{"id":"3f8cc8d1-98c9-42e3-9310-e7e1d8da3b80","arxiv_id":"2412.01842","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Visibility-graph criticality detection works for collective and k-core driven spreading, but strong localization produces false negatives, so missing the signature does not mean the system is off-critical.","lead":"The paper tests a method for detecting criticality in time series by mapping them into visibility graphs, and finds that strong localization of spreading around hubs or rare regions hides the criticality signature. The result is a needed caution for analyses of brain, biological, and epidemic data, where localized activity is common.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"False-negative claim may rest on hub-reactivation sampling artifacts and susceptibility-peak labels that are not valid critical labels under strong localization.","rationale":"The reader's weakest_assumption identified exactly this pair of issues: susceptibility peaks may be smeared or shifted by localization, and hub reactivation may alter critical time-series statistics. The present stress-test sharpens the mechanism: near a localized critical point, the absorbing state is visited often, and every hub restart creates a localized spike; in the RRN-with-hub example the hub is supercritical at the global susceptibility peak, so the restart process is not a neutral sampling device. The paper's own admission in Sec. III that hub reactivation can artificially localize subcritical dynamics makes this the most load-bearing threat to the central claim, because the VG signature is read from the very fluctuations that the sampler may contaminate. The proposed control is decisive and inexpensive: replacing hub restart with a reflecting protocol and labeling criticality by a method independent of the susceptibility peak would show whether the assortative VG persists at true criticality. If it does not persist, the paper's distinctive false-negative claim reduces to a sampling artifact; if it persists, the conclusion is robust. Since the reader already attached a CONDITIONAL verdict, the present read does not move the verdict, but it does identify the single experiment that should be required before the false-negative wording is kept.","tokens_in":13919,"tokens_out":7249,"duration_ms":80618,"concrete_test":"Recompute the RRN-with-hub case (Fig. 7) using a quasistationary protocol without hub restart, e.g., reflecting boundary conditions that return to the last active configuration, and determine the collective critical point independently via Binder-cumulant crossing or bulk-seeded survival-probability scaling. Compare the VG Knn curves at this independently identified lambda with those in the inset of Fig. 7. If the VG becomes disassortative at the independently labeled critical point while remaining assortative at the local-outlier peak, the claimed false negative is an artifact of the hub-reactivation sampler and susceptibility-peak label, not of localization itself. The same control should be repeated for the diluted CP Griffiths-phase case with the five disorder realizations of Fig. 8.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The false-negative conclusion presupposes that the time series fed to the VG are faithful samples of critical dynamics. Two linked assumptions carry the weight: (i) the susceptibility peak used to label a run 'critical' is the true collective critical point even under strong localization; (ii) restarting from the most-connected node after each absorbing event, the quasistationary protocol used throughout, does not inject the very localization the paper blames. The authors concede in Sec. III that hub reactivation 'artificially introduces strong localization in a single node irrespective of the model nature or network structure,' and dismiss this only for the deep subcritical regime. But in the RRN-with-hub setup of Sec. V.B and in the diluted CP Griffiths phase, the absorbing state is reached frequently near criticality and every restart is a delta-like localized excursion from the hub or a high-degree node. At the global susceptibility peak of Fig. 7, lambda = 0.345, the hub is already individually supercritical, so reactivation may dominate the fluctuation statistics that the VG reads. Likewise, in the gamma = 3.5 natural-cutoff UCM networks, the susceptibility peaks ticked in Fig. 6 include outlier activation events, so those runs may not be at the collective critical point at all. If either assumption fails, the assortative VG is not a false negative; it is the correct signature of the off-critical or sampling-contaminated series that was actually analyzed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14089,"tokens_out":4528,"duration_ms":44828,"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":[{"comment":"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.","section":"Sec. V.B and Sec. III"},{"comment":"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.","section":"Sec. IV, Eqs. (2)-(4), Fig. 5"},{"comment":"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.","section":"Sec. V.A, Fig. 6"},{"comment":"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.","section":"Sec. V.C, Fig. 8"}],"minor_comments":[{"comment":"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.","section":"Fig. 5 caption"},{"comment":"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).","section":"Sec. IV, after Eq. (4)"},{"comment":"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.'","section":"Fig. 