Strong localization blurs criticality of time series for spreading phenomena on networks

Juliane T. Moraes; Silvio C. Ferreira

REVIEW 4 major objections 5 minor 72 references

Strong localization blurs criticality of time series for spreading phenomena on networks

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Strong localization in network structure can blur or erase the disassortative visibility-graph signature that marks critical time series, so genuinely critical systems can look off-critical—while a clear signal always means true…

desk verdict A useful and mostly correct caveat about VG criticality detection under strong localization, but the false-negative claims need robustness checks on ground-truth labels and sampling. read the letter →

arxiv 2412.01842 v2 pith:ES7YEI6V submitted 2024-11-21 physics.bio-ph physics.soc-ph

Juliane T. Moraes , Silvio C. Ferreira This is my paper · ORCID

classification physics.bio-phphysics.soc-ph

keywords visibilitygraphcriticalitytimeseriesspreadingdynamicscomplexnetworkslocalizationGriffithsphasesabsorbingstatephasetransitions

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

The paper tests whether the visibility-graph (VG) method—a way of turning a time series into a network—can detect criticality in spreading dynamics on highly heterogeneous networks. It finds the method's disassortative degree-correlation signature reliably marks criticality for collective activation and for activation driven by a dense k-core, but is delayed to very large sizes for sparse-hub activation and disappears entirely under strong localization. In strongly localized regimes—degree outliers, a single hub in a random regular network, and diluted contact processes with Griffiths phases—critical time series come out looking off-critical, giving false negatives. Crucially, no false positives are observed: whenever the disassortative signature appears, the system is truly critical. The conclusion is that macroscopic time series stay good proxies for criticality unless localization is extreme, in which case the VG test can miss a real critical point.

What carries the argument

The natural visibility graph (VG) of a time series, defined by connecting two points if every intermediate point lies below the straight line joining them. The paper's diagnostic is the VG's degree correlation, measured by the average nearest-neighbor degree $K_{nn}(k_{vg})$ and a partial Pearson coefficient $r$: an asymptotically decreasing $K_{nn}$ (disassortativity, $r<0$) marks a critical series, while an increasing (assortative, $r>0$) pattern marks off-critical dynamics. The machinery is applied to time series of epidemic prevalence produced by SIS and contact-process simulations on random regular, annealed, uncorrelated scale-free (UCM), and diluted Erdős–Rényi networks, with critical points labeled by peaks of the dynamical susceptibility $\chi = N[\langle\rho^2\rangle-\langle\rho\rangle^2]/\langle\rho\rangle$ in quasistationary simulations.

What would settle it

Run the VG analysis on the same strongly localized models (e.g., the random regular network with a single hub, or the diluted contact process at its Griffiths phase) but label criticality by an independent method that does not rely on hub reactivation—for instance, by locating the epidemic threshold from the scaling of the order parameter with system size or from spectral analysis of the network's adjacency matrix. If a disassortative VG signature reappears at those independently located critical points, the false negatives would be artifacts of the labeling procedure rather than an intrinsic limit of the visibility-graph method.

Watch

Extended reading notes

Core claim

The central claim is that the asymptotic disassortative degree correlation of the visibility graph—previously established as a fingerprint of critical time series on regular substrates—remains a reliable marker of criticality on heterogeneous networks as long as the activation of the spreading process is collective or driven by a densely connected set of hubs (maximum k-core). When activation is driven by sparsely distributed hubs the criticality signature is postponed, emerging only in networks as large as $N \sim 10^8$ nodes, and in cases of strong structural localization (a degree outlier, a single hub immersed in a random regular network, or the rare-region disorder producing Griffiths phases) the critical time series yields an assortative VG pattern typical of off-critical dynamics. The authors stress that the VG method never produced a false positive for criticality; false negatives occur only under strong localization. They conclude that while macroscopic time series remain good proxies for criticality for collective or maximum k-core activation, systems under spatial localization can postpone the signatures of—or, in extreme localization, entirely hide—the criticality of the time series.

Load-bearing premise

The paper trusts that the susceptibility peaks from its hub-restarting simulation method mark the true critical points; if that restarting distorts the dynamics, the reported false negatives could be mislabeled series rather than genuine failures of the method.

Editorial extensions

If this is right

For spreading processes with collective activation, including SIS on annealed networks and the contact process on quenched scale-free networks, a disassortative VG signature reliably identifies the critical point regardless of network heterogeneity.
For activation driven by a maximum k-core of hubs, the VG signature remains as clear as in the collective case, so the method works when a subextensive but densely connected set drives the transition.
For activation driven by sparsely distributed hubs, the critical signature is delayed to very large system sizes ($N$ of order $10^8$), implying that typical finite-size simulations can miss it and mistake a critical system for an off-critical one.
Under strong localization—outliers, a single hub, or Griffiths-phase disorder—the critical time series produces the assortative pattern of an off-critical series; these are false negatives for the VG method.
No false positives were observed: an asymptotic disassortative VG correlation always corresponds to a truly critical system, making a positive VG signal a strong diagnostic even on heterogeneous structures.

