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L^2GC:Lorentzian Linear Graph Convolutional Networks for Node Classification

Qiuyu Liang, Weihua Wang, Feilong Bao, Guanglai Gao

TL;DR

The paper tackles the mismatch between Euclidean feature transformation and the hierarchical structure of many graphs by introducing a Lorentzian linear GCN that operates in hyperbolic space. It builds a three-stage pipeline consisting of parameter-free propagation in Euclidean space, a Lorentzian linear transformation in hyperbolic space, and prediction back in Euclidean space via logarithmic maps, intentionally omitting nonlinear activations between layers. The approach yields state-of-the-art semi-supervised results on Citeseer and PubMed, with Citeseer at $74.7\%$ and PubMed at $81.3\%$, and achieves substantial training speedups on PubMed compared to nonlinear GCNs, while using far fewer parameters than competing models. This work demonstrates that hyperbolic geometry, particularly the Lorentz model, can effectively encode hierarchical graph structure in a scalable, linear framework, and suggests promising directions toward mixed-space models for graphs with varying structural properties.

Abstract

Linear Graph Convolutional Networks (GCNs) are used to classify the node in the graph data. However, we note that most existing linear GCN models perform neural network operations in Euclidean space, which do not explicitly capture the tree-like hierarchical structure exhibited in real-world datasets that modeled as graphs. In this paper, we attempt to introduce hyperbolic space into linear GCN and propose a novel framework for Lorentzian linear GCN. Specifically, we map the learned features of graph nodes into hyperbolic space, and then perform a Lorentzian linear feature transformation to capture the underlying tree-like structure of data. Experimental results on standard citation networks datasets with semi-supervised learning show that our approach yields new state-of-the-art results of accuracy 74.7$\%$ on Citeseer and 81.3$\%$ on PubMed datasets. Furthermore, we observe that our approach can be trained up to two orders of magnitude faster than other nonlinear GCN models on PubMed dataset. Our code is publicly available at https://github.com/llqy123/LLGC-master.

L^2GC:Lorentzian Linear Graph Convolutional Networks for Node Classification

TL;DR

The paper tackles the mismatch between Euclidean feature transformation and the hierarchical structure of many graphs by introducing a Lorentzian linear GCN that operates in hyperbolic space. It builds a three-stage pipeline consisting of parameter-free propagation in Euclidean space, a Lorentzian linear transformation in hyperbolic space, and prediction back in Euclidean space via logarithmic maps, intentionally omitting nonlinear activations between layers. The approach yields state-of-the-art semi-supervised results on Citeseer and PubMed, with Citeseer at and PubMed at , and achieves substantial training speedups on PubMed compared to nonlinear GCNs, while using far fewer parameters than competing models. This work demonstrates that hyperbolic geometry, particularly the Lorentz model, can effectively encode hierarchical graph structure in a scalable, linear framework, and suggests promising directions toward mixed-space models for graphs with varying structural properties.

Abstract

Linear Graph Convolutional Networks (GCNs) are used to classify the node in the graph data. However, we note that most existing linear GCN models perform neural network operations in Euclidean space, which do not explicitly capture the tree-like hierarchical structure exhibited in real-world datasets that modeled as graphs. In this paper, we attempt to introduce hyperbolic space into linear GCN and propose a novel framework for Lorentzian linear GCN. Specifically, we map the learned features of graph nodes into hyperbolic space, and then perform a Lorentzian linear feature transformation to capture the underlying tree-like structure of data. Experimental results on standard citation networks datasets with semi-supervised learning show that our approach yields new state-of-the-art results of accuracy 74.7 on Citeseer and 81.3 on PubMed datasets. Furthermore, we observe that our approach can be trained up to two orders of magnitude faster than other nonlinear GCN models on PubMed dataset. Our code is publicly available at https://github.com/llqy123/LLGC-master.
Paper Structure (19 sections, 10 equations, 5 figures, 6 tables)

This paper contains 19 sections, 10 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Linear Graph Convolutional Networks
  4. Hyperbolic Graph Convolutional Networks
  5. Preliminaries
  6. The Lorentz Model
  7. Lorentzian version of Function
  8. Methodology
  9. Parameter-Free Propagation
  10. Lorentzian Linear Transformation
  11. Graph Node Prediction
  12. Experiments
  13. Datasets
  14. Baselines and Setup
  15. Performance on Semi-supervised Node Classification
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Visualization of a tree-like hierarchical structure citation network in Euclidean and hyperbolic space. In both graphs, nodes represent documents, and edges represent citations. Each color node represents a type of document.
  • Figure 2: The framework of Lorentzian Linear Graph Convolutional Networks for node Classification.
  • Figure 3: Comparison over training time on PubMed dataset.
  • Figure 4: Comparison node classification result with linear model ($\text{G}^{2}$CN). The color indicates the label category of nodes.
  • Figure 5: Comparison node classification result with hyperbolic model (HYBONET). The color indicates the label category of nodes.