GKAN: Graph Kolmogorov-Arnold Networks

TL;DR

This work introduces Graph Kolmogorov-Arnold Networks (GKANs) by extending Kolmogorov-Arnold Networks to graph-structured data and replacing fixed edge weights with learnable spline-based univariate functions. It proposes two architectures—one applying KAN layers after aggregation and another applying them before—and demonstrates on Cora that GKANs outperform a parameter-matched GCN, with Architecture 2 often delivering the best results. The study systematically analyzes how spline grid size, degree, and hidden layer width affect performance, revealing that intermediate grid sizes, linear splines, and moderate hidden widths yield optimal accuracy and convergence. The results suggest GKANs offer a more parameter-efficient approach to graph representation learning and open avenues for integrating KANs with other graph architectures in future work.

Abstract

We introduce Graph Kolmogorov-Arnold Networks (GKAN), an innovative neural network architecture that extends the principles of the recently proposed Kolmogorov-Arnold Networks (KAN) to graph-structured data. By adopting the unique characteristics of KANs, notably the use of learnable univariate functions instead of fixed linear weights, we develop a powerful model for graph-based learning tasks. Unlike traditional Graph Convolutional Networks (GCNs) that rely on a fixed convolutional architecture, GKANs implement learnable spline-based functions between layers, transforming the way information is processed across the graph structure. We present two different ways to incorporate KAN layers into GKAN: architecture 1 -- where the learnable functions are applied to input features after aggregation and architecture 2 -- where the learnable functions are applied to input features before aggregation. We evaluate GKAN empirically using a semi-supervised graph learning task on a real-world dataset (Cora). We find that architecture generally performs better. We find that GKANs achieve higher accuracy in semi-supervised learning tasks on graphs compared to the traditional GCN model. For example, when considering 100 features, GCN provides an accuracy of 53.5 while a GKAN with a comparable number of parameters gives an accuracy of 61.76; with 200 features, GCN provides an accuracy of 61.24 while a GKAN with a comparable number of parameters gives an accuracy of 67.66. We also present results on the impact of various parameters such as the number of hidden nodes, grid-size, and the polynomial-degree of the spline on the performance of GKAN.

Figures (10)

• Figure 1: Comparison of different model architectures.
• Figure 2: Accuracy comparison of GCN and GKAN architectures for $k=1$ and $g=3$.
• Figure 3: Loss value comparison of GCN and GKAN architectures for $k=1$ and $g=3$.
• Figure 4: Accuracy comparison of GKAN Architecture 2 for $g\in\{1,2,3\}$, $k=1$, and $h=16$.
• Figure 5: Loss value of GKAN Architecture 2 for $g\in \{3,7,11\}$ and $k=1$, and $h=16$.