Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs

Shiping Liu; Yunyan Yang

REVIEW 2 major objections 4 minor 13 references

Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read On every connected locally finite weighted graph, the p-Ollivier curvature equals (1−p) times the Lin–Lu–Yau curvature for all p ≥ 1/2.

desk verdict Theorem 1.1 is correct and genuinely extends the Bourne et al. result to weighted graphs, but the flow section leans on an unreviewed preprint and the abstract overclaims. read the letter →

arxiv 2608.05939 v1 pith:MMRMYBUX submitted 2026-08-06 math.DG math.APmath.CO

Shiping Liu , Yunyan Yang This is my paper · ORCID

classification math.DGmath.APmath.CO MSC 05C9905C2247A1049Q20

keywords OlliviercurvatureLin–Lu–YauweightedgraphsidlenessparameterWassersteindistanceoptimaltransportflowdiscreteRicci

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a scaling identity that unifies two standard notions of discrete Ricci curvature on graphs. On any connected weighted graph, for every pair of vertices and every admissible distance function, the authors show that the $p$-Ollivier curvature and the Lin–Lu–Yau curvature satisfy $\kappa_p(x,y)=(1-p)\kappa_{\mathrm{LLY}}(x,y)$ for all $p\in[1/2,1]$. In particular, $\kappa_{\mathrm{LLY}}(x,y)=2\kappa_{1/2}(x,y)$, so the two curvatures carry exactly the same information at idleness $1/2$. The threshold $1/2$ is sharp: on a two-vertex weighted graph the linear relation fails on every interval reaching below $1/2$. Because the equivalence holds for arbitrary distances and for all pairs of vertices, it transfers results between the two curvature theories, and in particular gives a direct route to global existence and uniqueness for the Lin–Lu–Yau curvature flow.

What carries the argument

The load-bearing object is the one-parameter idleness family in (1.4), $\mu^p_x=p\delta_x+(1-p)\mu^*_x$, in which the off-diagonal probability measure $\mu^*_x$ is fixed and only the self-mass at $x$ grows with $p$. This family makes $p\mapsto W_1(\mu^p_x,\mu^p_y)$ convex by optimal-transport duality and lets a single dual potential be optimal at more than one idleness. The proof uses three devices: complementary slackness, which turns the strict positivity of the direct mass $\pi(x,y)\ge 2p-1$ into the equality $\varphi(x)-\varphi(y)=\rho(x,y)$; Lemma 2.2, which says that if one potential is simultaneously optimal at both endpoints of an idleness interval, then $W_1$ is affine there; and the observation that the same potential is automatically optimal at $p=1$ once the equality holds. Together these make $\kappa_p$ affine on $[1/2,1]$, and the endpoint $\kappa_1=0$ converts that affine function into $\kappa_p=(1-p)\kappa_{\mathrm{LLY}}$.

What would settle it

To test the sharpness of the theorem's hypothesis, compute $W_1(\mu^p_x,\mu^p_y)$ for a three-vertex graph with $\rho(x,y)=1$, $\mu^p_y=p\delta_y+(1-p)\delta_x$, and $\mu^p_x=p\delta_x+(1-p)((2-2p)\delta_y+(2p-1)\delta_z)$ on $p\in[1/2,1]$. If the resulting curve is not affine on this interval, the identity $\kappa_p=(1-p)\kappa_{\mathrm{LLY}}$ fails once the off-diagonal measure is allowed to depend on $p$, isolating the fixed $\mu^*_x$ hypothesis in (1.4) as the load-bearing condition.

