Shuang Liu — Neckpinch

The Ollivier Ricci flow with prescribed curvature on infinite graphs

Mon, 08 Jun 2026 00:00:00 GMT

Bobo Hua, Yong Lin, Shuang Liu — In this paper, we consider the Ricci flow with prescribed curvature on infinite graphs, which reads as \beginequation* \fracddtω(t)=-(κ(t)-κ^*)ω(t), t>0, \endequation* where ω is the edge weight, κ and κ^* are Lin-Lu-Yau Ricci curvature and the prescribed curvature on the set of edges, respectively. First, we establish the existence and uniqueness of the solution to the Ricci flow. Furthermore, we prove the convergence of the Ricci flow for graphs with girth at least 6 under two different conditions. Our convergence result aligns with the conclusion of Rodin and Sullivan (J Differ Geom, 26(2) 1987) that a circle packing in the plane with the hexagonal pattern is the regular hexagonal packing.

The Ricci flow with prescribed curvature on graphs

Wed, 11 Mar 2026 00:00:00 GMT

Yong Lin, Shuang Liu — In this paper, we consider the Ricci flow with prescribed curvature on the finite graph G=(V,E). For any e in E, (dω(t,e))/dt = −(κ(t,e)−κ^*(e))ω(t,e), t > 0, where ω is the weight function, κ is Lin-Lu-Yau Ricci curvature, and κ^* is the prescribed curvature. By imposing invariance of the graph distance with respect to time t, the Ricci flow introduced above characterizes the weight evolution governed by the Lin-Lu-Yau curvature. We first establish the existence and uniqueness of the solution to this equation on general graphs. Furthermore, for graphs with girth of at least 6, we prove that the Ricci flow converges exponentially to weights of κ^* if and only if κ^* is attainable (namely, there exist weights realizing κ^*). In particular, we prove that the weights for constant curvature exist if and only if max_(∅ ≠ Ωsubsetneq V) (|E(Ω)|)/(|Ω|) < (|E|)/(|V|), where E(Ω) denotes the set of edges within the induced subgraph of Ω, and |A| is the cardinality of the set A. Viewing edge weights as metrics on surface tilings with girth of at least 5 or the duals of triangulations with vertex degrees exceeding 5, we demonstrate that our constant Lin-Lu-Yau curvature flow serves as an analog to the 2D combinatorial Ricci flow for piecewise constant curvature metrics, thereby providing an affirmative answer to Question 2 posed by Chow and Luo (J Differ Geom, 63(1) 2002).

The weighted Forman and Lin-Lu-Yau Ricci flow on graphs

Tue, 06 Jan 2026 00:00:00 GMT

Shuliang 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.

On the Ricci flow on Trees

Fri, 26 Sep 2025 00:00:00 GMT

Shuliang 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 Flow on Weighted Digraphs with Balancing Factor

Wed, 24 Sep 2025 00:00:00 GMT

Shuliang 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.