arxiv: 2604.27930 · v1 · submitted 2026-04-30 · ⚛️ physics.soc-ph

Recognition: unknown

Scale-freeness under node removal: a finite-size scaling perspective

Yeonsu Jeong , Deok-Sun Lee , Mi Jin Lee , Seung-Woo Son

Authors on Pith no claims yet

Pith reviewed 2026-05-07 07:39 UTC · model grok-4.3

classification ⚛️ physics.soc-ph

keywords scale-free networksnode removalfinite-size scalingKullback-Leibler divergencenetwork robustnessdegree distributionstructural stabilitynetwork degradation

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The pith

Under hub-preferential node removal, networks can match a scale-free degree distribution yet fail to show scale-invariant organization across different sizes.

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

The paper investigates how the scale-free property of networks changes when nodes are removed in ways that depend on their degrees, controlled by a parameter θ for hub protection. It compares two ways of checking if a network remains scale-free: one based on how close its degree distribution is to a reference using Kullback-Leibler divergence, and another using finite-size scaling to see if properties collapse consistently when system size changes. The authors find that for random or hub-protecting removals these two checks agree, but for removals that prefer hubs, networks often look scale-free by the first check but do not collapse properly under the second. This matters because many real systems like social or ecological networks rely on their scale-free structure for stability, and different removal sequences can lead to different collapse modes. If the finding holds, relying only on degree distribution similarity may not be enough to confirm that scale-free organization persists as networks grow or shrink.

Core claim

Starting from scale-free networks, the application of degree-dependent node removal with varying hub-protection strength θ shows that under hub-preferential removal (θ < 0) the networks satisfy the Kullback-Leibler criterion for scale-freeness but do not exhibit the data collapse required by finite-size scaling analysis. This reveals that resemblance to a reference degree distribution does not ensure the persistence of scale-invariant organization across system sizes. Consequently, the two diagnostic methods capture complementary aspects of network structure and should be used together for a fuller understanding of structural degradation under node removal.

What carries the argument

The finite-size scaling (FSS) analysis applied to the degree distributions or related quantities of node-removed networks, contrasted with Kullback-Leibler divergence from a reference scale-free distribution, under a removal process where node removal probability depends on degree raised to power θ.

If this is right

For random removal and hub-protecting removal, the KL and FSS criteria agree on whether the network remains scale-free.
The discrepancy under hub-preferential removal shows that matching a static degree distribution does not guarantee scale-invariance across sizes.
Structural stability assessments in networks should incorporate finite-size scaling tests in addition to distribution comparisons.
The joint use of both diagnostics offers a more complete picture of how scale-free organization degrades.

Where Pith is reading between the lines

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

In applications to real-world networks subject to targeted attacks on hubs, one should verify scale-freeness with FSS rather than relying solely on KL divergence.
This discrepancy might explain why some networks appear robust by degree stats but still show collapse when system sizes vary in simulations.
Future models could incorporate FSS directly into robustness metrics for predicting systemic stability under different removal regimes.
Testing on empirical networks from ecology or social systems could reveal whether observed scale-free traits survive removal processes in practice.

Load-bearing premise

The finite-size scaling procedure accurately detects the absence of scale-invariant organization without being affected by the details of how the networks are generated or nodes removed, and that the starting networks are genuinely scale-free.

What would settle it

If networks generated or modified under hub-preferential removal (θ < 0) exhibit successful data collapse in the finite-size scaling analysis across multiple system sizes while still passing the KL criterion, this would contradict the reported discrepancy between the two diagnostics.

Figures

Figures reproduced from arXiv: 2604.27930 by Deok-Sun Lee, Mi Jin Lee, Seung-Woo Son, Yeonsu Jeong.

**Figure 1.** Figure 1: FIG. 1. FSS tests on BA networks under a general node-removal process. (a) Schematic illustration of the subnetwork sampling view at source ↗

**Figure 2.** Figure 2: FIG. 2. Basic statistics of the available scaling region of the degree. (a) Removal fraction view at source ↗

**Figure 3.** Figure 3: FIG. 3. Measurements from the FSS analysis as a function of view at source ↗

**Figure 4.** Figure 4: FIG. 4. The KL divergence view at source ↗

**Figure 5.** Figure 5: FIG. 5. Breakdown behavior of the GCC under degree-dependent node removal. Relative size of the GCC versus the removal view at source ↗

