Depth-Adaptive Graph Neural Networks via Learnable Bakry-Émery Curvature

Graph Neural Networks (GNNs) have demonstrated strong representation learning capabilities for graph-based tasks. Recent advances on GNNs leverage geometric properties, such as curvature, to enhance their representation capabilities by modeling complex connectivity patterns and information flow within graphs. However, most existing approaches primarily focus on discrete graph topology, overlooking diffusion dynamics and task-specific dependencies essential for effective learning. To address this, we propose a learnable integration of Bakry-Émery curvature, which captures both structural and diffusion aspects of information propagation. We develop an efficient, learnable approximation strategy, making curvature computation scalable for large graphs. Furthermore, we introduce an adaptive depth mechanism that dynamically adjusts message-passing layers per vertex based on its curvature, ensuring efficient propagation. Our theoretical analysis establishes a link between curvature and feature distinctiveness, showing that high-curvature vertices require fewer layers, while low-curvature ones benefit from deeper propagation. Extensive experiments on diverse downstream tasks validate the effectiveness of our approach, showing that the proposed depth-adaptive mechanism consistently uplifts the performance of a wide range of GNN architectures.