  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.
