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arxiv: 2506.10492 · v1 · submitted 2025-06-12 · 🧮 math.SP

Repelling curvature via ε-repelling Laplacian on positive connected signed graphs

Yong Lin , Shi Wan This is my paper

Pith reviewed 2026-05-19 10:10 UTC · model grok-4.3

classification 🧮 math.SP
keywords signed graphsε-repelling Laplacianrepelling curvatureLin-Lu-Yau curvatureLichnerowicz inequalityresistance curvaturespectral graph theoryconsensus problem
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The pith

The edge ε-repelling curvature on positive connected signed graphs is at most the Lin-Lu-Yau curvature of the underlying graph computed using ε-repelling cost as the transport cost.

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

The paper defines an ε-repelling Laplacian as a positive semidefinite operator on positive connected signed graphs for sufficiently small positive ε. Using the pseudoinverse of this operator, it constructs an ε-repelling cost that serves as a distance on an associated simplex. The authors extend node and edge resistance curvatures to this new setting and establish corresponding Lichnerowicz inequalities. They prove that the edge ε-repelling curvature does not exceed the Lin-Lu-Yau curvature on the underlying graph when the transport cost is taken to be the ε-repelling cost instead of the shortest-path distance. This matters because it offers a way to incorporate repelling interactions from signed edges into curvature measures used in network analysis and geometry.

Core claim

The paper establishes that on any positive connected signed graph, for ε less than the critical value from the consensus problem, the edge ε-repelling curvature is no more than the Lin-Lu-Yau curvature of the underlying graph whose transport cost is the ε-repelling cost rather than the length of the shortest path.

What carries the argument

The ε-repelling Laplacian, a positive semidefinite operator on signed graphs that encodes repelling effects and whose pseudoinverse defines the ε-repelling cost used in the curvature comparison.

If this is right

  • The Lichnerowicz inequalities hold for both node and edge ε-repelling curvatures.
  • The second smallest eigenvalue of the ε-repelling Laplacian admits an upper bound derived from the curvature.
  • The square root of the ε-repelling cost is a distance function among the vertices of the constructed simplex.
  • Node and edge resistance curvatures extend naturally to the ε-repelling versions on signed graphs.

Where Pith is reading between the lines

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

  • This comparison suggests that using repelling costs instead of geodesic distances may systematically lower curvature estimates in signed networks.
  • The construction ties curvature to the consensus problem, potentially linking geometric properties to dynamical stability in signed systems.
  • Similar repelling operators could be explored on graphs with mixed positive and negative edges beyond the positive case.

Load-bearing premise

The signed graph must be positive and connected, and the parameter ε must be strictly less than the constant ε0 determined by the graph's consensus problem.

What would settle it

Direct calculation of the edge ε-repelling curvature and the corresponding Lin-Lu-Yau curvature with ε-repelling cost on a small example like a complete graph with all positive signs; if the inequality is violated for some ε < ε0, the claim would be falsified.

Figures

Figures reproduced from arXiv: 2506.10492 by Shi Wan, Yong Lin.

Figure 2
Figure 2. Figure 2: The corresponding 3−simplex Definition 2.11. Define a matrix Ω σ ϵ ∈ R (n+1)×(n+1), called ϵ−repelling matrix, as follows: Ω σ ϵ (i, j) := (⃗ei − ⃗ej ) TL † ϵ (⃗ei − ⃗ej ) We call Ω σ ϵ (i, j) the ϵ−repelling cost between i and j. Denote by L † ϵ the pseudoinverse of L+ − ϵL− and from the above theorem we know L † ϵ = SST where S is the vertex matrix of some n−simplex. Proposition 2.12. p Ωσ ϵ (i, j) is a … view at source ↗
read the original abstract

The paper defines a positive semidefinite operator called $\epsilon-$repelling Laplacian on a positive connected signed graph where $\epsilon$ is an arbitrary positive number less than a constant $\epsilon_0$ related to the graph's consensus problem. Then we investigate the upper bound of the second smallest eigenvalue of $\epsilon-$repelling Laplacian. Besides, we use the pseudoinverse of $\epsilon-$repelling Laplacian to construct a simplex as well as $\epsilon-$repelling cost whose square root turns out to be a distance among the vertices of the simplex. We also extend the node and edge resistance curvature proposed by K.Devriendt et al. to node and edge $\epsilon-$repelling curvature and derive the corresponding Lichnerowicz inequalities on any positive connected signed graph. Moreover, it turns out that edge $\epsilon-$repelling curvature is no more than the Lin-Lu-Yau curvature of the underlying graph whose transport cost is $\epsilon-$repelling cost rather than the length of the shortest path.

