Changing the ranking in eigenvector centrality of a weighted graph by small perturbations

Michele Benzi; Nicola Guglielmi

REVIEW 3 major objections 6 minor 1 cited by

Changing the ranking in eigenvector centrality of a weighted graph by small perturbations

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper shows that the robustness of eigenvector-centrality rankings can be quantified by the smallest weight perturbation, preserving the graph's edge structure, that makes the top m centrality scores exactly equal, and proposes a…

desk verdict A useful local-minimum algorithm for a new eigenvector-centrality nearness problem, but the abstract's global optimality claim is not supported and the gradient theory has a fixed-index gap. read the letter →

arxiv 2501.10745 v3 pith:66VWL4EZ submitted 2025-01-18 math.NA cs.NAmath.DSmath.OC

Michele Benzi , Nicola Guglielmi This is my paper · ORCID

classification math.NAcs.NAmath.DSmath.OC MSC 15A6005C5015A4247A55

keywords eigenvectorcentralityPerronmatrixnearnessproblemrobustnessradiusstructuredeigenvalueoptimizationgraphperturbationgradientflowrankingsensitivity

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

Eigenvector centrality ranks the nodes of a weighted graph by the entries of the Perron eigenvector of its adjacency matrix, and that ranking is only trustworthy if it survives realistic uncertainty in the edge weights. This paper tries to measure that trustworthiness directly: how far, in Frobenius norm, must the weights move, preserving the graph's edge structure, before the top m centrality scores become exactly equal and the ranking at the top turns ambiguous. The authors propose a two-level algorithm that alternates a constrained matrix gradient flow, which for a fixed perturbation size drives the spread of the top m scores to a minimum, with an outer one-dimensional search that shrinks the perturbation until the spread reaches zero. They derive an explicit formula for the gradient using first-order eigenvector perturbation theory, show that stationary perturbations have an underlying low-rank form, and demonstrate on sparse graphs with hundreds to ten thousand nodes that the approach is faster than generic constrained optimization. If the algorithm is correct, the computed perturbation size gives a robustness radius for the ranking, and the perturbing matrix itself identifies which edges are most able to change the ranking.

What carries the argument

The argument is carried by the dispersion functional $F_\varepsilon(E)=\frac{1}{2}\sum_{k=1}^m (v_{i_k}(A+\varepsilon E)-\langle v\rangle_m)^2$, which is zero exactly when the m largest entries of the Perron eigenvector are equal. The paper computes its structured gradient $G_\varepsilon(E)$ by projecting the free gradient, built from the group inverse $(A+\varepsilon E-\lambda I)^\#$ and rank-one outer products of the Perron eigenvector with the relevant coordinate vectors, onto the graph's edge pattern; this formula comes from first-order eigenvector perturbation theory. The inner iteration integrates the constrained gradient flow under the unit-Frobenius-norm and sparsity constraints, and the outer iteration uses the identity $f'(\varepsilon)=-\|G_\varepsilon(E(\varepsilon))\|$ to run a Newton-bisection root find. At stationary points the perturbation is proportional to the projected gradient, which is a projection of a rank-one matrix in the directed case and of a rank-two symmetric matrix in the undirected case, and the paper exploits this low-rank structure computationally.

What would settle it

Track which nodes occupy the top m positions during the inner gradient flow on a graph where the top set changes, as in the paper's own Example 1 with m=4, where node 109 enters the top set. If the dispersion functional $F_\varepsilon$ ever increases along the numerically integrated flow at the moment the index set changes, the gradient formula no longer describes the true functional and the monotone-descent theorem fails; alternatively, run a global search over admissible unit-norm perturbations $E$ for the smallest $\varepsilon$ with $F_\varepsilon(E)=0$ and compare with the reported value, since any smaller $\varepsilon$ found would disprove the claimed optimality.

Watch

Extended reading notes

Core claim

The paper's central claim is that the robustness of an eigenvector-centrality ranking can be quantified by a structured matrix nearness problem: find the smallest Frobenius-norm change to the weighted adjacency matrix, keeping zero entries fixed and keeping the matrix nonnegative, such that the m largest entries of the Perron eigenvector coalesce. The proposed solution is a nested iteration: an inner constrained gradient system minimizes the dispersion functional $F_\varepsilon(E)$ at fixed perturbation size $\varepsilon$, and an outer Newton-bisection iteration finds the smallest $\varepsilon$ for which that functional vanishes. The authors state that the algorithm produces the optimal perturbation, with the smallest Frobenius norm causing the coalescence, while also noting in Section 3.3 that the gradient-flow approach converges to local optima, so the computed value is in general an upper bound on the robustness radius $\varrho_m(G)$. Numerical experiments on realistic sparse graphs report coalescence achieved by perturbations of about one percent or less of the normalized adjacency matrix, with the gradient flow far outperforming a standard constrained-optimization solver.

