Continuous Geometry-Aware Graph Diffusion via Hyperbolic Neural PDE

We address scalability limitations in Hyperbolic Graph Neural Networks (HGNNs) by reformulating information propagation as a continuous partial differential equation. Treating network depth as temporal evolution and using node-wise attention mechanisms as diffusivity on non-Euclidean manifolds, we introduce the Hyperbolic Graph Diffusion Equation (HGDE)—a flexible vector flow function that can be integrated to obtain expressive hyperbolic node embeddings. HGDE handles both low- and high-order proximity through local-global diffusivity functions. Experiments on node classification, link prediction, and image-text classification demonstrate that the proposed method consistently outperforms competitive models by a significant margin.