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Extended reading notes

Core claim

The paper establishes Theorem 1.1: on a connected locally finite weighted graph $G=(V,E,w)$ with any distance function $\rho:V\times V\to[0,\infty)$, for all vertices $x,y\in V$ and all $p\in[1/2,1]$, $\kappa_p(x,y)=(1-p)\kappa_{\mathrm{LLY}}(x,y)$, where $\kappa_{\mathrm{LLY}}(x,y)=\lim_{p\to 1^-}\kappa_p(x,y)/(1-p)$. The proof fixes $p>1/2$, picks an optimal transport plan $\pi$ and an optimal dual potential $\varphi$ for the pair $(\mu^p_x,\mu^p_y)$ under the Wasserstein distance $W_1$, and observes that at least $2p-1$ units of mass must be sent directly from $x$ to $y$. Complementary slackness then forces $\varphi(x)-\varphi(y)=\rho(x,y)$, so the same potential remains optimal at $p=1$; a linearity lemma then makes $p\mapsto W_1(\mu^p_x,\mu^p_y)$, and hence $p\mapsto\kappa_p(x,y)$, affine on $[p,1]$. Passing to the limit $p\downarrow 1/2$ and using $\kappa_1(x,y)=0$ fixes the slope as $\kappa_{\mathrm{LLY}}(x,y)$, yielding the corollary $\kappa_{\mathrm{LLY}}(x,y)=2\kappa_{1/2}(x,y)$ for every pair of vertices.

Load-bearing premise

The identity rests on the convention that the idleness parameter p moves probability mass only between the vertex itself and a fixed off-diagonal distribution; if that distribution were allowed to change with p, the linearity argument would break down and the equivalence would not follow.

Editorial extensions

If this is right

The $1/2$-Ollivier curvature and the Lin–Lu–Yau curvature become interchangeable up to the factor $2$ on every weighted graph, so any theorem proved for one transfers to the other.
Because the identity works for any distance function and any pair of vertices, long-range curvature statements, not just edge curvature statements, inherit the equivalence.
The Lin–Lu–Yau curvature flow on finite weighted graphs can be run as the $1/2$-Ollivier curvature flow; the ODE estimates in Section 3 yield global existence and uniqueness of solutions, giving a simpler proof of the well-posedness of that flow.
The proof directly yields the Laplacian comparison formula $\kappa_{\mathrm{LLY}}(x,y)=\inf_{\varphi\in\mathrm{Lip}_1,\ \varphi(x)-\varphi(y)=\rho(x,y)}(\Delta\varphi(y)-\Delta\varphi(x))/\rho(x,y)$, tying the curvature to the graph Laplacian.
The two-vertex example pins the threshold: for every $c<1/2$ the function $p\mapsto\kappa_p(x,y)$ fails to be linear on $[c,1]$, so no extension of the identity below $p=1/2$ is possible in the general weighted-graph setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: because the proof uses only the metric properties of $\rho$ and the fixed off-diagonal measure $\mu^*_x$, the same scaling relation should hold for Ollivier curvature of Markov chains on arbitrary metric spaces with idleness entering as $p\delta_x+(1-p)\mu^*_x$; testing a non-graph metric space would indicate whether graph structure is essential.
Beyond the paper: a practical computational shortcut follows—solve the optimal transport problem once at $p=1/2$ and multiply by $2$ to obtain the Lin–Lu–Yau curvature, avoiding the limiting procedure $p\to 1^-$; on large networks this could reduce the cost of discrete Ricci curvature computation.
Beyond the paper: the piecewise linear behavior below the threshold, visible in the two-vertex example, suggests a family of sub-threshold curvature flows interpolating between Ollivier and Lin–Lu–Yau dynamics; the paper does not explore these flows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main theorem is the real thing, and it's correct. Theorem 1.1 proves κ_p(x,y) = (1-p) κ_LLY(x,y) for all p ≥ 1/2 on any connected locally finite weighted graph with any distance function, and the two-vertex example shows the threshold is sharp. The proof is a compact, self-contained transport argument: for p > 1/2 every coupling has positive mass on (x,y), so complementary slackness gives a Kantorovich potential with φ(x)-φ(y)=ρ(x,y); that same potential works at p=1, and the linearity lemma forces W1 to be linear on [p,1]. Continuity extends this to [1/2,1]. The argument is clean, and the generalization from Bourne et al.'s degree-bound to arbitrary weights and distances is a real step forward. The sharp-threshold remark is also to the point.