Reading between the lines

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

Beyond the paper: a practical reading is that the VG test is asymmetric—a clear disassortative signal is strong evidence for criticality, but a null or assortative signal is inconclusive unless strong localization can be excluded by independent structural analysis.
Beyond the paper: the delayed signature in sparse-hub networks suggests a crossover scale; one could test whether the required network size grows with a power of the hub degree or the rarity of hubs, giving a quantitative criterion for when the VG method will fail.
Beyond the paper: the dilution-induced false negatives imply that in real systems with quenched disorder—such as damaged brain networks or heterogeneous contact patterns—a failure to see the criticality signature should not be interpreted as evidence against criticality.
Beyond the paper: the same asymmetry might apply to other single-observable criticality markers based on time-series fluctuations; combining VG analysis with a localization-sensitive structural measure could yield a more reliable overall test.

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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

This is a solid, honest follow-up to the authors' 2023 VG paper. The new result is that strong localization—outliers, a single hub, or rare-region Griffiths dynamics—can wipe out the visibility-graph signature of criticality even when the spreading model is at a collective critical point. That is a genuinely new caveat, and it matters for anyone running VG on empirical brain or epidemic time series, where localization is common.

The paper earns its keep on several fronts. The false-negative claim is tested across three independent model families: SIS on UCM networks with different exponents, an RRN with one hub, and a diluted contact process. The k-core control case preserves the disassortative signature, showing the method is not failing everywhere. The partial Pearson coefficient is a reasonable tool for tracking asymptotic trends, and the authors are upfront in Sec. III that hub reactivation artificially injects localization into quasistationary series.

The soft spots are real, and the stress-test worry lands. Ground-truth critical labels come from susceptibility peaks, and in exactly the localized regimes studied those peaks are ambiguous. Fig. 6(c) shows two peaks, one from an outlier rather than the collective transition; if the series fed to the VG is not actually at the collective critical point, then calling the assortative result a false negative is wrong—it may be the correct reading of an off-critical series. Relatedly, the hub-reactivation artifact is dismissed only for the deep subcritical regime. In the RRN-with-hub case, at the global susceptibility peak the hub is already individually supercritical, so the restart protocol may be shaping the fluctuations the VG reads, not just keeping the system alive. The authors need to show the false-negative picture survives alternative QS schemes or alternative criticality markers.

The quantitative details are also weaker than the qualitative message. The q0 thresholds for the partial Pearson coefficient are post hoc, with no sensitivity analysis. The finite-size extrapolation for γ = 2.75 gives r∞ ≈ −0.023 with no error bars; that is statistically indistinguishable from neutral. And the "no false positives" claim is only as strong as the set of off-critical series tested; it should be phrased as "no false positives in the scenarios considered."

Net: the qualitative conclusion is probably right—a disassortative VG is a reliable positive signal, but its absence does not rule out criticality in strongly localized systems. The paper deserves a serious referee, but it needs a robustness pass before the quantitative details are taken at face value.

Referee Report

4 major / 5 minor

Summary. The manuscript studies time series of the epidemic prevalence (order parameter) from SIS and contact-process simulations on highly heterogeneous networks, maps them to visibility graphs (VG), and tests whether the VG degree-correlation signature of criticality (asymptotic disassortativity) remains valid under different activation mechanisms. The authors report that the disassortative signature is robust for collective activation (annealed networks, CP on quenched UCM networks) and for maximum-k-core activation (UCM with γ=2.25), but is postponed or absent for activation by sparsely distributed hubs (UCM γ=2.75 and γ=3.5 with rigid cutoff), for strong localization due to degree outliers (UCM γ=3.5 natural cutoff), for an RRN with a single hub, and for a diluted contact process in a Griffiths phase. They conclude that strong localization can blur or even erase the VG criticality signature, producing false negatives but no false positives.

Significance. If the conclusions hold, the paper provides a practically important caveat for the use of visibility-graph degree correlations as a proxy for criticality in real-world time series, especially in biological systems where strong heterogeneity and rare regions are common. The strength of the evidence is the use of several independent model families (UCM networks with different exponents, an RRN with an outlier hub, and a diluted contact process), the consistent qualitative picture across these models, and the explicit finite-size analyses. The absence of any reported false positive is a useful asymmetry for practitioners. However, the quantitative support for 'postponed' criticality rests on a partial Pearson coefficient with an ad hoc cutoff, and the false-negative claim depends on the reliability of susceptibility-peak labels and on the quasistationary sampling protocol. These issues require additional controls before the conclusion can be considered fully established.

major comments (4)

