GKAN: Graph Kolmogorov-Arnold Networks

The paper presents Graph Kolmogorov-Arnold Networks (GKAN), a new neural network architecture that adapts the concept of learnable univariate functions from KAN to graph data, showing improved performance over traditional GCNs in semi-supervised learning tasks on the Cora dataset.

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.

Method

The authors used a semi-supervised learning approach to evaluate the performance of Graph Kolmogorov-Arnold Networks (GKAN) on the Cora dataset. They compared two different GKAN architectures, which apply learnable functions either before or after feature aggregation, against traditional Graph Convolutional Networks (GCNs). The evaluation focused on classification accuracy, and the authors also explored the effects of varying parameters such as the number of hidden nodes, grid size, and spline polynomial degree on the performance of GKANs.

Main Finding

The authors discovered that Graph Kolmogorov-Arnold Networks (GKANs) outperform traditional Graph Convolutional Networks (GCNs) in semi-supervised learning tasks on the Cora dataset. They found that GKANs achieve higher classification accuracy with a comparable number of parameters to GCNs. Specifically, when considering 100 features, GCN provided an accuracy of 53.5% while GKAN achieved 61.76%, and with 200 features, GCN provided 61.24% accuracy while GKAN achieved 67.66%. Additionally, the authors observed that the performance of GKANs is influenced by the choice of parameters such as the number of hidden nodes, grid size, and the degree of the spline polynomial used in the learnable univariate functions.

Conclusion

The conclusion of the paper is that Graph Kolmogorov-Arnold Networks (GKANs) offer a promising new approach for graph representation learning, demonstrating superior performance in terms of classification accuracy compared to traditional Graph Convolutional Networks (GCNs) on the Cora dataset. The authors suggest that GKANs could serve as a foundation for various graph deep learning methods, potentially replacing MLPs at the core of approaches like GCNs, GAT, Graph Autoencoders, and Graph Transformers. However, they acknowledge that the current limitation of slower training times for GKANs is an area for future improvement.

Keywords

Graph Kolmogorov-Arnold Networks (GKAN), Kolmogorov-Arnold Networks (KAN), Graph Convolutional Networks (GCNs), semi-supervised learning, Cora dataset, learnable univariate functions, spline-based functions, neural network architecture, classification accuracy, parameter efficiency, hidden nodes, grid size, spline polynomial degree, graph representation learning, deep learning, machine learning, neural networks, graph-structured data.