As on arXiv
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.
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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The weighted Forman and Lin-Lu-Yau Ricci flow on graphs
Topics Discrete & graph Ricci, Stability & uniqueness · Same authors Shuliang Bai, Shuang Liu, Xin Lai
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On the Ricci flow on Trees
Topics Discrete & graph Ricci · Same authors Shuliang Bai, Shuang Liu · Shared terms “edge weights”
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The Ollivier Ricci flow with prescribed curvature on infinite graphs
Topics Discrete & graph Ricci, Stability & uniqueness · Same author Shuang Liu · Shared terms “edge weights”
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Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees
Topics Discrete & graph Ricci · Same author Shuliang Bai · Shared terms “edge weights”
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The Ricci flow with prescribed curvature on graphs
Topics Discrete & graph Ricci, Stability & uniqueness · Same author Shuang Liu · Shared terms “edge weights”, “demonstrate”
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Core detection via Ricci curvature flows on weighted graphs
Topics Discrete & graph Ricci · Shared terms “weighted graphs”, “detection”, “undirected graph”
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Evolving edge weights via local entropy flow and cohesion flow on graphs
Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “edge weights”, “community detection”, “weighted graphs”
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A modified Ricci flow on arbitrary weighted graph
Topics Discrete & graph Ricci, Stability & uniqueness · Shared terms “weighted graphs”, “community detection”, “algorithm”
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6D.11Computing Geometric Flows
Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.