The soft spots are in Section 3, not in the main theorem. The curvature-flow results import a Lipschitz estimate for W1 in the edge weights from an unreviewed preprint by Ma and Yang ([7]), and the existence in Theorem 3.2 is only "up to γ-surgeries." The abstract says the equivalence yields a simple proof of Bai et al.'s global existence result, but the correspondence is not spelled out: the reader cannot tell whether this flow, with its surgery caveat, actually recovers the full Bai et al. statement. That is an overreach. The dependence on an unpublished preprint for the key estimate makes the flow section non-independent; the authors should either prove the estimate or cite a published source. Also, Lemma 2.2's proof has typographical errors in the subscripts of the W1 terms; the intended inequality is clear, so that is a minor fix.

One modeling assumption worth flagging: the equivalence holds for the one-parameter family (1.4) in which the off-diagonal measure μ*_x is independent of p. If μ*_x changed with p, the linearity argument would break. This is stated in the definition, not a hidden gap, but readers should treat it as a condition.

The paper is for people who use Ollivier or Lin-Lu-Yau curvature on weighted graphs—they get a canonical bridge and a sharper threshold. I would cite Theorem 1.1. For a journal, I would send it to a referee: the main theorem deserves scrutiny, and the flow section needs either expansion or trimming to match what is actually proved.

Referee Report

2 major / 4 minor

Summary. The paper studies the relation between p-Ollivier curvature and Lin-Lu-Yau curvature on connected locally finite weighted graphs. For probability measures of the form (1.4), in which the off-diagonal kernel µ*_x is fixed and only the idleness p varies, Theorem 1.1 states that κ_p(x,y)=(1-p)κ_LLY(x,y) for every pair of vertices and every p in [1/2,1]. The proof fixes p>1/2, shows any optimal transport plan has positive mass on (x,y), uses complementary slackness to obtain an optimal potential φ with φ(x)-φ(y)=ρ(x,y), observes that φ is also optimal for the Dirac pair at p=1, and applies a simultaneous-optimality lemma to conclude that p ↦ W1 is affine on [p,1]; continuity then covers p=1/2. A two-vertex example shows the threshold is sharp. Section 3 applies the equivalence to p-Ollivier curvature flows and, via κ_LLY=2κ_{1/2}, to the Lin-Lu-Yau curvature flow of Bai et al.

Significance. If Theorem 1.1 holds, it is a clean and useful generalization of Bourne et al.'s result from combinatorial graphs to arbitrary locally finite weighted graphs with arbitrary distance functions. The equivalence κ_LLY=2κ_{1/2} implies that the two curvature notions carry identical information for p≥1/2, and the sharp threshold example is convincing. The proof itself is elegant: it uses only complementary slackness, Kantorovich duality, and convexity of the Wasserstein distance in the idleness parameter, and the central theorem is self-contained. The curvature-flow application is potentially valuable, but its current presentation relies on an imported estimate from an unreviewed preprint and on an imprecise surgery procedure; these issues do not affect Theorem 1.1 but need to be addressed before the flow claims can be accepted as written.

major comments (2)

[Section 3, Eq. (3.2)] The key estimate |W1(µ_p_x,µ_p_y)−W1(µ~_p_x,µ~_p_y)|≤C|w⃗−w~⃗| is not proved in the manuscript; it is asserted by reference to [7, Lemma 3.1], an unreviewed preprint by the same group. Theorems 3.1 and 3.2 and the abstract's claim that the paper gives 'a simple proof' of the Bai et al. curvature flow all depend on this estimate. Please either prove (3.2) directly from hypotheses (i)-(iii), which appears feasible by a triangle-inequality argument, or explicitly state that Theorems 3.1-3.2 are consequences of [7] and separate them from the paper's original contribution.
[Section 3, Theorem 3.2] The phrase 'up to γ-surgeries' is not made precise. When an edge satisfies w_xy/ρ(x,y)>γ, the graph, the dimension of the ODE system, and the hypotheses (i)-(iii) all change discontinuously, but the proof does not define a solution concept through such events or explain whether the hypotheses are preserved after surgery. Please give a precise definition of the surgery procedure and of the piecewise solution, or state Theorem 3.2 as a direct corollary of [7] where these details may already be settled.

