Continuous Geometry-Aware Graph Diffusion via Hyperbolic Neural PDE

TL;DR

This paper introduces a continuous-time hyperbolic PDE framework for graph neural networks, enhancing scalability and expressiveness in hierarchical graph data modeling.

Contribution

It proposes the Hyperbolic Graph Diffusion Equation (HGDE) with theoretical grounding and numerical schemes, improving deep hyperbolic GNN performance.

Findings

Outperforms existing models on node classification tasks.

Effectively models both local and global graph structures.

Demonstrates superior link prediction and image-text classification results.

Abstract

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.