2(a) caption"},{"comment":"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.'","section":"Sec. IV, paragraph on q0"},{"comment":"The network name is given as 'Erdos-Renyi' in the text and caption; use 'Erdős–Rényi' for consistency with the rest of the manuscript.","section":"Sec. V.C"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a direct follow-up to the authors' own previous work (Ref. [45]) that defines the VG criticality signature, and the central conclusion depends heavily on that signature without an independent replication. That is not by itself disqualifying, but it raises the stakes for the reliability of the ground-truth labels and sampling protocol. I would also note that the paper does not include a data-availability or code-availability statement; for a largely numerical study, providing the simulation and VG code would substantially increase the reproducibility of the finite-size extrapolations. The journal fit for physics.bio-ph is appropriate given the potential relevance to brain-criticality time series."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\"\n\nNet: 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.","headline":"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.","tokens_in":14691,"tokens_out":6351,"would_cite":true,"duration_ms":66654,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["visibility graph","criticality","time series","spreading dynamics","complex networks","localization","Griffiths phases","absorbing state phase transitions"],"falsifier":"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.","tokens_in":13622,"feed_emoji":"🕸️","tokens_out":7423,"duration_ms":60434,"temperature":0.7,"pith_summary":"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.","feed_headline":"Network localization can erase criticality's time-series fingerprint","feed_subtitle":"The method never fakes criticality, but strong localization can hide true critical points.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the visibility-graph degree-correlation signature of criticality (disassortative vs assortative) that the paper extends to heterogeneous networks and tests under localization.","marker":"[45]"},{"why":"Defines the natural visibility graph mapping used to transform each prevalence time series into a network.","marker":"[46]"},{"why":"Identifies the competing activation mechanisms (k-core vs. sparsely distributed hubs) that structure the paper's analysis.","marker":"[34]"},{"why":"Introduces Griffiths phases on complex networks, the strong-disorder scenario used for the diluted contact process tests.","marker":"[28]"},{"why":"Provides the quasistationary sampling method with hub reactivation used to generate time series, including the acknowledged caveat that it can artificially localize subcritical dynamics.","marker":"[48]"},{"why":"Introduces the random regular network with a single hub, the controlled model of strong localization that produces a clear false negative.","marker":"[72]"},{"why":"Supplies the dynamical susceptibility and the threshold-estimation procedure used to label the critical points.","marker":"[60]"}],"fun_headline_variants":["Strong localization blurs critical time-series signatures on networks","Network localization can hide true critical points from time series","Visibility graph criticality test fails under strong localization","Extreme localization erases criticality fingerprint in dynamics","Localized hubs can delay or erase criticality signals in networks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Strong localization blurs critical time-series signatures on networks","Network localization can hide true critical points from time series","Visibility graph criticality test fails under strong localization","Extreme localization erases criticality fingerprint in dynamics","Localized hubs can delay or erase criticality signals in networks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00025,"raw_usage":{"total_tokens":1574,"prompt_tokens":989,"completion_tokens":585,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":508}},"tokens_in":605,"tokens_out":585,"duration_ms":5686,"temperature":1.0,"reasoning_tokens":508,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:54:53.435645+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the visibility-graph degree-correlation signature of criticality (disassortative vs assortative) that the paper extends to heterogeneous networks and tests under localization."},{"cited_title":"Lacasa, B","cited_arxiv_id":null,"evidence_quote":"Defines the natural visibility graph mapping used to transform each prevalence time series into a network."},{"cited_title":"Castellano and R","cited_arxiv_id":null,"evidence_quote":"Identifies the competing activation mechanisms (k-core vs. sparsely distributed hubs) that structure the paper's analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces Griffiths phases on complex networks, the strong-disorder scenario used for the diluted contact process tests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the quasistationary sampling method with hub reactivation used to generate time series, including the acknowledged caveat that it can artificially localize subcritical dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the random regular network with a single hub, the controlled model of strong localization that produces a clear false negative."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the dynamical susceptibility and the threshold-estimation procedure used to label the critical points."}],"review_version":1}