minor comments (4)

[Section 2, Lemma 2.2] Inequality (2.4) contains a typographical error: the terms should be W1(µ^{p1}_x, µ^{p1}_y) and W1(µ^{p2}_x, µ^{p2}_y), not W1(µ^{p1}_x, µ^{p2}_x) and W1(µ^{p2}_x, µ^{p2}_y). As printed, the inequality does not state the convexity upper bound.
[Proof of Theorem 1.1] The step 'By continuity, the function p↦W1(...) is linear on [1/2,1]' skips a small argument. Since for every p>1/2 the function is affine on [p,1], one should explicitly note that overlapping intervals force a common slope, and then use continuity at p=1/2; please add a sentence.
[Section 3, Remark 3.3] Because κ_LLY=2κ_{1/2}, the Lin-Lu-Yau flow d/dt w_xy=-κ_LLY w_xy coincides with the 1/2-Ollivier flow (3.5) only after rescaling time by a factor 2. Global existence and uniqueness are invariant under this constant time change, but the equivalence should be stated explicitly rather than as 'becomes'.
[General] There are several minor typographical issues: 'Lipshitz' should be 'Lipschitz', 'solutiona' should be 'solution', 'We will forward the proof' should be 'We will provide the proof', and the phrase in Theorem 3.1 'µ_x and µ*_x are two probability measures on V and V\{x} respectively as in (1.4)' should be clarified, since (1.4) defines µ^p_x rather than µ_x.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: Theorem 1.1 is proved from optimal transport duality and complementary slackness; the flow section cites a same-group preprint for a Lipschitz estimate, but this is not a circular input and does not feed back into the main theorem.

full rationale

Theorem 1.1 is not circular. The Lin–Lu–Yau curvature is defined by the limit in (1.7), while the theorem establishes genuine linearity of p ↦ W_1(µ^p_x, µ^p_y) on [1/2,1]. For p > 1/2 the proof shows any optimal plan has π(x,y) ≥ 2p − 1 > 0, so complementary slackness forces ϕ(x) − ϕ(y) = ρ(x,y); the same ϕ is then optimal for the Dirac pair at p = 1, and Lemma 2.2 (proved from convexity and simultaneous optimality, apart from harmless subscript typos in the displayed inequalities) gives linearity on [p,1], extended to [1/2,1] by continuity. This does not assume the identity it proves. The sharpness example in Remark 2.3 is independent. Remark 2.4 re-derives the Münch–Wojciechowski formula as a consequence of the theorem rather than using it as input. The only self-citation concern is in Section 3: the local Lipschitz estimate (3.2) is quoted from [7, Lemma 3.1], a same-group preprint, and Theorems 3.1–3.2 are said to follow as in [7, Theorem 2.1]. This makes the flow existence proof depend on an external/same-group source, but the cited estimate is logically prior to and independent of Theorem 1.1; it is a missing-proof/self-citation issue, not a circular reduction of the central claim.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central theorem has no fitted parameters and no invented objects. The flow section introduces an unproved Lipschitz estimate as an external input, which is the main unseen dependency.