Load-bearing premise

The load-bearing assumption is that the identities of the top m nodes do not change while the perturbation moves; if a new node enters the top m, the gradient formula no longer describes the true dispersion functional and the guaranteed descent can break.

Editorial extensions

If this is right

For any weighted graph, the method yields a concrete robustness radius: a perturbation smaller than the computed value cannot make the top m eigenvector centralities tie, so a small radius signals that the ranking is fragile under weight noise.
The perturbing matrix found at the optimum reveals which edges have the strongest influence on the top of the ranking, since its largest entries are the weights whose adjustment most shrinks the gap.
Because the formulation only needs a nonnegative matrix, the same radius and closest-ambiguous-matrix construction apply to any dominant-eigenvector ranking, including those arising in population models, consensus dynamics, and economic input-output analysis.
For undirected graphs, the stationary perturbations are projections of rank-two symmetric matrices, which keeps the per-step cost dominated by one Perron eigenvector computation plus m solves of a bordered linear system, so large sparse graphs are tractable.
On the tested sparse graphs, the two-level gradient-flow method reached the coalescing perturbation in seconds to minutes, whereas a generic constrained-optimization solver required orders of magnitude more time and many more function evaluations.

Reading between the lines

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

The same gradient-flow machinery could be applied to PageRank and Katz centrality by replacing the Perron eigenvector with the corresponding resolvent-based centrality vector; the paper notes the kinship of these measures but does not derive the analogous gradient or robustness radius.
One testable refinement is to make the inner iteration detect index switches, for example by checking whether the top m set has changed and restarting the flow with the updated set, since the paper's own Example 1 shows such switches do occur.
The robustness radius could be used as a ranking-confidence diagnostic: compare the radius to the estimated uncertainty in the edge weights, and if the radius is smaller, the top m order is not statistically distinguishable from ambiguous, a decision rule the paper does not spell out.
A randomized or rank-restricted variant of the gradient flow could extend the method to very large networks, since the extremizers are known to be projections of low-rank matrices.

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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

First thing you should know: this is a real contribution to a niche but useful problem—how fragile is eigenvector centrality ranking, and what is the nearest graph that makes the top m ambiguous. The authors formulate a new structured matrix nearness problem, derive explicit gradient formulas via first-order perturbation theory, and uncover a low-rank (rank-1/rank-2) structure in the stationary points. The numerical experiments on realistic graphs (A27, nopoly) are convincing: the method converges quickly and beats fmincon by orders of magnitude in CPU time. That part is solid.

The soft spots are real, though, and they sit precisely where the authors overclaim. The abstract promises 'the optimal perturbation' with smallest Frobenius norm. Section 3.3 concedes the gradient-flow inner iteration only globally converges to local minima, so the computed ε* is an upper bound on ϱ_m(G), not the exact robustness radius. No global optimality certificate is supplied. That mismatch should be fixed by either proving global optimality (unlikely for this nonconvex problem) or systematically softening the claims in the abstract and text.

The second issue is the fixed-index assumption in Lemma 5.1. The proof differentiates F_ε(E(t)) treating the top-m indices as constant, but those indices are defined as the current top-m set. If the set changes along the path—and the paper's own Example 1 (m=4) shows node 109 replacing node 29—the functional is nonsmooth and formula (16) is not the gradient of the actual functional. The theorem's monotonicity conclusion doesn't follow without handling index switches. This is also addressable: one can add an explicit regularity assumption, or track switch events and restart the flow. As written, it's a gap in the theory, though the numerics suggest it doesn't break the method in practice.

Minor: Lemma 6.1's proof is terse—the εE' term vanishes because E ∝ G at a minimizer, which is true but should be said. Also, no code or data is released, so the CPU-time claims and the fmincon comparison are hard to reproduce. That's a normal request for a numerical paper.

Overall: the core idea is good, the derivation is mostly sound, and the local-minimum algorithm is genuinely useful for sensitivity analysis and manipulation studies. The paper deserves a proper referee. I would send it out, but the referee should insist on resolving the local-global language and the index-set stability issue before publication. I would not cite the global optimality claim as is, but I would cite the algorithm and gradient formulas once cleaned up.

