KAGNNs: Kolmogorov-Arnold Networks meet Graph Learning (2406.18380v4)

Abstract: In recent years, Graph Neural Networks (GNNs) have become the de facto tool for learning node and graph representations. Most GNNs typically consist of a sequence of neighborhood aggregation (a.k.a., message-passing) layers, within which the representation of each node is updated based on those of its neighbors. The most expressive message-passing GNNs can be obtained through the use of the sum aggregator and of MLPs for feature transformation, thanks to their universal approximation capabilities. However, the limitations of MLPs recently motivated the introduction of another family of universal approximators, called Kolmogorov-Arnold Networks (KANs) which rely on a different representation theorem. In this work, we compare the performance of KANs against that of MLPs on graph learning tasks. We implement three new KAN-based GNN layers, inspired respectively by the GCN, GAT and GIN layers. We evaluate two different implementations of KANs using two distinct base families of functions, namely B-splines and radial basis functions. We perform extensive experiments on node classification, link prediction, graph classification and graph regression datasets. Our results indicate that KANs are on-par with or better than MLPs on all tasks studied in this paper. We also show that the size and training speed of RBF-based KANs is only marginally higher than for MLPs, making them viable alternatives. Code available at https://github.com/RomanBresson/KAGNN.

• The paper introduces two new architectures—KAGIN and KAGCN—that replace conventional MLPs with KANs in graph neural networks.
• It leverages B-spline-based learnable activations to improve function approximation and interpretability in diverse graph tasks.
• Empirical results show significant gains in graph regression and mixed outcomes in node and graph classification, highlighting contextual strengths.

An Expert Overview of "KAGNNs: Kolmogorov-Arnold Networks meet Graph Learning"

In "KAGNNs: Kolmogorov-Arnold Networks meet Graph Learning," the authors propose novel Graph Neural Network (GNN) architectures that utilize Kolmogorov-Arnold Networks (KANs) for transforming node features as an alternative to the traditionally employed Multi-Layer Perceptrons (MLPs). The paper addresses a significant limitation in current GNN methodologies and introduces empirical results to demonstrate the potential advantages of KANs over MLPs in various graph learning tasks.

Introduction

The integration of GNNs in various domains such as social networks, chemo-informatics, and quantum mechanical property estimations has been profound. Traditional GNNs capitalize on message-passing frameworks where each node's representation is iteratively updated based on its neighbors' features, typically using MLPs for transformation. However, the limitations of MLPs—including non-convex loss functions and poor interpretability—prompted investigation into alternative methods like KANs, which are grounded in the Kolmogorov-Arnold representation theorem.

Key Contributions

The paper presents two novel architectures: KAGIN (Kolmogorov-Arnold Graph Isomorphism Network) and KAGCN (Kolmogorov-Arnold Graph Convolution Network). These are modifications of the well-established GIN and GCN models, respectively, where KANs replace the MLPs typically employed in these networks. The universal approximation capabilities of KANs theoretically place them on par with MLPs, albeit with distinct operational characteristics derived from their reliance on B-spline-based learnable activations.

Methodology and Model Architectures

KAGIN Layer

The KAGIN model maintains the expressive power of GIN by leveraging KANs as function approximators:

$$\mathbf{h}^{(\ell)}_v = KAN^{(\ell)}\left((1+\epsilon) \cdot \mathbf{h}^{(\ell-1)}_v + \sum_{u \in \mathcal{N}(v)} \mathbf{h}^{(\ell-1)}_u\right)$$

KAGCN Layer

In the KAGCN model, the GCN's feature transformation is redefined to use a KAN layer:

$$\mathbf{h}^{(\ell)}_v = \Phi^{(\ell)} \left( \sum_{u \in \mathcal{N}(v) \cup \{v\} } \frac{\mathbf{h}_u^{(\ell-1)}}{\sqrt{(\deg(v)+1) (\deg(u)+1)}} \right)$$

Experimental Evaluation

The empirical evaluation consists of rigorous experiments on node classification, graph classification, and graph regression tasks using well-known datasets. The results show that KAN-based GNNs can offer competitive performance, particularly on graph regression tasks, where KAGIN models significantly outperform GIN models.

Node Classification

While KAGIN models demonstrated noticeable improvements over GINs in node classification tasks, KAGCN’s performance relative to GCNs was mixed. This suggests that the architectural compatibility between KANs and the underlying GNN structure may impact performance in node classification scenarios.

Graph Classification

In graph classification, KAGIN models generally outperformed GINs, validating the potential of KANs in these tasks. However, performance was contingent on dataset characteristics, such as the nature of node features and the number of target classes. The findings indicate that KANs may struggle with continuous features without proper normalization.

Graph Regression

The superiority of KANs was most evident in graph regression tasks. On large datasets like ZINC-12K and QM9, KAGIN models achieved substantially lower mean absolute errors (MAE) than GIN models, highlighting the efficacy of KANs in approximating complex functions inherent in regression problems.

Implications and Future Directions

The results suggest several implications for both practical applications and theoretical explorations in graph learning. Practically, the adoption of KANs could enhance the performance of GNNs in tasks where the relationships between variables exhibit regular patterns. Theoretically, the interpretability of KANs offers a promising avenue for developing more transparent models, crucial for applications requiring model explainability.

Future research could further explore:

• The robustness of KANs in handling datasets with varying types of node features.
• Optimizing the computational efficiency of KANs to mitigate their higher computational costs relative to MLPs.
• Expanding the empirical evaluation to a wider array of graph learning tasks to establish more comprehensive benchmarks.

In conclusion, while the introduction of KANs into GNN architectures presents some challenges, particularly in terms of computational efficiency, their potential advantages in terms of performance and interpretability warrant further investigation and development within the graph learning community.