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

Jhony H. Giraldo, Aref Einizade, Andjela Todorovic, Jhon A. Castro-Correa, Mohsen Badiey, Thierry Bouwmans, Fragkiskos D. Malliaros

TL;DR

This work proposes a novel graph convolutional operator, known as Sparse Sobolev GNN (S2-GNN), which employs Hadamard products between matrices to maintain the sparsity level in graph representations and theoretically analyzes the stability of S2-GNN to show the robustness of the model against possible graph perturbations.

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.

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

TL;DR

This work proposes a novel graph convolutional operator, known as Sparse Sobolev GNN (S2-GNN), which employs Hadamard products between matrices to maintain the sparsity level in graph representations and theoretically analyzes the stability of S2-GNN to show the robustness of the model against possible graph perturbations.

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.

Paper Structure

This paper contains 27 sections, 7 theorems, 44 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Mathematical Notation
  5. Graph Signals
  6. Graph Convolutional Filters
  7. Sparse Sobolev Graph Neural Network
  8. Sobolev Norm
  9. Sparse Sobolev Norm
  10. Graph Sparse Sobolev Layer
  11. Re-normalization Trick and Sobolev Term
  12. Relationship with Graph Convolutional Network and Computational Complexity
  13. Theoretical Stability Analysis
  14. Experimental Evaluation
  15. Datasets
  16. ...and 12 more sections

Key Result

Theorem 1

The sparse Sobolev norm $\Vert \mathbf{x} \Vert_{(\rho),\epsilon} \triangleq \Vert (\mathbf{L}+\epsilon \mathbf{I})^{(\rho/2)} \mathbf{x} \Vert$ satisfies the basic properties of vector norms for $\epsilon>0$ (for $\epsilon=0$, we obtain a semi-norm). Proof: See Appendix app:proof_SSob_norm.

Figures (7)

  • Figure 1: The pipeline of our S2-GNN algorithm. S2-GNN can be used in a broad range of data such as images, text, and videos, among others. However, the step of mapping the original dataset to the data matrix $\mathbf{X}\in \mathbb{R}^{N\times M}$ could be different in each case. Our framework is composed of (a) inference of the graph topology and (b) the S2-GNN architecture.
  • Figure 2: Process of using higher-order relationships in GNNs vs. higher-order sparse convolutions.
  • Figure 3: Eigenvalues penalization for the non-sparse and sparse matrix multiplications of the combinatorial Laplacian matrix.
  • Figure 4: Basic configuration of our S2-GNN architecture with $n$ layers and $\alpha$ branches per layer.
  • Figure 5: Comparison of inference times (in seconds) and memory consumption (in terms of GB) on the underlying ER graphs with varying number of nodes $N\in\{500,1000,5000,10000\}$ and edge probabilities $p_{ER}\in\{0.03,0.04,0.05,0.06\}$.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1
  • Definition 2: Pesenson pesenson2009variational
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Definition 4
  • Theorem 3
  • proof
  • proof
  • Theorem 4
  • ...and 6 more