhopadhyay

TL;DR

This work reframes oversmoothing in deep graph neural networks as a dynamical-systems problem and introduces DYNAMO-GAT, a noise-informed, Anti-Hebbian pruning strategy that adaptively prunes attention weights to maintain feature diversity. By combining noise-driven covariance analysis with gradual pruning and dynamic thresholding, the method disrupts convergent fixed points that cause homogenization while preserving the expressive capacity of deep GNNs. The authors provide theoretical results showing reduced Jacobian spectral radii at oversmoothing fixed points and rank preservation of layer covariances, complemented by extensive experiments on real and synthetic datasets that demonstrate improved accuracy and efficiency over GCN, GAT, and G2GAT across depths. The approach offers a principled, dynamically adjustable mechanism to stabilize deep GNNs, with potential broader impact on stability and expressiveness in deep learning architectures handling complex graph-structured data.

Abstract

Oversmoothing in Graph Neural Networks (GNNs) poses a significant challenge as network depth increases, leading to homogenized node representations and a loss of expressiveness. In this work, we approach the oversmoothing problem from a dynamical systems perspective, providing a deeper understanding of the stability and convergence behavior of GNNs. Leveraging insights from dynamical systems theory, we identify the root causes of oversmoothing and propose \textbf{\textit{DYNAMO-GAT}}. This approach utilizes noise-driven covariance analysis and Anti-Hebbian principles to selectively prune redundant attention weights, dynamically adjusting the network's behavior to maintain node feature diversity and stability. Our theoretical analysis reveals how DYNAMO-GAT disrupts the convergence to oversmoothed states, while experimental results on benchmark datasets demonstrate its superior performance and efficiency compared to traditional and state-of-the-art methods. DYNAMO-GAT not only advances the theoretical understanding of oversmoothing through the lens of dynamical systems but also provides a practical and effective solution for improving the stability and expressiveness of deep GNNs.

A Dynamical Systems-Inspired Pruning Strategy for Addressing Oversmoothing in Graph Neural Networks

TL;DR

This work reframes oversmoothing in deep graph neural networks as a dynamical-systems problem and introduces DYNAMO-GAT, a noise-informed, Anti-Hebbian pruning strategy that adaptively prunes attention weights to maintain feature diversity. By combining noise-driven covariance analysis with gradual pruning and dynamic thresholding, the method disrupts convergent fixed points that cause homogenization while preserving the expressive capacity of deep GNNs. The authors provide theoretical results showing reduced Jacobian spectral radii at oversmoothing fixed points and rank preservation of layer covariances, complemented by extensive experiments on real and synthetic datasets that demonstrate improved accuracy and efficiency over GCN, GAT, and G2GAT across depths. The approach offers a principled, dynamically adjustable mechanism to stabilize deep GNNs, with potential broader impact on stability and expressiveness in deep learning architectures handling complex graph-structured data.

Abstract

Oversmoothing in Graph Neural Networks (GNNs) poses a significant challenge as network depth increases, leading to homogenized node representations and a loss of expressiveness. In this work, we approach the oversmoothing problem from a dynamical systems perspective, providing a deeper understanding of the stability and convergence behavior of GNNs. Leveraging insights from dynamical systems theory, we identify the root causes of oversmoothing and propose \textbf{\textit{DYNAMO-GAT}}. This approach utilizes noise-driven covariance analysis and Anti-Hebbian principles to selectively prune redundant attention weights, dynamically adjusting the network's behavior to maintain node feature diversity and stability. Our theoretical analysis reveals how DYNAMO-GAT disrupts the convergence to oversmoothed states, while experimental results on benchmark datasets demonstrate its superior performance and efficiency compared to traditional and state-of-the-art methods. DYNAMO-GAT not only advances the theoretical understanding of oversmoothing through the lens of dynamical systems but also provides a practical and effective solution for improving the stability and expressiveness of deep GNNs.

Paper Structure

This paper contains 29 sections, 10 theorems, 48 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dynamical Systems View of Oversmoothing
  3. Theoretical Analysis of Oversmoothing in GATs
  4. DYNAMO-GAT Algorithm
  5. Covariance Matrix and Noise Injection
  6. Anti-Hebbian Pruning Criterion
  7. Gradual Pruning Process and Update Rule
  8. Recalibration of Attention Weights
  9. Theoretical Results
  10. Experimental Results
  11. Experimental Setup
  12. Performance on Real-World Datasets
  13. Performance Comparison Across Datasets
  14. Synthetic Dataset Results
  15. Conclusion
  16. ...and 14 more sections

Key Result

Lemma 1

(Existence of Fixed Points in GATs). Consider a GAT with an update rule $f: \mathbb{R}^{N \times d} \rightarrow \mathbb{R}^{N \times d}$, where $\mathbf{X}(t) \in \mathbb{R}^{N \times d}$ represents the node features at layer $t$. The update rule for the node features can be expressed as: where $\sigma$ is a nonlinear activation function, $\mathbf{W}$ is a weight matrix, and $\alpha_{ij}(t)$ are

Figures (3)

  • Figure 1: As the number of layers $k$ in a GNN increases, oversmoothing causes node embeddings to converge towards a single attractor state, resulting in the loss of node feature diversity. Pruning mitigates this effect by maintaining multiple attractor states, thereby preserving the distinctiveness of node embeddings and preventing the detrimental effects of oversmoothing.
  • Figure 2: Comparison of oversmoothing coefficient ($\mu(X)$) and test accuracy across layers for Citeseer, Cora, and Cornell datasets. DYNAMO-GAT consistently outperforms both GCN, GAT and G2GAT maintaining high accuracy across all layers.
  • Figure 3: Performance of DYNAMO-GAT, G2GAT, GCN, and GAT on the Syn_Products dataset. (a) Oversmoothing vs. layers: DYNAMO-GAT shows the least oversmoothing. Comparing test accuracy (b) vs. number of layers (c) vs. homophily for sparse graph (Avg. Degree=11.93) (d) vs. homophily for dense graph (Avg. Degree=68.75)

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 5 more