  While Hyperbolic Graph Neural Network (HGNN) has recently emerged as a
powerful tool dealing with hierarchical graph data, the limitations of
scalability and efficiency hinder itself from generalizing to deep models. In
this paper, by envisioning depth as a continuous-time embedding evolution, we
decouple the HGNN and reframe the information propagation as a partial
differential equation, letting node-wise attention undertake the role of
diffusivity within the Hyperbolic Neural PDE (HPDE). By introducing theoretical
principles \textit{e.g.,} field and flow, gradient, divergence, and diffusivity
on a non-Euclidean manifold for HPDE integration, we discuss both implicit and
explicit discretization schemes to formulate numerical HPDE solvers. Further,
we propose the Hyperbolic Graph Diffusion Equation (HGDE) -- a flexible vector
flow function that can be integrated to obtain expressive hyperbolic node
embeddings. By analyzing potential energy decay of embeddings, we demonstrate
that HGDE is capable of modeling both low- and high-order proximity with the
benefit of local-global diffusivity functions. Experiments on node
classification and link prediction and image-text classification tasks verify
the superiority of the proposed method, which consistently outperforms various
competitive models by a significant margin.
