Title: Ricci Flow on Weighted Digraphs with Balancing Factor Show Chinese title

Title: 带平衡因子的加权有向图上的里奇流

Abstract: Ricci curvature and Ricci flow have proven to be powerful tools for analyzing the geometry of discrete structures, particularly on undirected graphs, where they have been applied to tasks ranging from community detection to graph representation learning. However, their development on directed graphs remains limited, with Ricci flow being especially underexplored. In this work, we introduce a rigorous formulation of Ricci flow on directed weighted graphs, which evolves edge weights while preserving distances, and establish both the existence and uniqueness of its solutions. To capture the essence of asymmetry in directed networks and to enhance the capability of modeling more flexible structures, we incorporate a node-wise balancing factor that regulates between outflow and inflow. Building on the continuous Ricci flow evolution framework, we propose a discrete Ricci flow algorithm that is applicable to numerical computing. Numerical studies on various directed graph examples demonstrate the capacity of the proposed flow to reveal structural asymmetry and dynamic evolutions. Show Chinese abstract

Abstract: 黎曼曲率和黎曼流已被证明是分析离散结构几何的强大工具，特别是在无向图上，它们被应用于从社区检测到图表示学习的各种任务。 然而，它们在有向图上的发展仍然有限，尤其是黎曼流方面尤其缺乏探索。 在本工作中，我们引入了有向加权图上的黎曼流的严格公式，该公式在保持距离的同时演变边权重，并建立了其解的存在性和唯一性。 为了捕捉有向网络中的不对称性并增强建模更灵活结构的能力，我们引入了一个节点级的平衡因子，用于调节流出和流入之间的关系。 基于连续黎曼流演化框架，我们提出了一种适用于数值计算的离散黎曼流算法。 在各种有向图示例上的数值研究展示了所提出的流揭示结构不对称性和动态演化的潜力。