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GGBall: Graph Generative Model on Poincaré Ball

Tianci Bu, Chuanrui Wang, Hao Ma, Haoren Zheng, Xin Lu, Tailin Wu

TL;DR

GGBall is introduced, a novel hyperbolic framework for graph generation that integrates geometric inductive biases with modern generative paradigms and reduces degree MMD by over 75% on Community-Small and over 40% on Ego-Small compared to state-of-the-art baselines, demonstrating an improved ability to preserve topological hierarchies.

Abstract

Generating graphs with hierarchical structures remains a fundamental challenge due to the limitations of Euclidean geometry in capturing exponential complexity. Here we introduce \textbf{GGBall}, a novel hyperbolic framework for graph generation that integrates geometric inductive biases with modern generative paradigms. GGBall combines a Hyperbolic Vector-Quantized Autoencoder (HVQVAE) with a Riemannian flow matching prior defined via closed-form geodesics. This design enables flow-based priors to model complex latent distributions, while vector quantization helps preserve the curvature-aware structure of the hyperbolic space. We further develop a suite of hyperbolic GNN and Transformer layers that operate entirely within the manifold, ensuring stability and scalability. Empirically, our model reduces degree MMD by over 75\% on Community-Small and over 40\% on Ego-Small compared to state-of-the-art baselines, demonstrating an improved ability to preserve topological hierarchies. These results highlight the potential of hyperbolic geometry as a powerful foundation for the generative modeling of complex, structured, and hierarchical data domains. Our code is available at \href{https://github.com/AI4Science-WestlakeU/GGBall}{here}.

GGBall: Graph Generative Model on Poincaré Ball

TL;DR

GGBall is introduced, a novel hyperbolic framework for graph generation that integrates geometric inductive biases with modern generative paradigms and reduces degree MMD by over 75% on Community-Small and over 40% on Ego-Small compared to state-of-the-art baselines, demonstrating an improved ability to preserve topological hierarchies.

Abstract

Generating graphs with hierarchical structures remains a fundamental challenge due to the limitations of Euclidean geometry in capturing exponential complexity. Here we introduce \textbf{GGBall}, a novel hyperbolic framework for graph generation that integrates geometric inductive biases with modern generative paradigms. GGBall combines a Hyperbolic Vector-Quantized Autoencoder (HVQVAE) with a Riemannian flow matching prior defined via closed-form geodesics. This design enables flow-based priors to model complex latent distributions, while vector quantization helps preserve the curvature-aware structure of the hyperbolic space. We further develop a suite of hyperbolic GNN and Transformer layers that operate entirely within the manifold, ensuring stability and scalability. Empirically, our model reduces degree MMD by over 75\% on Community-Small and over 40\% on Ego-Small compared to state-of-the-art baselines, demonstrating an improved ability to preserve topological hierarchies. These results highlight the potential of hyperbolic geometry as a powerful foundation for the generative modeling of complex, structured, and hierarchical data domains. Our code is available at \href{https://github.com/AI4Science-WestlakeU/GGBall}{here}.

Paper Structure

This paper contains 121 sections, 1 theorem, 60 equations, 4 figures, 12 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Definition
  3. Hyperbolic Geometry
  4. Riemannian Manifold
  5. Hyperbolic Space and Poincaré Ball Model
  6. Graphs in Hyperbolic Space
  7. Graph Generation in Hyperbolic Space
  8. Problem definition.
  9. Latent factorization.
  10. Two-stage paradigm.
  11. Method
  12. Poincaré Network Architecture
  13. Poincaré Graph Neural Network
  14. Tangent Space Aggregation.
  15. Distance-Modulated Message Function.
  16. ...and 106 more sections

Key Result

Theorem 1

(de Gromov's approximation theorem) For any $\delta$-hyperbolic space, there exists a continuous mapping from the space to an $R$-Tree such that the distances between points are approximately preserved with a small error term. More formally, the mapping $\Phi: \mathcal{X}\rightarrow T$ satisfies the where $\mathcal{X}$ is the $\delta$-hyperbolic space, $T$ is the $R$-Tree, and $k$ is the number of

Figures (4)

  • Figure 1: Degree similarity and edge reconstruction accuracy on reconstructed dataset. Hyperbolic models consistently outperform Euclidean baselines.
  • Figure 2: Overview of our hyperbolic graph generation framework. We encode graphs into a hyperbolic latent space using a Poincaré GNN and geodesic-attention Transformer. The latent representations are quantized via a Poincaré codebook and modeled with a Poincaré flow prior. A hyperbolic Transformer then decodes the latent code to reconstruct or generate graphs, enabling structure-aware generation in non-Euclidean geometry.
  • Figure 3: Geodesic interpolation in hyperbolic latent space between molecular graphs. Top row shows 10-step latent transitions with 2D-projected node embeddings. Lower rows decode these into molecules, with source and target at ends. Green labels mark valid intermediates; red indicates invalid ones. Chemical formulas and ring counts are annotated.
  • Figure 4: Training dynamics of the hyperbolic VAE (HVAE). Left: KL divergence exhibits large oscillations, indicating numerical instability. Right: Edge reconstruction accuracy improves slowly and remains suboptimal. These results highlight the limitations of using continuous probabilistic models in curved latent spaces.

Theorems & Definitions (1)

  • Theorem 1