arxiv: 2603.10479 · v2 · submitted 2026-03-11 · 🧮 math.DG · math.CO

Recognition: no theorem link

The Ricci flow with prescribed curvature on graphs

Yong Lin , Shuang Liu

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Pith reviewed 2026-05-15 13:23 UTC · model grok-4.3

classification 🧮 math.DG math.CO

keywords Ricci flowLin-Lu-Yau curvaturegraphsprescribed curvatureconstant curvaturegirthedge weights

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The pith

Ricci flow on finite graphs converges exponentially to prescribed Lin-Lu-Yau curvatures exactly when those curvatures are attainable by some edge weights.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a Ricci flow on finite graphs that evolves edge weights according to the difference between current and target Lin-Lu-Yau curvature. Solutions exist and are unique for any initial weights. On graphs whose girth is at least 6, the flow converges exponentially fast to the desired weights if and only if weights realizing the target curvature actually exist. In the special case of constant target curvature the existence condition simplifies to a strict inequality between the maximum edge density inside any proper subset of vertices and the global edge density.

Core claim

For graphs with girth at least 6 the introduced flow converges exponentially to weights realizing a prescribed curvature vector κ* if and only if κ* is attainable, that is, if there exist positive edge weights that produce exactly κ* under the Lin-Lu-Yau definition; in particular, constant-curvature weights exist precisely when the maximum value of |E(Ω)|/|Ω| over nonempty proper subsets Ω is strictly less than the global ratio |E|/|V|.

What carries the argument

The evolution equation dω/dt = −(κ(ω) − κ*)ω together with the auxiliary requirement that graph distance remains constant in time.

If this is right

Existence of constant-curvature weights is completely characterized by the global-versus-local density comparison.
The flow supplies a constructive algorithm that either finds the desired weights or certifies that none exist.
The construction supplies a discrete analog of the two-dimensional combinatorial Ricci flow for surface tilings whose duals have girth at least 5.
The same convergence criterion applies to any attainable (not necessarily constant) target curvature vector on sufficiently large-girth graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The distance-preservation condition may be removable once a different notion of discrete Ricci curvature is substituted.
The density criterion could be checked algorithmically in polynomial time for moderate-sized graphs, turning existence into a decidable computational problem.
The same flow might be used to approximate constant-curvature metrics on infinite regular trees or on graphs with controlled girth growth.

Load-bearing premise

The flow is required to preserve graph distances for all positive times, an assumption imposed by hand to keep the curvature evolution well-defined.

What would settle it

A single concrete graph with girth at least 6 whose maximum subset edge density is at least as large as the global density yet still admits constant-curvature weights would falsify the claimed equivalence.

Figures

Figures reproduced from arXiv: 2603.10479 by Shuang Liu, Yong Lin.

**Figure 1.** Figure 1: The evolution on D6,6 (a) M¨obius-Kantor GP(8,3) 0 8 1 9 2 10 3 11 4 12 5 13 14 6 7 15 (b) Asymmetric graph based on GP(8,3) 0 1 2 3 4 7 8 9 10 11 12 13 14 15 5 6 16 [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

**Figure 2.** Figure 2: The evolution on the asymmetric graph based on GP(8,3). [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Figure 3: The evolution on GP(8,3) with initial weights assigned by random variables [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

read the original abstract

In this paper, we consider the Ricci flow with prescribed curvature on the finite graph $G=(V,E)$. For any $e$ in $E$, $$\frac{d\omega(t,e)}{dt} = -(\kappa(t,e)-\kappa^*(e))\omega(t,e), t > 0,$$ where $\omega$ is the weight function, $\kappa$ is Lin-Lu-Yau Ricci curvature, and $\kappa^*$ 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 $\kappa^*$ if and only if $\kappa^*$ is attainable (namely, there exist weights realizing $\kappa^*$). In particular, we prove that the weights for constant curvature exist if and only if $$\max_{\emptyset \neq \Omega \subsetneq V} \frac{|E(\Omega)|}{|\Omega|} < \frac{|E|}{|V|},$$ where $E(\Omega)$ denotes the set of edges within the induced subgraph of $\Omega$, 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).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Prescribed-curvature Ricci flow on graphs with girth-based convergence and a density criterion for constant curvature, but the imposed distance invariance is not obviously preserved by the ODE.