**Figure 6.** Figure 6: FIG. 6. Scale-freeness in the view at source ↗

read the original abstract

In heterogeneous network systems such as ecological and social networks, structural stability depends on how connectivity changes under node removal, as different removal sequences can trigger distinct modes of systemic collapse. While robustness to random failures and targeted attacks has been extensively studied, most analyses have focused on connectivity loss or degree distribution, rather than on how scale-invariant organization emerges and evolves with system size. Here we examine how scale-free structure evolves under progressive degree-dependent node removal, systematically varying the hub-protection strength $\theta$. Starting from scale-free networks, we apply the recently developed finite-size scaling (FSS) analysis to node-removed networks and compare the results with those from Kullback-Leibler (KL) divergence-based classification. We find that under random ($\theta=0$) and hub-protecting removal ($\theta>0$), the two criteria largely agree, whereas under hub-preferential removal ($\theta<0$), networks may appear scale-free according to the KL criterion while failing the FSS test of scaling collapse. This discrepancy indicates that similarity to a reference degree distribution does not guarantee the persistence of scale-invariant organization across system sizes. The two diagnostics thus probe complementary aspects of network structure, and their joint use provides a more complete characterization of structural degradation.

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

The paper finds that KL divergence and finite-size scaling disagree on scale-freeness after hub-preferential removal, with FSS flagging loss of invariance where KL does not.

read the letter

The main thing to know is that under hub-preferential node removal (negative θ) the networks can still look scale-free by KL divergence to a reference distribution, yet fail the finite-size scaling collapse test. The authors treat this as evidence that the two diagnostics are complementary rather than interchangeable. That discrepancy is the concrete new observation they report, and it follows directly from running both checks on the same sequence of removed networks. For random or hub-protecting removal the methods agree, which lines up with what one would expect. The work is straightforward simulation: generate scale-free graphs, apply degree-biased removal with tunable θ, then compare the two criteria on the resulting degree distributions across remaining sizes. The comparison itself is useful because many robustness studies still rely on matching a power-law shape without checking whether the scaling behavior survives. The paper stays within its lane and does not overreach. The soft spots are mostly about missing detail and limited scope. The description gives no numbers on network sizes, number of realizations, or exact removal probabilities, so it is difficult to judge how strong the collapse failure really is or whether it is sensitive to the initial model. The stress-test worry that the FSS ansatz itself might be violated by the removal rule is reasonable and not obviously addressed by extra checks on alternative generators or correlation effects. Still, the central simulation result holds up on its own terms and does not rest on circular fitting. This is for network scientists who already work with scale-free diagnostics and robustness questions. A reader who knows both KL and FSS will get the most from the side-by-side application. It deserves peer review because the finding is specific, falsifiable in simulation, and could change how people choose between these tools in practice. I would send it with requests for method parameters, run counts, and at least one robustness test on a different initial network family.

Referee Report

2 major / 2 minor

Summary. The manuscript examines how scale-free organization in networks evolves under degree-dependent node removal, parameterized by a hub-protection strength θ. Starting from scale-free networks, the authors apply finite-size scaling (FSS) analysis to test for data collapse of the degree distribution P(k) across remaining system sizes after removal, and compare this to classification by Kullback-Leibler (KL) divergence to a reference power-law distribution. They report agreement between the two diagnostics for random (θ=0) and hub-protecting (θ>0) removal, but a discrepancy for hub-preferential removal (θ<0), where networks can satisfy the KL criterion while failing FSS scaling collapse. The central conclusion is that KL similarity to a reference distribution does not guarantee persistence of scale-invariant structure across sizes, so the two diagnostics are complementary for characterizing structural degradation.

Significance. If the reported discrepancy is robust, the result would be significant for network robustness studies in social, ecological, and infrastructure systems, as it shows that static distributional similarity (KL) can be insufficient to confirm scale-freeness under perturbation, while FSS provides a stricter test of scale-invariance across sizes. The joint use of two independent diagnostics is a clear strength, moving beyond single-metric analyses common in the field. The simulation-driven approach with tunable θ allows systematic exploration, but the absence of detailed methodological parameters limits generalizability.

major comments (2)

[Methods] Methods section: The text provides no quantitative details on initial network sizes N, the algorithm used to generate the starting scale-free networks (e.g., Barabási–Albert parameters or configuration-model exponents), the exact functional form of the removal probability p(k) ∝ k^θ, the number of independent realizations per θ value, or any statistical measures (error bars, p-values) for the KL/FSS agreement or discrepancy. These omissions are load-bearing for the central claim, because without them it cannot be verified that the FSS collapse failure under θ<0 is not an artifact of the specific generation or removal implementation, as raised in the stress-test note.
[Results] Results section (FSS analysis): The scaling-collapse procedure is described only at a high level; the manuscript does not specify the assumed scaling ansatz (e.g., P(k) = k^{-α} f(k/N^β) or alternative form), the range of post-removal system sizes used for collapse, or the quantitative criterion for declaring 'failure' of collapse versus visual inspection. This is critical because the discrepancy claim for θ<0 rests on FSS correctly diagnosing loss of scale-invariance rather than a mismatch between the removal-induced correlations and the FSS assumptions.