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.

Referee Report

0 major / 4 minor

Summary. The paper defines the ε-repelling Laplacian on positive connected signed graphs for ε < ε0 (where ε0 relates to the graph's consensus problem). It establishes an upper bound on the second smallest eigenvalue of this operator. Using the pseudoinverse, it constructs a simplex and an ε-repelling cost, showing that the square root of this cost defines a distance on the simplex vertices. The work extends node and edge resistance curvatures to their ε-repelling analogues and derives the associated Lichnerowicz inequalities. It further proves that edge ε-repelling curvature is at most the Lin-Lu-Yau curvature of the underlying graph when the transport cost is taken to be the ε-repelling cost rather than shortest-path length.

Significance. If the derivations hold, the manuscript supplies a new operator and curvature framework adapted to positive signed graphs that incorporates repelling effects. This extends classical resistance curvature and Lichnerowicz-type results to a signed setting and supplies a concrete comparison with the Lin-Lu-Yau curvature under a modified transport cost. The distance property obtained from the pseudoinverse is a technically useful construction that may find use in network analysis and consensus dynamics.

minor comments (4)
  1. The abstract would benefit from a single sentence indicating the principal techniques (e.g., variational characterization or matrix perturbation) used to obtain the eigenvalue bound and the curvature comparison.
  2. Clarify the precise construction of the simplex from the pseudoinverse of the ε-repelling Laplacian and verify that the square-root cost satisfies the triangle inequality; this step is load-bearing for the subsequent curvature definitions.
  3. Add a short remark on how the positivity and connectedness assumptions are used to guarantee that the ε-repelling cost is finite and positive for distinct vertices.
  4. The reference list should include the full bibliographic details for the Devriendt et al. paper on resistance curvature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its contributions. We are pleased that the referee finds the introduction of the ε-repelling Laplacian and the associated curvature framework on positive connected signed graphs to be of interest, and we appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins with the definition of the ε-repelling Laplacian as a positive semidefinite operator on positive connected signed graphs for ε < ε0 (determined by the consensus problem), proceeds to construct a simplex and ε-repelling cost via the pseudoinverse (with square root shown to be a distance), extends the resistance curvature of Devriendt et al. to the ε-repelling setting, and derives Lichnerowicz inequalities. The central comparison—that edge ε-repelling curvature is at most the Lin-Lu-Yau curvature of the underlying graph under the ε-repelling transport cost—is obtained by applying standard optimal-transport and curvature inequalities to these independently defined objects. No step reduces the target inequality to a definitional identity, a fitted parameter renamed as a prediction, or a self-citation chain; the constructions are self-contained extensions of spectral graph theory and prior resistance-curvature work without internal collapse.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The central claims rest on the domain assumption that the signed graph is positive and connected, on the existence of a consensus-related threshold ε0, and on standard facts from linear algebra and graph theory; no numerical fitting parameters or new physical entities are introduced.

free parameters (1)
  • ε
    Arbitrary positive scalar required to be smaller than the consensus threshold ε0; its value is not fitted to data but chosen to guarantee positive-semidefiniteness.
axioms (2)
  • domain assumption The signed graph is positive and connected.
    Invoked to ensure the ε-repelling Laplacian is positive semidefinite and to support the subsequent curvature definitions.
  • standard math Standard spectral properties of graph Laplacians and pseudoinverses hold.
    Used implicitly when constructing the simplex and defining distances from the pseudoinverse.
invented entities (2)
  • ε-repelling Laplacian no independent evidence
    purpose: Positive semidefinite operator on signed graphs for small ε
    Newly defined operator whose properties are studied; no independent existence claim outside the definition.
  • ε-repelling curvature no independent evidence
    purpose: Extension of resistance curvature to the signed setting
    Defined via the new Laplacian; no external falsifiable prediction supplied.

pith-pipeline@v0.9.0 · 5701 in / 1689 out tokens · 34729 ms · 2026-05-19T10:10:59.922995+00:00 · methodology

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Reference graph

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