Semi-Supervised Classification with Graph Convolutional Networks (1609.02907v4)

Abstract: We present a scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks which operate directly on graphs. We motivate the choice of our convolutional architecture via a localized first-order approximation of spectral graph convolutions. Our model scales linearly in the number of graph edges and learns hidden layer representations that encode both local graph structure and features of nodes. In a number of experiments on citation networks and on a knowledge graph dataset we demonstrate that our approach outperforms related methods by a significant margin.

Semi-Supervised Classification with Graph Convolutional Networks (GCNs)

Thomas N. Kipf and Max Welling propose an innovative approach for semi-supervised learning on graph-structured data using an efficient implementation of convolutional neural networks operating directly on graphs. Their method introduces a layer-wise propagation rule, motivated through a localized first-order approximation of spectral graph convolutions, offering both computational efficiency and robust performance.

Introduction

The core focus of this paper is the classification of nodes within graph structures, such as citation networks, where labels are sparse. The method incorporates the graph structure directly within a convolutional neural network (CNN), thus effectively utilizing both the node features and their structural context. This is a departure from traditional graph-based semi-supervised learning techniques that typically rely on explicit regularization using a graph Laplacian.

Methodology

Graph Convolutional Network (GCN)

The GCN presented in the paper employs a specific propagation rule:

$$H^{(l+1)}= \sigma\left(\tilde{D}^{-1/2} \tilde{A}\tilde{D}^{-1/2}H^{(l)} W^{(l)} \right),$$

where $\tilde{A} = A + I_N$ denotes the adjacency matrix with added self-loops, and $\tilde{D}$ is the corresponding degree matrix. This rule stems from a first-order Chebyshev polynomial approximation of spectral graph convolutions, simplifying computational complexity to $\mathcal{O}(|\mathcal{E}|FC)$ per layer.

Spectral Graph Convolutions

The model leverages spectral graph convolutions, defining them as the multiplication of a node feature signal with a filter in the graph's Fourier domain. This operation is approximated using Chebyshev polynomials, ensuring that computational complexity remains linear with respect to the number of graph edges.

Experiments and Results

The GCN was tested against several benchmark datasets: Citeseer, Cora, and Pubmed citation networks, and a knowledge graph dataset NELL. Results indicated that the GCN significantly outperformed traditional methods, with particular improvements noted in classification accuracy and computational efficiency.

Numerical Results

• Citeseer: Achieved an accuracy of 70.3% with a training time of 7 seconds per epoch.
• Cora: Achieved an accuracy of 81.5% with a training time of 4 seconds per epoch.
• Pubmed: Achieved an accuracy of 79.0% with a training time of 38 seconds per epoch.
• NELL: Achieved an accuracy of 66.0% with a training time of 48 seconds per epoch.

This demonstrates a substantial improvement over competing methods like Planetoid, which recorded lower accuracy and higher training times.

Implications

Practical Implications

GCNs scale linearly with the number of graph edges, making it viable for large-scale applications in various domains like social networks, citation networks, and knowledge graphs. This approach mitigates the usual limitations of semi-supervised learning methods that either excessively rely on the assumption that connected nodes are similar or require complex multi-step optimization procedures as seen in models like DeepWalk or PLANETOID.

Theoretical Implications

From a theoretical perspective, the paper ties the GCN framework to the Weisfeiler-Lehman (WL) graph isomorphism test, showing that the proposed model can be seen as a differentiable, parameterized extension of the WL algorithm. This connection not only provides a deeper understanding of GCN's capability in capturing graph structural properties but also underlines its potential in distinguishing non-isomorphic graphs.

Future Directions

The paper opens several promising directions for future research:

1. Scalability: Implementing mini-batch stochastic gradient descent could significantly enhance scalability.
2. Directed Graphs and Edge Features: Extending the GCN to naturally incorporate directed edges and edge features would broaden its applicability.
3. Complexity Reduction: Further approximations and optimizations could reduce computational load for extremely large graphs.
4. Extended Architectures: Exploring deeper and more complex GCN architectures potentially equipped with residual connections or adaptive gating mechanisms could enhance learning in broader contexts.

Conclusion

Kipf and Welling introduce an efficient, scalable GCN model that combines spectral graph theory with deep learning. Through rigorous experimentation and comparative analysis, the paper establishes the GCN's superiority in a semi-supervised learning context, paving the way for robust, scalable graph-based machine learning models. This research marks a significant step forward in leveraging graph structures directly in neural network models, thereby promising substantial advancements in both computational efficiency and predictive performance.