assumptions (4)

standard math Standard optimal transport duality and complementary slackness apply to finite-support measures on locally finite metric graphs.
Used in Lemma 2.1 with citations to Minoux and Villani; this is foundational background.
domain assumption The random walk measures are of the form mu^p_x = p delta_x + (1-p) mu^*_x, with mu^*_x fixed as p varies.
Equation (1.4) defines the measures; the proof of Theorem 1.1 needs the off-diagonal part to be independent of p.
domain assumption In the flow theorems, rho(x,y) and mu^*_x are locally Lipschitz in the edge weight vector w (hypotheses (i)-(iii) of Theorem 3.1).
These hypotheses are assumed and are needed for the local existence argument.
ad hoc to paper The Wasserstein distance is Lipschitz in the edge weights, giving (3.2).
This estimate is taken from [7, Lemma 3.1], a preprint by the same group, and is not proved in this paper.

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Cite this review

Pith. "Pith review of Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs." pith.science (2026). https://pith.science/paper/MMRMYBUX

@misc{pith2026260805939,
  author       = {Pith},
  title        = {Pith review of: Equivalence of Lin--Lu--Yau curvature and 1/2-Ollivier curvature on weighted graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MMRMYBUX}},
  note         = {Machine review of arXiv:2608.05939}
}

abstract

In this note, we prove that, on weighted graphs, the Lin--Lu--Yau curvature coincides with the $p$-Ollivier curvature up to scaling whenever the idleness parameter $p\geq 1/2$. Moreover, the threshold $1/2$ is sharp. This extends an earlier result of Bourne et al. (Ollivier--Ricci idleness functions of graphs, SIAM J. Discrete Math., 32 (2018), no. 2, 1408-1424), where combinatorial graphs were considered. This observation yields a simple proof for the global existence and uniqueness of solutions of the Lin--Lu--Yau curvature flow in Bai et al. (Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages

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arXiv 2024
[1]

S. Bai, Y . Lin, L. Lu, Z. Wang, S. Yau,Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747

work page 2024
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D. P. Bourne, D. Cushing, S. Liu, F. Münch, N. Peyerimhoff,Ollivier–Ricci idleness func- tions of graphs, SIAM Journal on Discrete Mathematics 32 (2018), no. 2, 1408-1424

work page 2018
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Cushing, S

D. Cushing, S. Kamtue,Long-scale Ollivier Ricci Curvature of graphs, Anal. Geom. Metr. Spaces 7 (2019), 22-44

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[9]

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F. Münch, R. K. Wojciechowski,Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds, Adv. Math. 356 (2019), 106759

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2026

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [7]

J. Ma, Y . Yang,Evolution of weights on a connected finite graph, arXiv: 2411.06393, 2024

arXiv 2024

[2] [1]

S. Bai, Y . Lin, L. Lu, Z. Wang, S. Yau,Ollivier Ricci-flow on weighted graphs, Amer. J. Math. 146 (2024), 1723-1747

work page 2024

[3] [2]

D. P. Bourne, D. Cushing, S. Liu, F. Münch, N. Peyerimhoff,Ollivier–Ricci idleness func- tions of graphs, SIAM Journal on Discrete Mathematics 32 (2018), no. 2, 1408-1424

work page 2018

[4] [3]

Cushing, S

D. Cushing, S. Kamtue,Long-scale Ollivier Ricci Curvature of graphs, Anal. Geom. Metr. Spaces 7 (2019), 22-44

work page 2019

[5] [4]

X. Lai, S. Bai, Y . Lin,Normalized discrete Ricci flow used in community detection, Phys. A 597 (2022), 127251

work page 2022

[6] [5]

Y . Lin, L. Lu, S. T. Yau,Ricci curvature of graphs, Tohoku Math. J. 63 (2011), 605-627

work page 2011

[7] [6]

J. Ma, Y . Yang,A modified Ricci flow on arbitrary weighted graph, J. Geom. Anal. 35 (2025), 332

work page 2025

[8] [8]

Minoux,Mathematical programming: theory and algorithms, John Wiley & Sons, 1986

M. Minoux,Mathematical programming: theory and algorithms, John Wiley & Sons, 1986. 8

work page 1986