Referee Report

3 major / 6 minor

Summary. The paper studies the robustness of eigenvector centrality rankings with respect to weighted perturbations that preserve the graph's sparsity pattern. It defines a robustness radius ϱ_m(G) as the minimal Frobenius-norm perturbation that makes the m largest entries of the Perron eigenvector coalesce, proposes a two-level algorithm (an inner constrained gradient flow that minimizes a dispersion functional for a fixed perturbation norm, and an outer Newton-bisection that seeks the minimal such norm), derives an explicit structured gradient formula via first-order eigenvector perturbation theory, shows that stationary points have a low-rank structure, and reports numerical experiments on a water-network graph and on a large sparse graph. The abstract claims global optimality of the resulting perturbation, while Section 3.3 concedes that only local optima are guaranteed and that the computed matrix provides an upper bound to the problem.

Significance. The problem is well motivated and the paper contains several useful contributions: a clean formulation of the nearness problem, a closed-form gradient that avoids numerical differentiation, a characterization of stationary points as projections of low-rank matrices, a practical inner-outer algorithm that compares favorably with generic constrained optimization in the reported examples, and reproducible-looking experiments on realistic graphs. If the global-optimality gap were closed, or the claims appropriately weakened, the robustness radius would be a valuable addition to the network-sensitivity literature. The derivation of the gradient and the low-rank reduction are the strongest parts of the manuscript.

major comments (3)

[Abstract and Section 3.3] The Abstract states that the algorithm produces 'the optimal perturbation (i.e., the one with smallest Frobenius norm)' causing coalescence, but Section 3.3 ('Existence of local minima') explicitly says that 'in general we cannot be sure to compute global optima, but only to globally converge to local optima, which means that the final computed matrix provides an upper bound to the solution of (8).' These two statements are inconsistent. Since (8) defines the robustness radius as the infimum over all admissible E, a locally minimizing branch can make the returned ε* strictly larger than ϱ_m(G). No global optimality certificate, convexity argument, or uniqueness proof is supplied. The Abstract and the discussion in Sections 3.3 and 8 should be revised to present the computed quantity as an upper bound, or the claimed global optimality must be proved.
[Lemma 5.1, Eqs. (16)-(17)] Lemma 5.1 derives the derivative d/dt F_ε(E(t)) under the convention that i1,...,im are the indices of the m largest entries of v(t) at each t. In the proof, the identities (17) differentiate v_{i_k}(t) while treating i_k as constant. If the set of top-m indices changes along the path E(t), F_ε is continuous but not differentiable at the switching point, and formula (16) is not the gradient of the actual top-m dispersion functional. The paper's own Example 1 (Section 7.1, m=4) exhibits such a change: node 109 enters the top set and node 29 leaves it. No mechanism for detecting or handling these index switches is described. Consequently, Theorem 5.1's monotonicity statement is not established for the full flow, and the 'gradient descent' interpretation of (25) is not rigorous in general. The authors should either restrict the flow to intervals where the index set is constant and add a switch-detection/reinitialization strategy, or prove that the formula remains valid in a suitable weaker sense.
[Lemma 6.1, Eqs. (37)-(38)] Lemma 6.1 concludes f'(ε) = -||G_ε(E(ε))||. The proof shows f' = ⟨G,E⟩ and then invokes 'property no. 3' of Theorem 5.1, which says E ∝ G. Since ||E|| = 1, this only gives ⟨G,E⟩ = ±||G||; the sign is not determined. The negative sign would follow if f(ε) were the value of a global minimum that is non-increasing in ε, but neither global optimality nor monotonicity of f has been established. If the sign is wrong, the Newton step (39) in the outer iteration points in the wrong direction. This is load-bearing for the outer iteration's correctness and for the claim that the computed root is the minimal ε. The authors should provide a correct proof of the sign, or use a derivative-free outer iteration.

minor comments (6)

