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Hyperbolic Geometric Latent Diffusion Model for Graph Generation

Xingcheng Fu, Yisen Gao, Yuecen Wei, Qingyun Sun, Hao Peng, Jianxin Li, Xianxian Li

TL;DR

HypDiff introduces diffusion in a hyperbolic latent space to generate graphs with complex non-Euclidean topologies. By coupling a hyperbolic autoencoder with a radial–angular constrained diffusion process, it addresses Gaussian additivity limitations in hyperbolic space and preserves topology through tangent-plane diffusion guided by cluster structure. Empirical results on synthetic and real-world datasets show improved node classification and graph-generation fidelity and efficiency compared with state-of-the-art baselines. The approach demonstrates that hyperbolic geometry provides natural, interpretable priors for topology-preserving graph generation with practical computational benefits.

Abstract

Diffusion models have made significant contributions to computer vision, sparking a growing interest in the community recently regarding the application of them to graph generation. Existing discrete graph diffusion models exhibit heightened computational complexity and diminished training efficiency. A preferable and natural way is to directly diffuse the graph within the latent space. However, due to the non-Euclidean structure of graphs is not isotropic in the latent space, the existing latent diffusion models effectively make it difficult to capture and preserve the topological information of graphs. To address the above challenges, we propose a novel geometrically latent diffusion framework HypDiff. Specifically, we first establish a geometrically latent space with interpretability measures based on hyperbolic geometry, to define anisotropic latent diffusion processes for graphs. Then, we propose a geometrically latent diffusion process that is constrained by both radial and angular geometric properties, thereby ensuring the preservation of the original topological properties in the generative graphs. Extensive experimental results demonstrate the superior effectiveness of HypDiff for graph generation with various topologies.

Hyperbolic Geometric Latent Diffusion Model for Graph Generation

TL;DR

HypDiff introduces diffusion in a hyperbolic latent space to generate graphs with complex non-Euclidean topologies. By coupling a hyperbolic autoencoder with a radial–angular constrained diffusion process, it addresses Gaussian additivity limitations in hyperbolic space and preserves topology through tangent-plane diffusion guided by cluster structure. Empirical results on synthetic and real-world datasets show improved node classification and graph-generation fidelity and efficiency compared with state-of-the-art baselines. The approach demonstrates that hyperbolic geometry provides natural, interpretable priors for topology-preserving graph generation with practical computational benefits.

Abstract

Diffusion models have made significant contributions to computer vision, sparking a growing interest in the community recently regarding the application of them to graph generation. Existing discrete graph diffusion models exhibit heightened computational complexity and diminished training efficiency. A preferable and natural way is to directly diffuse the graph within the latent space. However, due to the non-Euclidean structure of graphs is not isotropic in the latent space, the existing latent diffusion models effectively make it difficult to capture and preserve the topological information of graphs. To address the above challenges, we propose a novel geometrically latent diffusion framework HypDiff. Specifically, we first establish a geometrically latent space with interpretability measures based on hyperbolic geometry, to define anisotropic latent diffusion processes for graphs. Then, we propose a geometrically latent diffusion process that is constrained by both radial and angular geometric properties, thereby ensuring the preservation of the original topological properties in the generative graphs. Extensive experimental results demonstrate the superior effectiveness of HypDiff for graph generation with various topologies.
Paper Structure (40 sections, 2 theorems, 55 equations, 9 figures, 7 tables, 2 algorithms)

This paper contains 40 sections, 2 theorems, 55 equations, 9 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Graph Generative Diffusion Model
  4. Hyperbolic Graph Learning
  5. Methodology
  6. Hyperbolic Geometric Autoencoding
  7. Hyperbolic Geometric Latent Diffusion Process
  8. Experiment
  9. Datasets
  10. Experimental Setup
  11. Performance Evaluation
  12. Analysis of HypDiff
  13. Conclusion
  14. Acknowledgments
  15. Impact Statements
  16. ...and 25 more sections

Key Result

Theorem 3.1

Given the hyperbolic clustering parameter $k \in [1,n]$, which represents the number of sectors dividing the hyperbolic space (disk). The hyperbolic anisotropic diffusion is equivalent to directional diffusion in the Klein model $\mathbb{K}^{n}_{c}$ with multi-curvature ${c_{i\in{|k|}}}$, which is a

Figures (9)

  • Figure 1: Visualization of node embeddings by singular value decomposition (SVD); (a) Original structure visualization of the NCAA football graph and different colors indicate different labels(teams); (b) Visualization of node embeddings in 2D Euclidean space and planar projection; (c) Visualization of node embeddings in 2D hyperbolic space and Poincaré disk projection.
  • Figure 2: (a) Geometric interpretation of the hyperbolic geometry, which unifies the radius and angle measurements in polar coordinates and interprets as popularity and similarity respectively; (b) Hyperbolic latent diffusion processing with isotropic/anisotropic noise;
  • Figure 3: An illustration of HypDiff architecture.
  • Figure 4: Ablation study results.
  • Figure 5: Sensitivity analysis of geometric constraints.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • proof
  • proof
  • proof