Hyperbolic Graph Diffusion Model

TL;DR

HGDM integrates hyperbolic geometry into diffusion-based graph generation by learning hyperbolic node representations with a Hyperbolic VAE and performing reverse diffusion in hyperbolic space for node features, while modeling adjacency in Euclidean space. The model introduces a novel Hyperbolic Graph Attention layer to maintain efficiency and proposes a hyperbolic score-based framework with encoders/decoders that operate in tangent spaces and then warp back to the hyperbolic manifold. Empirical results on generic graphs and molecular datasets show HGDM outperforms strong baselines, especially for highly hierarchical graphs, with a notable 48% gain in generation quality under hierarchical structure. The work demonstrates the practical value of matching diffusion dynamics to the intrinsic geometry of graph data, and provides code to enable reproducibility and further development.

Abstract

Diffusion generative models (DMs) have achieved promising results in image and graph generation. However, real-world graphs, such as social networks, molecular graphs, and traffic graphs, generally share non-Euclidean topologies and hidden hierarchies. For example, the degree distributions of graphs are mostly power-law distributions. The current latent diffusion model embeds the hierarchical data in a Euclidean space, which leads to distortions and interferes with modeling the distribution. Instead, hyperbolic space has been found to be more suitable for capturing complex hierarchical structures due to its exponential growth property. In order to simultaneously utilize the data generation capabilities of diffusion models and the ability of hyperbolic embeddings to extract latent hierarchical distributions, we propose a novel graph generation method called, Hyperbolic Graph Diffusion Model (HGDM), which consists of an auto-encoder to encode nodes into successive hyperbolic embeddings, and a DM that operates in the hyperbolic latent space. HGDM captures the crucial graph structure distributions by constructing a hyperbolic potential node space that incorporates edge information. Extensive experiments show that HGDM achieves better performance in generic graph and molecule generation benchmarks, with a $48\%$ improvement in the quality of graph generation with highly hierarchical structures.

Figures (8)

• Figure 1: The number of points in 2D hyperbolic and Euclidean space that can be placed at radius $r$ from the center point while maintaining a distance of at least $s = 0.1$ from each other.
• Figure 2: The average distance of the hyperbolic embedding of the nodes from the origin in different datasets with the degree of the nodes.
• Figure 3: Random samples taken from the HGDM trained on QM9.
• Figure 4: Random samples taken from the HGDM trained on ZINC250k.
• Figure 5: Visualization of the graphs from the Ego small dataset and the generated graphs of HGDM.