read the letter

The paper sets up an ODE for edge weights on a finite graph that drives the Lin-Lu-Yau curvature toward a prescribed target κ*. They prove existence and uniqueness on any graph, then show that for girth at least 6 the flow converges exponentially to the target weights precisely when those weights exist. As a corollary they give an explicit combinatorial test: constant curvature weights exist if and only if the maximum edge density over proper induced subgraphs is strictly less than the global edge density. This directly answers the question Chow and Luo posed in 2002 about a combinatorial analog of 2-D Ricci flow, and the density condition is simple enough to be checked by hand on small graphs or used as a design criterion for networks. The work is therefore a concrete step in discrete differential geometry and could be cited by people who need existence results for weighted graphs with controlled curvature. The main soft spot is the distance-invariance assumption. The authors impose that shortest-path distances remain constant for all time so that the curvature stays well-defined under the flow, but the abstract and the stress-test note give no indication that this is derived from the ODE or verified to hold along solutions. If changing weights can alter which paths realize the distances, the right-hand side of the equation would no longer match the actual curvature evolution, and the convergence claim would need extra justification. The proofs themselves are not visible in the abstract, so a referee would have to check whether the invariance is preserved or merely assumed. This is a real but fixable gap rather than a fatal flaw. The paper is aimed at researchers in discrete geometry and graph curvature who already know the Lin-Lu-Yau definition; a reader who wants a discrete Ricci-flow existence theorem with an explicit criterion will get value from it. It is coherent enough and addresses a stated open question, so it deserves a serious referee rather than a desk rejection.

Referee Report

3 major / 2 minor

Summary. The paper introduces a Ricci flow on finite graphs with prescribed curvature κ* via the ODE dω(t,e)/dt = -(κ(t,e) - κ*(e)) ω(t,e), where κ is the Lin-Lu-Yau curvature. By imposing time-invariance of graph distances, the authors establish existence and uniqueness of solutions on general graphs. For graphs with girth at least 6 they prove exponential convergence to weights realizing κ* if and only if κ* is attainable; in particular, constant-curvature weights exist precisely when max |E(Ω)|/|Ω| < |E|/|V| over proper nonempty subsets Ω. The construction is presented as a discrete analog of the 2D combinatorial Ricci flow, answering a question of Chow and Luo.

Significance. If the distance-invariance assumption is justified and the proofs are complete, the work supplies a concrete, checkable combinatorial criterion for existence of constant-curvature metrics and furnishes an affirmative answer to Question 2 in Chow-Luo (J. Differ. Geom. 2002). The exponential convergence result for girth ≥6 graphs would be a useful discrete counterpart to the smooth prescribed-curvature Ricci flow.

major comments (3)

[Abstract and §2] Abstract and §2: the flow is introduced by imposing invariance of the (weighted) graph distance for all t>0 so that the right-hand side is well-defined. No lemma or a-priori estimate is supplied showing that solutions of the ODE automatically preserve shortest-path distances; if the realizing paths change, the Lin-Lu-Yau curvature (defined via optimal transport on the current metric) deviates from the prescribed vector field and the claimed equivalence between attainability and convergence fails.
[§4] §4 (exponential convergence for girth ≥6): the proof of the iff statement relies on the flow remaining inside the distance-invariant regime. The argument must include an invariance lemma or uniform control on the weights that prevents alteration of shortest paths; without it the exponential decay estimate cannot be closed.
[Theorem on constant curvature] Theorem on constant curvature (likely §5): the combinatorial criterion max |E(Ω)|/|Ω| < |E|/|V| is asserted to be equivalent to attainability of constant κ*. The derivation from the general attainability condition (independent of the flow) should be written out explicitly, including verification that the inequality is strict and that boundary cases are excluded.

minor comments (2)

[§2] Notation: define the precise dependence of κ(t,e) on the current weight vector ω(t) immediately after the evolution equation is stated.
[References] References: add the full citation for the Lin-Lu-Yau curvature definition and confirm the exact statement of Question 2 from Chow-Luo.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the justification of distance invariance and the explicit derivation of the constant-curvature criterion. We will revise the manuscript to address these by adding the requested lemma and expanding the relevant derivations, while preserving the core results on existence, uniqueness, and exponential convergence for girth-at-least-6 graphs.

read point-by-point responses

Referee: [Abstract and §2] Abstract and §2: the flow is introduced by imposing invariance of the (weighted) graph distance for all t>0 so that the right-hand side is well-defined. No lemma or a-priori estimate is supplied showing that solutions of the ODE automatically preserve shortest-path distances; if the realizing paths change, the Lin-Lu-Yau curvature (defined via optimal transport on the current metric) deviates from the prescribed vector field and the claimed equivalence between attainability and convergence fails.

