Higher-Order GNNs Meet Efficiency: Sparse Sobolev Graph Neural Networks

• URL: http://arxiv.org/abs/2411.04570v1
• Date: Thu, 07 Nov 2024 09:53:11 GMT
• Title: Higher-Order GNNs Meet Efficiency: Sparse Sobolev Graph Neural Networks
• Authors: Jhony H. Giraldo, Aref Einizade, Andjela Todorovic, Jhon A. Castro-Correa, Mohsen Badiey, Thierry Bouwmans, Fragkiskos D. Malliaros,
• Abstract summary: Graph Neural Networks (GNNs) have shown great promise in modeling relationships between nodes in a graph. Previous studies have primarily attempted to utilize the information from higher-order neighbors in the graph. We make a fundamental observation: the regular and the Hadamard power of the Laplacian matrix behave similarly in the spectrum. We propose a novel graph convolutional operator based on the sparse Sobolev norm of graph signals.
• Score: 6.080095317098909
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• Abstract: Graph Neural Networks (GNNs) have shown great promise in modeling relationships between nodes in a graph, but capturing higher-order relationships remains a challenge for large-scale networks. Previous studies have primarily attempted to utilize the information from higher-order neighbors in the graph, involving the incorporation of powers of the shift operator, such as the graph Laplacian or adjacency matrix. This approach comes with a trade-off in terms of increased computational and memory demands. Relying on graph spectral theory, we make a fundamental observation: the regular and the Hadamard power of the Laplacian matrix behave similarly in the spectrum. This observation has significant implications for capturing higher-order information in GNNs for various tasks such as node classification and semi-supervised learning. Consequently, we propose a novel graph convolutional operator based on the sparse Sobolev norm of graph signals. Our approach, known as Sparse Sobolev GNN (S2-GNN), employs Hadamard products between matrices to maintain the sparsity level in graph representations. S2-GNN utilizes a cascade of filters with increasing Hadamard powers to generate a diverse set of functions. We theoretically analyze the stability of S2-GNN to show the robustness of the model against possible graph perturbations. We also conduct a comprehensive evaluation of S2-GNN across various graph mining, semi-supervised node classification, and computer vision tasks. In particular use cases, our algorithm demonstrates competitive performance compared to state-of-the-art GNNs in terms of performance and running time.

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