[Sec. V.B and Sec. III] In Sec. III the authors concede that the hub-reactivation method 'artificially introduces strong localization in a single node irrespective of the model nature or network structure.' This concession is directly relevant to the RRN-with-hub experiment in Sec. V.B: at the global susceptibility peak λ=0.345 (Fig. 7) the hub has degree khub=√N with N=10^7, so λ khub ≫ 1. Every reactivation of the most connected node after an absorbing event therefore injects a localized outbreak that is not representative of the collective critical fluctuations. The VG may then be correctly reading the sampling-contaminated series, not providing a false negative for criticality. Please provide a control using a sampling protocol that does not restart from the hub (e.g., the reflecting boundary condition with discontinuity removal discussed in Ref. [48]) and verify that the χ peak coincides with the collective critical point via finite-size scaling of the peak height and width.
[Sec. IV, Eqs. (2)-(4), Fig. 5] The quantitative claim that criticality is 'postponed' to very large system sizes relies on the partial Pearson coefficient r defined in Eqs. (2)-(4), with a lower cutoff q0 chosen separately for each degree exponent (q0=2⟨kvg⟩ for γ=2.25 and 2.75, q0=6⟨kvg⟩ for γ=3.5). No error bars or sensitivity analysis with respect to q0 are reported, and the extracted asymptotic values differ considerably across cases (r∞≈-0.023 for γ=2.75 versus r∞≈-0.087 for γ=3.5 with rigid cutoff). Since the fitting exponent b is stated to be non-universal, the extrapolation is meaningful only if the q0 dependence is shown to be weak. Please report r as a function of q0/⟨kvg⟩ for each case and include confidence intervals or bootstrap estimates; otherwise the different r∞ values cannot be distinguished from fitting artifacts.
[Sec. V.A, Fig. 6] The susceptibility peaks in Fig. 6 are used as ground-truth labels of criticality, but the text itself notes that for the natural-cutoff γ=3.5 networks one peak is related to activation of an outlier and another to global activation. At finite N=10^7 these peaks can be smeared or shifted by rare regions, so the time series at the arrowed λ values may not actually correspond to the collective critical point of the infinite system. To make the false-negative conclusion robust, the chosen λ values should be validated against an independent determination of the asymptotic critical point (e.g., finite-size scaling of the order parameter, Binder cumulant, or spectral gap analysis), and the analysis should be repeated for at least two system sizes to show that the assortative pattern does not drift toward disassortative behavior.
[Sec. V.C, Fig. 8] The diluted-contact-process experiment is presented as another instance of strong localization producing false negatives, but the quasistationary sampling protocol is not specified in this section. If the same hub-reactivation method is used, the most connected node in a realization may lie inside a rare region, and the repeated restart may dominate the low-activity dynamics, again contaminating the time series with artificially localized excursions. Please state the sampling protocol explicitly and, if it is hub reactivation, provide a control with an alternative protocol or with a clear demonstration that the VG result is insensitive to the restart rule.

minor comments (5)

[Fig. 5 caption] The inset of Fig. 5 is described as 'the VG corresponding to UCM network with γ=3.5 and natural cutoff,' but it actually shows the Pearson coefficient r as a function of N for that case; please correct the caption.
[Sec. IV, after Eq. (4)] The sentence 'The standard Pearson coefficient is recovered when q = q_min' should read 'when q0 = q_min' to match the notation of the partial average in Eq. (4).
[Fig. 2(a) caption] The parenthetical phrase 'the susceptibility as a function of the infection rate which presents a peak' is grammatically garbled; please rephrase, e.g., 'the inset shows the susceptibility as a function of the infection rate, which presents a peak at λ=λc.'
[Sec. IV, paragraph on q0] The clause 'The choice of this lower bound was based on the change of the pattern shown in Fig.4' would be clearer as 'based on the change in the pattern shown in Fig. 4.'
[Sec. V.C] The network name is given as 'Erdos-Renyi' in the text and caption; use 'Erdős–Rényi' for consistency with the rest of the manuscript.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the VG false-negative claim is tested against independent susceptibility-peak and Griffiths-phase labels, not derived from the signature itself.

full rationale

The paper's central claim is that the VG degree-correlation signature of criticality (disassortativity) can be blurred or turn assortative under strong localization, producing false negatives. This claim is not derived from the signature itself: the critical labels are assigned independently, either by the quasistationary dynamical susceptibility peak χ = N[<ρ^2>−<ρ>^2]/<ρ> (Sec. II) or, for the diluted contact process, by the power-law decay of the Griffiths phase taken from Ref. [28]. The VG signature is not defined in terms of these labels, and no parameter is fitted to the susceptibility and then renamed a prediction; the VG and the susceptibility are distinct observables computed from the same simulations but with no equation linking them by construction. The only self-citation, Ref. [45], supplies the method being tested rather than the grounds for declaring false negatives, and the paper independently re-validates the signature on new heterogeneous substrates (Figs. 3–5) before turning to localization. The explicit caveat in Sec. III that hub reactivation 'artificially introduces strong localization in a single node' is a stated limitation about the benchmark protocol, and therefore a correctness/robustness concern rather than a circular step. The derivation chain is thus self-contained against external benchmarks; score 1 reflects only the modest reliance on the authors' own earlier VG calibration, which is not load-bearing in a circular sense.