[Equation (8)] The display 'eεm − →arg min ε>0 F_ε(E(ε)) = 0' is not standard mathematical notation; please rewrite it as a precise optimization or root-finding statement.
[Table 1] The column 'fc(εk)' is not defined in the caption; it appears to denote F^c_ε(E_k), but this should be spelled out.
[Section 5.6] The statement that 'we can prove' a one-to-one correspondence of stationary points and local convergence of the rank-1 ODE, followed by 'we omit a full derivation', is not sufficient for a journal paper unless the omitted result is readily available in [18]; please either include the proof or mark the statement explicitly as a conjecture with numerical support.
[Section 7.2] The percentages quoted for the perturbation norms appear to be off by several orders of magnitude: for a matrix with ||A||_F = 1, ε2 = 0.00383633 is about 0.38%, not 0.00008%, and ε5 = 0.0161 is about 1.6%, not 0.00016%. Please correct these values or clarify what they are relative to.
[Section 7.1, Example 1] The sentence 'It is interesting to note the the node 109 was not at the highest rank but replaced the node 29 after the perturbation' contains a typographical error ('the the') and also demonstrates the index-switching issue discussed above; please correct the typo and comment on the implications for the fixed-index assumption in Lemma 5.1.
[Section 3.2] The set Ω in (4) is defined using the indices i1,...,im of the m largest entries of u, but it would be clearer to specify how ties are handled, especially because the numerical examples intentionally create ties at coalescence.

Circularity Check

0 steps flagged · score 1.0 of 10

Central derivation is self-contained; the only self-citation (Section 5.6, rank-1 ODE to [18]) is not load-bearing, and the abstract's global-optimality claim is weakened by the paper's own Section 3.3 local-minimum caveat, an overclaim rather than a circular reduction.

full rationale

The derivation chain is self-contained at its core. Lemma 5.1 obtains the structured gradient Gε(E) by differentiating Fε via Theorem 4.3 (Meyer–Stewart) and the group inverse, rather than importing the target result; Theorem 5.1 monotonicity follows algebraically from the gradient system (25) and Cauchy–Schwarz; Lemma 6.1 derives the outer derivative f′(ε) = −‖Gε(E(ε))‖ from Theorem 5.1's characterization of minimizers. The robustness radii in Section 7 (ε* ≈ 0.0093, 0.0038, etc.) are outputs of the outer root-finding loop, not fitted constants later relabelled as predictions. The one reliance on the authors' prior work is Section 5.6, where the rank-1 ODE reduction for H is asserted and referred to [18] with the proof omitted; however, the numerical integration in Section 6.2 solves the gradient system (25) directly, and the reported experiments never invoke the rank-1 ODE, so the central claim does not reduce to that self-citation. Two non-circular weaknesses remain: (a) Section 3.3 explicitly concedes that the two-level method only globally converges to local optima and that the computed matrix 'provides an upper bound to the solution of (8)', which contradicts the abstract's unqualified 'optimal perturbation' statement; this is an unsupported global-optimality claim, not a circular one. (b) Lemma 5.1 differentiates v_{i_k}(t) while treating i_k as fixed, although i_1,...,i_m are defined as the current top-m indices; Example 1 (m=4) shows node 109 entering the top set, so the fixed-index gradient can fail at index switches. These are correctness limitations to weigh separately from circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central computation depends on standard Perron-Frobenius theory, first-order perturbation theory, and an implicit smoothness/local-constancy assumption about the top-m index set. The only hand-chosen parameters are numerical tolerances (delta, stopping tolerances, step-size parameters), which do not enter the mathematical formulation of the robustness radius. No new entities are postulated.

free parameters (3)

delta (positivity tolerance) = not specified (stated 1 >> delta > 0)
Replaces strict positivity of perturbed edge weights in (3); never active in the reported experiments.
coalescence tolerance tol = 1e-5 to 1e-10 depending on example
Declares coalescence when the dispersion or entry difference falls below tol; affects the reported values of epsilon* at the tolerance level.
step-size and Armijo parameters h, theta = not specified
Determine the explicit Euler step and step-size adaptation in (34)-(36); influence convergence speed but not the mathematical optimum.

assumptions (6)

standard math Perron-Frobenius theorem for nonnegative irreducible matrices
Invoked in Theorem 3.1 to guarantee a simple positive Perron eigenvalue and positive Perron eigenvector.
domain assumption Strong connectivity (irreducibility) of the graph
Assumption 3.1 throughout the paper; needed for existence and uniqueness of the Perron eigenvector and to rule out eigenvalue coalescence of type (II).
standard math First-order perturbation formulas for simple eigenvalues and eigenvectors
Theorems 4.1 and 4.3 (derived from Kato and Meyer-Stewart) are the backbone of the gradient computation in Lemma 5.1.
domain assumption Smoothness of the minimizer path E(epsilon)
Assumption 6.1, needed for the differentiability of f(epsilon) and the Newton step in the outer iteration; if it fails, the paper falls back to bisection.
domain assumption Fixed set of top-m indices during inner iterations
Implicit in Lemma 5.1: the gradient formula assumes i1,...,im do not change along E(t). Not stated as an assumption; Example 1 with m=4 shows index changes can occur.
domain assumption Existence of a perturbation causing coalescence
Section 3.3 notes the infimum may be infinite; the algorithm assumes a finite solution exists, otherwise the robustness radius is infinite and the outer loop has no root.

