Geometric deep learning (GDL) models have demonstrated a great potential for the analysis of non-Euclidian data. They are developed to incorporate the geometric and topological information of non-Euclidian data into the end-to-end deep learning architectures. Motivated by the recent success of discrete Ricci curvature in graph neural network (GNNs), we propose TorGNN, an analytic Torsion enhanced Graph Neural Network model. The essential idea is to characterize graph local structures with an analytic torsion based weight formula. Mathematically, analytic torsion is a topological invariant that can distinguish spaces which are homotopy equivalent but not homeomorphic. In our TorGNN, for each edge, a corresponding local simplicial complex is identified, then the analytic torsion (for this local simplicial complex) is calculated, and further used as a weight (for this edge) in message-passing process. Our TorGNN model is validated on link prediction tasks from sixteen different types of networks and node classification tasks from three types of networks. It has been found that our TorGNN can achieve superior performance on both tasks, and outperform various state-of-the-art models. This demonstrates that analytic torsion is a highly efficient topological invariant in the characterization of graph structures and can significantly boost the performance of GNNs.

翻译：几何深度学习（GDL）模型在分析非欧几里得数据方面展现出巨大潜力。这类模型旨在将非欧几里得数据的几何与拓扑信息融入端到端深度学习架构中。受近期离散里奇曲率在图神经网络（GNN）中成功应用的启发，我们提出TorGNN——一种基于解析扭结的增强型图神经网络模型。其核心思想是通过基于解析扭结的权重公式来表征图的局部结构。从数学角度看，解析扭结是一种拓扑不变量，能够区分同伦等价但非同胚的空间。在TorGNN中，每条边对应一个局部单纯复形，通过计算该局部单纯复形的解析扭结，并将其作为消息传递过程中的边权重。我们在来自16种不同类型网络的链接预测任务和三种类型网络的节点分类任务上验证了TorGNN模型。实验结果表明，TorGNN在这两类任务上均取得了优越性能，并超越了多种当前最优模型。这表明解析扭结是一种高效的表征图结构的拓扑不变量，能显著提升GNN的性能。