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

This is a simulation study with no closed-form derivation, so the ledger records the analysis choices and background assumptions that the conclusions depend on. The most important entries are the post hoc q0 cutoff used to compute partial Pearson coefficients, the fixed time-series length, and the assumption that susceptibility peaks and hub-reactivation sampling provide trustworthy ground-truth critical time series.

free parameters (4)

q0 cutoff for partial Pearson coefficient = q0 = 2<k_vg> for gamma=2.25,2.75; q0 = 6<k_vg> for gamma=3.5
Chosen after viewing Fig. 4 to exclude the initial assortative regime; the computed r(N) and extrapolated r_infinity depend on this cutoff.
time series length N_vg = 10^6 points
Fixed for computational feasibility; because criticality signatures are asymptotic, this finite length can suppress weak disassortative trends and contributes to the blurred observations.
finite-size scaling exponent b = not stated numerically; nonuniversal
Used to extrapolate r to r_infinity for gamma=2.75 and 3.5 with few data points and no error bars; conclusion of slow convergence to r_infinity about -0.023 is fragile.
hub degree in RRN-with-hub network = k_hub = sqrt(N)
Selected to mimic the degree outlier regime of UCM networks with natural cutoff; the resulting single-node localization drives the false-negative in Fig. 7.

assumptions (4)

domain assumption Disassortative asymptotic VG degree correlation is the signature of critical time series; assortativity indicates off-critical dynamics.
Taken from the authors' prior paper [45]; not rederived here, and is the methodological premise under test.
domain assumption The maximum of dynamic susceptibility chi = N[<rho^2>-<rho>^2]/<rho> estimates the critical or finite-size pseudo-critical point.
Standard practice, but susceptibility can have multiple peaks under localization (Sec V A), making the ground-truth label of a time series ambiguous.
domain assumption Quasistationary sampling with reactivation of the most connected node preserves the relevant critical time-series statistics.
The authors note the gold-standard quasistationary method produces discontinuous series unsuitable for VG and that hub reactivation can artificially localize subcritical dynamics (Sec III); the assumption is therefore only partially checked.
domain assumption The UCM and ER network ensembles with chosen cutoffs realize the claimed activation mechanisms (collective, k-core, hub, Griffiths phase).
Classification of mechanisms relies on prior literature [34,63,28]; if those classifications are wrong for the chosen parameters, the interpretation of VG outcomes changes.

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Cite this review

Pith. "Pith review of Strong localization blurs criticality of time series for spreading phenomena on networks." pith.science (2026). https://pith.science/paper/ES7YEI6V

@misc{pith2026241201842,
  author       = {Pith},
  title        = {Pith review of: Strong localization blurs criticality of time series for spreading phenomena on networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ES7YEI6V}},
  note         = {Machine review of arXiv:2412.01842}
}

read the original abstract

We analyze critical time series of the order parameter generated with active to inactive phase transitions of spreading dynamics running on the top of heterogeneous networks. Different activation mechanisms that govern the dynamics near the critical point were investigated. The time series were analyzed using the visibility graph (VG) method where a disassortative degree correlation of the VG is a signature of criticality. In contrast, assortative correlation is associated with offcritical dynamics. The signature of criticality given by the VG is confirmed for collective activation phenomena, as in the case of homogeneous networks. Similarly, for a localized activation driven by a densely connected set of hubs, identified by a maximum k-core decomposition, critical times series were also successfully identified by the VG method. However, in the case of activation driven by sparsely distributed hubs, the time series criticality is blurred, being observable only for huge systems. In the case of strong structural localization induced by the presence of rare regions, an assortative VG degree correlation, typical of off-critical series, is observed. We conclude that while macroscopic times series remain good proxies for the analysis of criticality for collective or maximum k-core activation, systems under spatial localization can postpone the signatures of or, in case of extreme localization, lead to false negatives for criticality of time series.

Figures

Figures reproduced from arXiv: 2412.01842 by the authors.

**Figure 2.** FIG. 2. Analysis of the SIS model in a RRN of size [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Average degree of the nearest neighbors of the VG [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Average degree of the nearest neighbors for the VG generated from critical time series of prevalence for the SIS model [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Pearson correlation coefficient of the VG as a function [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Top panels: susceptibility as a function of infection rate for SIS model in 3 different samples of UCM networks with [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Main plot: susceptibility as a function of the infection [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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