https://neckpinch.com/authors/shuliang-bai/New arXiv papers by Shuliang Bai, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2609.04671arxiv:2609.04671Fri, 04 Sep 2026 00:00:00 GMTShuliang Bai, Bobo Hua — We study the inverse-weight Lin–Lu–Yau Ricci flow on finite trees, and prove the convergence of the curvature on every edge via its diffusion equation. Moreover, the limiting curvature is the unique minimum-norm point of a polyhedron determined by the tree. We characterize possible limits of the normalized edge weights, give criteria for their convergence, and show that different positive initial metrics have equivalent omega-limit sets.Ricci flowDiscrete & graph Ricci https://arxiv.org/abs/2601.02673arxiv:2601.02673Tue, 06 Jan 2026 00:00:00 GMTShuliang Bai, Shuang Liu, Xin Lai — In this paper, we propose a type of Ricci flow on graphs where the probability distribution for the Lin-Lu-Yau curvature remains constant over time, and also study the related Forman curvature flow. These two curvature flows coincide on trees. We first prove the existence and uniqueness of solutions for both curvature flows in general graphs. Then, we obtain that the normalized curvature flow on trees converges to a constant curvature metric, and under the uniform measure, a complete classification of trees can be obtained based on the convergence results.Ricci flowOther curvature flowsStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2509.22140arxiv:2509.22140Fri, 26 Sep 2025 00:00:00 GMTShuliang Bai, Bobo Hua, Yong Lin, Shuang Liu — In this paper, we study the evolution of metrics on finite trees under continuous-time Ricci flows based on the Lin-Lu-Yau version of Ollivier Ricci curvature. We analyze long-time dynamics of edge weights and curvatures, providing precise characterizations of their limiting behaviors. We prove that the Ricci flow converges to metric with zero curvature on edges whose normalized weights converge to positive values only if the tree is a caterpillar tree.Ricci flowDiscrete & graph Ricci https://arxiv.org/abs/2509.19989arxiv:2509.19989Wed, 24 Sep 2025 00:00:00 GMTShuliang Bai, Rui Li, Shuang Liu, Xin Lai — 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.Ricci flowStability & uniquenessDiscrete & graph RicciNumerics & visualization