GGBall: Graph Generative Model on Poincar'e Ball

2025-06-08

AI Summary

To address the limitation of Euclidean space in modeling exponentially hierarchical graph structures, this paper introduces the first flow-matching generative framework based on the Poincaré ball model. Methodologically: (1) we design a hyperbolic flow-matching prior with closed-form geodesic parameterization; (2) we develop fully manifold-intrinsic hyperbolic GNN and Transformer layers operating entirely within the hyperbolic space; and (3) we propose HVQVAE—the first vector-quantized variational autoencoder for hyperbolic latent spaces—ensuring geometric consistency. Evaluated on Community-Small and Ego-Small datasets, our method reduces degree distribution MMD by 75.3% and 42.7%, respectively, outperforming state-of-the-art approaches. This work establishes the first stable and scalable hyperbolic graph generation framework, introducing a novel paradigm for hierarchical graph representation and synthesis.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Natural Language Processing: Generation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Generating graphs with hierarchical structures remains a fundamental challenge due to the limitations of Euclidean geometry in capturing exponential complexity. Here we introduce extbf{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}.

Problem

• Generating hierarchical graphs in non-Euclidean space
• Modeling complex latent distributions with hyperbolic geometry
• Preserving topological hierarchies in graph generation

Innovation

• Hyperbolic framework for hierarchical graph generation
• HVQVAE with Riemannian flow matching prior
• Hyperbolic GNN and Transformer layers in manifold