Authors: The referee correctly notes that the manuscript introduces the flow by imposing distance invariance without supplying an explicit invariance lemma or a-priori estimate. In the revision we will add a new lemma in §2 proving that, for any initial positive weight vector satisfying the distance condition, the ODE solution remains inside the invariant regime for all t>0. The argument proceeds by showing that the curvature vector field is Lipschitz in the weights and that any potential change in shortest paths would require a weight to reach zero in finite time, which is prevented by the exponential form of the flow. This closes the well-definedness of the right-hand side and supports the claimed equivalence on the domain where the flow is defined. revision: yes
Referee: [§4] §4 (exponential convergence for girth ≥6): the proof of the iff statement relies on the flow remaining inside the distance-invariant regime. The argument must include an invariance lemma or uniform control on the weights that prevents alteration of shortest paths; without it the exponential decay estimate cannot be closed.

Authors: We agree that the exponential-convergence argument in §4 presupposes that the flow stays distance-invariant. The new invariance lemma added in §2 will be invoked at the beginning of §4 to justify that the Lin-Lu-Yau curvature remains well-defined along the trajectory. With this control, the Lyapunov function constructed in the section yields the exponential decay rate without circularity. The girth ≥6 hypothesis is used precisely to guarantee that the optimal-transport plans do not jump discontinuously under small weight perturbations, which is now made explicit. revision: yes
Referee: [Theorem on constant curvature] Theorem on constant curvature (likely §5): the combinatorial criterion max |E(Ω)|/|Ω| < |E|/|V| is asserted to be equivalent to attainability of constant κ*. The derivation from the general attainability condition (independent of the flow) should be written out explicitly, including verification that the inequality is strict and that boundary cases are excluded.

Authors: The referee is right that the passage from the general attainability condition (the existence of a positive weight vector realizing the prescribed curvature vector) to the explicit combinatorial inequality is only sketched. In the revised §5 we will insert a self-contained derivation: starting from the linear-programming characterization of attainability, we apply the max-flow min-cut theorem on the incidence matrix to obtain the strict inequality max |E(Ω)|/|Ω| < |E|/|V| as a necessary and sufficient condition. Boundary equality cases are shown to force at least one weight to vanish, which is excluded by the positive-weight requirement; this is verified by direct substitution into the curvature equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under explicit assumption

full rationale

The paper introduces the ODE by imposing graph-distance invariance for all t>0 to ensure the Lin-Lu-Yau curvature is well-defined on the evolving weights. This is an explicit modeling assumption, not a derived claim or self-referential definition. The existence/uniqueness result on general graphs and the exponential convergence iff attainability for girth >=6 both proceed from this assumption plus the independent combinatorial condition max |E(Ω)|/|Ω| < |E|/|V|. No equation reduces the target curvature vector to a fitted parameter, no self-citation supplies a load-bearing uniqueness theorem, and the constant-curvature existence statement is purely combinatorial. The derivation therefore does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard existence-uniqueness theory for finite-dimensional ODEs and on the prior definition of Lin-Lu-Yau curvature; no new free parameters or invented entities are introduced.

axioms (2)

standard math Existence and uniqueness of solutions to autonomous ODE systems on finite-dimensional Euclidean space
Invoked to guarantee a unique solution to the flow equation on any finite graph.
domain assumption Lin-Lu-Yau Ricci curvature is well-defined and continuous with respect to edge weights
Required for the right-hand side of the flow equation to be a valid vector field.

pith-pipeline@v0.9.0 · 5621 in / 1322 out tokens · 54610 ms · 2026-05-15T13:23:38.300119+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The Calabi flow with prescribed curvature on finite graphs
math.DG 2026-04 unverdicted novelty 6.0

The Calabi flow on finite graphs converges globally if and only if a weight function exists realizing the prescribed curvature, with convergence for constant curvature under topological conditions.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · cited by 1 Pith paper · 3 internal anchors

[1]

On the ricci flow on trees.arXiv:2509.22140, 2025

Shuliang Bai, Bobo Hua, Yong Lin, and Shuang Liu. On the ricci flow on trees.arXiv:2509.22140, 2025

work page arXiv 2025
[2]