minor comments (2)

[Abstract] Abstract: The phrase 'the recently developed finite-size scaling (FSS) analysis' is used without a citation; adding the reference would allow readers to locate the precise method being applied.
[Figures] Figure captions (throughout): Captions should explicitly state the values of θ, N, and number of realizations shown, as well as the metric used to assess collapse quality, to make the visual evidence self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We have revised the manuscript to address the concerns regarding insufficient methodological details and the high-level description of the finite-size scaling procedure. Our point-by-point responses to the major comments follow.

read point-by-point responses

Referee: [Methods] Methods section: The text provides no quantitative details on initial network sizes N, the algorithm used to generate the starting scale-free networks (e.g., Barabási–Albert parameters or configuration-model exponents), the exact functional form of the removal probability p(k) ∝ k^θ, the number of independent realizations per θ value, or any statistical measures (error bars, p-values) for the KL/FSS agreement or discrepancy. These omissions are load-bearing for the central claim, because without them it cannot be verified that the FSS collapse failure under θ<0 is not an artifact of the specific generation or removal implementation, as raised in the stress-test note.

Authors: We agree that the original Methods section omitted several quantitative parameters essential for reproducibility and independent verification. In the revised manuscript we have expanded the Methods section to specify the initial network sizes N, the algorithm and parameters used to generate the starting scale-free networks, the exact functional form of the removal probability p(k) ∝ k^θ (including normalization), the number of independent realizations per θ value, and the statistical measures (error bars and p-values) employed for the KL and FSS comparisons. These additions allow readers to confirm that the reported discrepancy for θ < 0 is not an artifact of the particular implementation. revision: yes
Referee: [Results] Results section (FSS analysis): The scaling-collapse procedure is described only at a high level; the manuscript does not specify the assumed scaling ansatz (e.g., P(k) = k^{-α} f(k/N^β) or alternative form), the range of post-removal system sizes used for collapse, or the quantitative criterion for declaring 'failure' of collapse versus visual inspection. This is critical because the discrepancy claim for θ<0 rests on FSS correctly diagnosing loss of scale-invariance rather than a mismatch between the removal-induced correlations and the FSS assumptions.

Authors: We acknowledge that the original description of the finite-size scaling analysis was presented at a high level. In the revised manuscript we have added a detailed subsection that explicitly states the scaling ansatz employed, the specific range of post-removal system sizes used for the collapse, and the quantitative criterion (combining visual assessment with a measure of curve variance) used to determine success or failure of collapse. This clarification supports the conclusion that the lack of collapse for θ < 0 reflects a genuine loss of scale-invariance rather than a mismatch with the assumed scaling form. revision: yes

Circularity Check

0 steps flagged

No circularity: simulation-based comparison of two independent diagnostics

full rationale

The paper generates scale-free networks, applies degree-dependent node removal parameterized by θ, and then applies two separate, externally defined diagnostics (KL divergence to a reference P(k) and finite-size scaling collapse test for scale-freeness). The reported discrepancy for θ<0 is an empirical observation from these simulations rather than a mathematical reduction. No equation or derivation equates the FSS outcome to the KL outcome or to any fitted input by construction. The FSS method is invoked as a pre-existing tool (cited as 'recently developed'), but its application here does not make the central claim tautological; the result remains falsifiable by the simulation data itself. This matches the default expectation for a simulation-driven study with no self-definitional or fitted-input circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard scale-free network models and numerical experiments; θ is a control parameter varied across simulations rather than a fitted constant that defines the claim.

free parameters (1)

θ (hub-protection strength)
Tunable parameter that sets the degree dependence of the removal probability; varied systematically to explore regimes.

axioms (2)

domain assumption Initial networks follow a power-law degree distribution and are scale-free
Starting condition stated in the abstract for all removal experiments.
domain assumption Finite-size scaling collapse is a valid test for persistence of scale-invariant organization
Core premise of the FSS diagnostic used to challenge the KL verdict.

pith-pipeline@v0.9.0 · 5525 in / 1535 out tokens · 74326 ms · 2026-05-07T07:39:22.769101+00:00 · methodology

discussion (0)

Reference graph

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