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KAA: Kolmogorov-Arnold Attention for Enhancing Attentive Graph Neural Networks

Taoran Fang, Tianhong Gao, Chunping Wang, Yihao Shang, Wei Chow, Lei Chen, Yang Yang

TL;DR

This work addresses the limited expressiveness of neighbor scoring in attentive GNNs by introducing Kolmogorov-Arnold Attention (KAA), which embeds a Kolmogorov-Arnold Network (KAN) into the score mapper of attention mechanisms. It unifies scoring functions via s(h_i,h_j)=Ψ(AF(h_i,h_j)) and introduces Maximum Ranking Distance (MRD) to quantify the expressive power of scoring functions. Theoretical analyses show that, under parameter constraints, KAA achieves near-unlimited ranking power (MRD ≤ δ for any δ>0) and outperforms linear and MLP-based scoring across backbone models. Extensive node- and graph-level experiments demonstrate consistent gains, with some tasks seeing improvements over 20%, highlighting KAA’s practical impact for enhancing attentive GNNs.

Abstract

Graph neural networks (GNNs) with attention mechanisms, often referred to as attentive GNNs, have emerged as a prominent paradigm in advanced GNN models in recent years. However, our understanding of the critical process of scoring neighbor nodes remains limited, leading to the underperformance of many existing attentive GNNs. In this paper, we unify the scoring functions of current attentive GNNs and propose Kolmogorov-Arnold Attention (KAA), which integrates the Kolmogorov-Arnold Network (KAN) architecture into the scoring process. KAA enhances the performance of scoring functions across the board and can be applied to nearly all existing attentive GNNs. To compare the expressive power of KAA with other scoring functions, we introduce Maximum Ranking Distance (MRD) to quantitatively estimate their upper bounds in ranking errors for node importance. Our analysis reveals that, under limited parameters and constraints on width and depth, both linear transformation-based and MLP-based scoring functions exhibit finite expressive power. In contrast, our proposed KAA, even with a single-layer KAN parameterized by zero-order B-spline functions, demonstrates nearly infinite expressive power. Extensive experiments on both node-level and graph-level tasks using various backbone models show that KAA-enhanced scoring functions consistently outperform their original counterparts, achieving performance improvements of over 20% in some cases.

KAA: Kolmogorov-Arnold Attention for Enhancing Attentive Graph Neural Networks

TL;DR

This work addresses the limited expressiveness of neighbor scoring in attentive GNNs by introducing Kolmogorov-Arnold Attention (KAA), which embeds a Kolmogorov-Arnold Network (KAN) into the score mapper of attention mechanisms. It unifies scoring functions via s(h_i,h_j)=Ψ(AF(h_i,h_j)) and introduces Maximum Ranking Distance (MRD) to quantify the expressive power of scoring functions. Theoretical analyses show that, under parameter constraints, KAA achieves near-unlimited ranking power (MRD ≤ δ for any δ>0) and outperforms linear and MLP-based scoring across backbone models. Extensive node- and graph-level experiments demonstrate consistent gains, with some tasks seeing improvements over 20%, highlighting KAA’s practical impact for enhancing attentive GNNs.

Abstract

Graph neural networks (GNNs) with attention mechanisms, often referred to as attentive GNNs, have emerged as a prominent paradigm in advanced GNN models in recent years. However, our understanding of the critical process of scoring neighbor nodes remains limited, leading to the underperformance of many existing attentive GNNs. In this paper, we unify the scoring functions of current attentive GNNs and propose Kolmogorov-Arnold Attention (KAA), which integrates the Kolmogorov-Arnold Network (KAN) architecture into the scoring process. KAA enhances the performance of scoring functions across the board and can be applied to nearly all existing attentive GNNs. To compare the expressive power of KAA with other scoring functions, we introduce Maximum Ranking Distance (MRD) to quantitatively estimate their upper bounds in ranking errors for node importance. Our analysis reveals that, under limited parameters and constraints on width and depth, both linear transformation-based and MLP-based scoring functions exhibit finite expressive power. In contrast, our proposed KAA, even with a single-layer KAN parameterized by zero-order B-spline functions, demonstrates nearly infinite expressive power. Extensive experiments on both node-level and graph-level tasks using various backbone models show that KAA-enhanced scoring functions consistently outperform their original counterparts, achieving performance improvements of over 20% in some cases.
Paper Structure (30 sections, 4 theorems, 52 equations, 1 figure, 9 tables)

This paper contains 30 sections, 4 theorems, 52 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Attentive Graph Neural Networks
  4. Kolmogorov–Arnold Networks
  5. Preliminary
  6. Methodology
  7. Kolmogorov-Arnold Attention
  8. Comparison of Expressive Power
  9. Limitation of Existing Measurement
  10. Designing Comprehensive Measurement
  11. Theoretical Comparison
  12. Experiment
  13. Experiment Setup
  14. Performance on Node-level Tasks
  15. Performance on Graph-level Tasks
  16. ...and 15 more sections

Key Result

Proposition 1

Given the alignment matrix $\mathbf{P}\in \mathbb{R}^{N\times d}$, for a scoring function in the form of $\text{\rm s}(h_i, h_j) = \mathbf{W} \cdot \text{\rm AF}(h_i, h_j)$, where $\mathbf{W} \in \mathbb{R}^{d \times 1}$, its MRD satisfies the following inequality: where $\mathcal{S}_{\text{\rm LT}}$ is the set of all candidate linear transformation-based scoring functions.

Figures (1)

  • Figure 1: The alignment of our proposed KAA and other applications of KAN. (a) Symbiotic regression and PDE-solving tasks, where KAN achieves strong performance. (b) These tasks utilize KAN to handle multi-dimensional inputs and one-dimensional outputs. (c) Since the score mapping in attentive GNNs follows a similar form, we replace it with KAN.

Theorems & Definitions (9)

  • Definition 1: Importance Ranking
  • Definition 2: Ranking Distance
  • Definition 3: Maximum Ranking Distance
  • Proposition 1: MRD of Linear Transformation-Based Attention
  • Proposition 2: MRD of MLP-Based Attention
  • Proposition 3: MRD of Kolmogorov-Arnold Attention
  • Theorem 1
  • proof
  • proof