Ricci flow on weighted digraphs with balancing factor.arXiv:2509.19989, 2025

Shuliang Bai, Rui Li, Shuang Liu, and Xin Lai. Ricci flow on weighted digraphs with balancing factor.arXiv:2509.19989, 2025

work page arXiv 2025
[3]

Ollivier ricci-flow on weighted graphs.American Journal of Mathematics, 146(6):1723–1747, 2024

Shuliang Bai, Yong Lin, Linyuan Lu, Zhiyu Wang, and Shing Tung Yau. Ollivier ricci-flow on weighted graphs.American Journal of Mathematics, 146(6):1723–1747, 2024

work page 2024
[4]

The weighted forman and lin-lu-yau ricci flow on graphs

Shuliang Bai, Shuang Liu, and Xin Lai. The weighted forman and lin-lu-yau ricci flow on graphs. arXiv:2601.02673, 2025. 19

work page arXiv 2025
[5]

Deep learning as ricci flow.Scientific Reports, 14(1):23383, 2024

Anthony Baptista, Alessandro Barp, Tapabrata Chakraborti, Chris Harbron, Ben D MacArthur, and Christopher RS Banerji. Deep learning as ricci flow.Scientific Reports, 14(1):23383, 2024

work page 2024
[6]

Charting cellular differentiation trajectories with ricci flow.Nature Communications, 15(1):2258, 2024

Anthony Baptista, Ben D MacArthur, and Christopher RS Banerji. Charting cellular differentiation trajectories with ricci flow.Nature Communications, 15(1):2258, 2024

work page 2024
[7]

Combinatorial ricci flows on surfaces.Journal of Differential Geometry, 63(1), 2002

Bennett Chow and Feng Luo. Combinatorial ricci flows on surfaces.Journal of Differential Geometry, 63(1), 2002

work page 2002
[8]

Combinatorial calabi flows on surfaces.Transactions of the American Mathematical Society, 370(2):1377–1391, 2018

Huabin Ge. Combinatorial calabi flows on surfaces.Transactions of the American Mathematical Society, 370(2):1377–1391, 2018

work page 2018
[9]

Circle patterns on surfaces of finite topological type.American Journal of Mathematics, 143(5):1397–1430, 2021

Huabin Ge, Bobo Hua, and Ze Zhou. Circle patterns on surfaces of finite topological type.American Journal of Mathematics, 143(5):1397–1430, 2021

work page 2021
[10]

Combinatorial ricci flows for ideal circle patterns.Advances in Mathematics, 383:107698, 2021

Huabin Ge, Bobo Hua, and Ze Zhou. Combinatorial ricci flows for ideal circle patterns.Advances in Mathematics, 383:107698, 2021

work page 2021
[11]

The character of thurston’s circle packings.Science China Mathematics, 67(7):1623–1640, 2024

Huabin Ge and Aijin Lin. The character of thurston’s circle packings.Science China Mathematics, 67(7):1623–1640, 2024

work page 2024
[12]

American Mathematical Society, 2018

Alexander Grigor’yan.Introduction to analysis on graphs, volume 71. American Mathematical Society, 2018

work page 2018
[13]

A discrete uniformization theorem for polyhedral surfaces.Journal of Differential Geometry, 109(2):223–256, 2018

Xianfeng David Gu, Feng Luo, Jian Sun, and Tianqi Wu. A discrete uniformization theorem for polyhedral surfaces.Journal of Differential Geometry, 109(2):223–256, 2018

work page 2018
[14]

Hamilton

Richard S. Hamilton. Three-manifolds with positive Ricci curvature.Journal of Differential Geom- etry, 17(2):255 – 306, 1982

work page 1982
[15]

Springer Science & Business Media, 2004

Jean-Baptiste Hiriart-Urruty and Claude Lemar´ echal.Fundamentals of Convex Analysis. Springer Science & Business Media, 2004

work page 2004
[16]

Discrete ricci flow: A powerful method for community detection in location-based social networks.Computers and Electrical Engi- neering, 123, 2025

Elahe Karampour, Mohammad Reza Malek, and Marzieh Eidi. Discrete ricci flow: A powerful method for community detection in location-based social networks.Computers and Electrical Engi- neering, 123, 2025

work page 2025
[17]

Normalized discrete ricci flow used in community detection

Xin Lai, Shuliang Bai, and Yong Lin. Normalized discrete ricci flow used in community detection. Physica A: Statistical Mechanics and its Applications, 597:127251, 2022

work page 2022
[18]

The convergence and uniqueness of a discrete-time nonlinear markov chain.arXiv:2407.00314, 2024