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Pith. "Pith review of Changing the ranking in eigenvector centrality of a weighted graph by small perturbations." pith.science (2026). https://pith.science/paper/66VWL4EZ

@misc{pith2026250110745,
  author       = {Pith},
  title        = {Pith review of: Changing the ranking in eigenvector centrality of a weighted graph by small perturbations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66VWL4EZ}},
  note         = {Machine review of arXiv:2501.10745}
}

read the original abstract

In this article, we consider eigenvector centrality for the nodes of a graph and study the robustness (and stability) of this popular centrality measure. For a given weighted graph {\mathcal G} (both directed and undirected), we consider the associated weighted adjacency matrix A, which by definition is a non-negative matrix. The eigenvector centralities of the nodes of {\mathcal G} are the entries of the Perron eigenvector of A, which is the (positive) eigenvector associated with the eigenvalue with largest modulus. They provide a ranking of the nodes according to the corresponding centralities. An indicator of the robustness of eigenvector centrality consists in looking for a nearby perturbed graph \widetilde{\mathcal G}, with the same structure as {\mathcal G} (i.e., with the same vertices and edges), but with a weighted adjacency matrix \widetilde A such that the highest m entries (m \ge 2) of the Perron eigenvector of \widetilde A coalesce, making the ranking at the highest level ambiguous. To compute a solution to this matrix nearness problem, a nested iterative algorithm is proposed that makes use of a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The proposed algorithm produces the {\em optimal} perturbation (i.e., the one with smallest Frobenius norm) of the A which causes the looked-for coalescence, which is a measure of the sensitivity of the graph. Our numerical experiments indicate that the proposed strategy outperforms more standard approaches based on algorithms for constrained optimization. The methodology is formulated in terms of graphs but applies to any nonnegative matrix, with potential applications in fields like population models, consensus dynamics, economics, etc.

Figures

Figures reproduced from arXiv: 2501.10745 by the authors.

**Figure 2.** Perturbed graph of [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Modified graph of [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Behavior of f(ε) for Example in Section 6.1. 7. Numerical tests In this section we present numerical experiments illustrating the behavior of our algorithm on a few undirected and directed sparse weighted graphs arising in applications. We also perform a comparison wit…

**Figure 5.** Figure 5: Patterns of the adjacency matrix for the graph example A27 (left) and for the graph example nopoly from Gaertner collection (right) 0 100 200 300 400 500 600 700 i 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 v i 0 100 200 300 400 500 600 700 i 0 0.05 0.1 0.15 0.2 0.25 0.3 0…

**Figure 6.** Figure 6: Distribution of the entries of Perron eigenvector for the example A27. Left picture: original matrix. Right picture: perturbed matrix (for m = 2). The norm of the perturbation which determines the coalescence of the two main entries of the Perron eigenvector is εe2 = 0…

**Figure 7.** Figure 7: Distribution of the entries of Perron eigenvector of the matrix obtained for m = 10 for the example nopoly from Gaertner collection. 8. Conclusions In this paper we have presented a methodology for finding a minimal perturbation to the adjacency matrix of a weighted (…

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The rate of change of a graph's Kemeny constant when an edge weakens gives an always-positive, cut-edge-safe edge centrality, and its unweighted version also scores non-edges for link prediction.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Andreotti, D

E. Andreotti, D. Edelmann, N. Guglielmi and C. Lubich, Constrained graph partitioning via matrix differential equations , SIAM J. Matrix Anal. Appl., 40(1):1–22, 2019

work page 2019

[2] [2]

Benner and M

P. Benner and M. Voigt, A structured pseudospectral method for H∞-norm computation of large-scale descriptor systems , Math. Control Signals Systems, 26(2):303–338, 2014. CHANGING RANKING IN A GRAPH 29

work page 2014

[3] [3]

Benzi, A direct projection method for Markov chains , Linear Algebra Appl., 386:27–49, 2004

M. Benzi, A direct projection method for Markov chains , Linear Algebra Appl., 386:27–49, 2004

work page 2004

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