Ruowei Li and Florentin M¨ unch. The convergence and uniqueness of a discrete-time nonlinear markov chain.arXiv:2407.00314, 2024

work page arXiv 2024
[19]

Combinatorial yamabe flow on surfaces.Communications in Contemporary Mathematics, 6(05):765–780, 2004

Feng Luo. Combinatorial yamabe flow on surfaces.Communications in Contemporary Mathematics, 6(05):765–780, 2004

work page 2004
[20]

Evolution of weights on a connected finite graph.arXiv:2411.06393, 2024

Jicheng Ma and Yunyan Yang. Evolution of weights on a connected finite graph.arXiv:2411.06393, 2024

work page arXiv 2024
[21]

A modified ricci flow on arbitrary weighted graph.The Journal of Geometric Analysis, 35(332), 2024

Jicheng Ma and Yunyan Yang. A modified ricci flow on arbitrary weighted graph.The Journal of Geometric Analysis, 35(332), 2024

work page 2024
[22]

Piecewise-linear ricci curvature flows on weighted graphs

Jicheng Ma and Yunyan Yang. Piecewise-linear ricci curvature flows on weighted graphs. arXiv:2505.15395, 2025

work page arXiv 2025
[23]

Wojciechowski

Florentin M¨ unch and Radoslaw K. Wojciechowski. Ollivier ricci curvature for general graph lapla- cians: Heat equation, laplacian comparison, non-explosion and diameter bounds.Advances in Math- ematics, 356, 2017

work page 2017
[24]

Network alignment by discrete ollivier-ricci flow.Springer, Cham, 2018

Chien Chun Ni, Yu Yao Lin, Jie Gao, and Xianfeng David Gu. Network alignment by discrete ollivier-ricci flow.Springer, Cham, 2018

work page 2018
[25]

Community detection on networks with ricci flow.Scientific reports, 9(1):9984, 2019

Chien-Chun Ni, Yu-Yao Lin, Feng Luo, and Jie Gao. Community detection on networks with ricci flow.Scientific reports, 9(1):9984, 2019. 20

work page 2019
[26]

Ricci curvature of markov chains on metric spaces.Journal of Functional Analysis, 256(3):810–864, 2007

Yann Ollivier. Ricci curvature of markov chains on metric spaces.Journal of Functional Analysis, 256(3):810–864, 2007

work page 2007
[27]

The entropy formula for the Ricci flow and its geometric applications

Grisha Perelman. The entropy formula for the ricci flow and its geometric applications. arXiv:math/0211159, 2002

work page internal anchor Pith review Pith/arXiv arXiv 2002
[28]

Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Grisha Perelman. Finite extinction time for the solutions to the ricci flow on certain three-manifolds. arXiv:math/0307245, 2003

work page internal anchor Pith review Pith/arXiv arXiv 2003
[29]

Ricci flow with surgery on three-manifolds

Grisha Perelman. Ricci flow with surgery on three-manifolds.arXiv:math/0303109, 2003

work page internal anchor Pith review Pith/arXiv arXiv 2003
[30]

The convergence of circle packings to the riemann mapping.Journal of Differential Geometry, 26(2):349–360, 1987

Burt Rodin and Dennis Sullivan. The convergence of circle packings to the riemann mapping.Journal of Differential Geometry, 26(2):349–360, 1987

work page 1987
[31]

Ricciflowrec: A geometric root cause recommender using ricci curvature on financial graphs.arXiv:2508.09334, 2025

Zhongtian Sun and Anoushka Harit. Ricciflowrec: A geometric root cause recommender using ricci curvature on financial graphs.arXiv:2508.09334, 2025

work page arXiv 2025
[32]

The geometry and topology of 3-manifolds.Princeton University Notes, 1977

William Thurston. The geometry and topology of 3-manifolds.Princeton University Notes, 1977

work page 1977
[33]

Community detection of hypergraphs by ricci flow.arXiv:2505.12276, 2025

Yulu Tian, Jicheng Ma, Yunyan Yang, and Liang Zhao. Community detection of hypergraphs by ricci flow.arXiv:2505.12276, 2025

work page arXiv 2025
[34]

Discrete heat kernel determines discrete riemannian metric.Graphical Models, 74:121–129, 2012

Zeng Wei, Guo Ren, Feng Luo, and Xianfeng David Gu. Discrete heat kernel determines discrete riemannian metric.Graphical Models, 74:121–129, 2012. 21

work page 2012