{"id":"f258a921-ba90-41d1-aadd-bc7260c2f540","arxiv_id":"2411.13511","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Wigner delay time and resonance width distributions of Erdos-Renyi, random geometric, and bipartite random geometric graphs collapse onto a universal curve when scaled by xi = <k> N^{-alpha} with model-dependent alpha.","lead":"This numerical study computes Wigner delay times and resonance widths for three families of random graphs with a single attached lead, and finds that a scaling parameter based on average degree and graph size collapses the statistical distributions. The apparent universality could be tested in microwave photonic graphs and may help characterize random scattering media without direct access to wavefunctions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The xi-scaling claim rests on visual collapse and an imported alpha; without a quantitative collapse metric or an independent refit, invariance under Eq. (19) is unverified, and residual N-dependence appears in the insulator regime.","rationale":"The reader's conditional verdict already identifies missing collapse metrics, absent code/data, imported fits, and the BRGG(4N/5) perfect-coupling limitation. My stress-test focuses on the scaling parameter itself: the universality claim depends on alpha from prior spectral fits being the correct exponent for delay-time and width distributions, and the published support for that transfer is visual overlay rather than a quantitative collapse test. This is not a contradiction of consensus; it is an internal-support gap. I would keep the manuscript CONDITIONAL rather than reject it, because the proposed KS-based refit is a feasible and decisive check: if the KS distances are within sampling error and alpha refits match, the claim is strong; if not, the main universal-scaling conclusion fails. The complete-graph agreement with RMT is on firmer ground because a complete random-weight graph approximates a GOE Hamiltonian, so the RMT reference is expected; the novel scaling part is what most needs quantitative validation. This is why I set agreement_with_reader to partial rather than agree: the reader's weakest assumption centered on the coupling formula and BRGG asymmetry, while I view the unverified transfer of alpha to the delay-time/width collapse as the most load-bearing single condition.","tokens_in":16495,"tokens_out":6227,"duration_ms":73331,"concrete_test":"For each model and each xi value in Figs. 6-8, compute Kolmogorov-Smirnov (or integrated L2) distances between the N = 100, 200, and 400 empirical distributions of ln(tau^-1/<tau^-1>), ln(tau/tau_typ), and ln(Gamma/Gamma_typ). Then independently refit alpha for the delay-time and width distributions by minimizing the average cross-size histogram distance as a function of xi = <k> N^{-alpha}, and check whether the best-fit values agree with 0.075, 0.26, and 0.3429 within the stated uncertainties. Extend one case to N = 800 to confirm that residual differences decrease with N; if the best-fit alpha shifts outside the quoted errors or KS distances do not shrink, the invariance claim in Eq. (19) is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, stated in the abstract and in Sec. IV, is that the distributions of Wigner delay times and resonance widths are invariant under xi = <k> N^{-alpha}, Eq. (19), and that mostly connected graphs match RMT distributions (11), (12), and (17). The load-bearing evidence for the scaling part is qualitative: Figs. 6-8 present superposed histograms with no KS/L2 distances, no error quantification for the collapse, and no demonstration that residuals shrink with N. The exponents alpha = 0.075, 0.26, 0.3429 are imported from previous spectral fits [26] rather than fitted to, or derived for, the delay-time and width distributions; since alpha controls the N-dependence of the connectivity (e.g., p ~ xi N^{alpha-1} for ERGs), even a modest error in alpha would destroy collapse at larger N. The visible residual N-dependence in the insulator regime, e.g., Fig. 6(c) for BRGGs at xi = 0.2, shows the claimed collapse is not uniform across the regimes used to advertise universality. Additionally, the RMT comparison uses k1, k2, k3 computed only at N = 100 (Tables I and II), so the claim that agreement improves with N is not directly tested; and the paper itself states that BRGG(4N/5) does not attain the perfect-coupling/RMT regime, which already restricts the RMT leg of the claim to ERGs and RGGs. These issues are addressable, but they are exactly where the central claim is least secure.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Wigner delay times and resonance widths of tight-binding random graphs (Erdős–Rényi graphs, random geometric graphs, and bipartite random geometric graphs) opened to a single-channel lead in the perfect-coupling regime. Using the scattering-matrix formalism, the authors compute numerically the distributions of the inverse Wigner delay time, the delay time normalized by its typical value, and the resonance widths, for graph sizes N = 100, 200, 400 with 10^6 samples each. Their central claim is that these distributions are invariant under the scaling parameter ξ = ⟨k⟩ N^{-α}, Eq. (19), with model-dependent α, and that for mostly connected graphs the distributions coincide with the RMT forms (11), (12), and (17). The perfect-coupling strength is set by the empirical formula (18), and the exponents α are imported from Ref. [26].","tokens_in":16849,"tokens_out":2856,"duration_ms":29944,"significance":"If the universal-collapse claim can be made quantitative, the paper would extend RMT-based scattering universality from fully connected random matrices to sparse random graphs and provide a useful single parameter, ξ, for classifying transport regimes in microwave or photonic realizations of tight-binding graphs. The numerical effort is substantial (10^6 realizations), and the reproduction of the known RMT distributions for complete ERGs and RGGs is a useful reference result. The main weaknesses are that the collapse under Eq. (19) is assessed visually, with no statistical metric, and that several load-bearing ingredients (α and the perfect-coupling formula) are imported from earlier fits rather than derived or independently verified here.","major_comments":[{"comment":"The central claim of the paper, that the distributions are invariant under ξ = ⟨k⟩ N^{-α}, is supported only by visual superposition of histograms. No quantitative collapse measure is provided, such as Kolmogorov–Smirnov distances, L2 differences, or scaling residuals as a function of N at fixed ξ. Because α is imported from spectral fits in Ref. [26] and is not fitted to or derived for delay times and widths, a small error in α would destroy the collapse at larger N. I ask the authors to quantify the collapse, for example by computing pairwise distances between empirical distributions at different N for fixed ξ and comparing them to the statistical fluctuations of the ensembles, and ideally to re-fit α directly from the delay-time and width distributions to check consistency with the spectral values.","section":"Section IV, Eq. (19), Figs. 6–8"},{"comment":"The RMT reference distributions (11), (12), and (17) are evaluated using parameters k1, k2, and k3 computed only for N = 100 (Tables I and II). The text states that agreement with RMT improves as graphs become more connected, but the plotted comparisons at N = 200 and N = 400 reuse the N = 100 parameters without comment. If k1, k2, k3 vary with N, the comparison at larger N is not the true RMT limit for that N; if they are stable, this should be stated and demonstrated. The authors should either recompute the table parameters at each N or show explicitly that the parameter values are independent of N within the statistical accuracy.","section":"Section IV, Tables I and II"},{"comment":"The authors acknowledge a residual graph-size dependence in the insulator regime, particularly for BRGGs in panel (c) of Fig. 6 and in the upper panels of Fig. 8. Since the claimed universality is meant to hold across regimes, this size dependence should be characterized quantitatively. The paper should state over which range of ξ the collapse holds to within a specified tolerance, and quantify the deviations in the insulator regime, rather than relying on the qualitative phrase 'slight dependence'.","section":"Section IV, Fig. 6(c) and accompanying text"},{"comment":"The perfect-coupling condition, which is the precondition for all the reported comparisons, is set by the empirical formula (18) imported from Refs. [25, 26]. The paper does not provide error estimates for the coefficients in (18), nor does it test the formula for the values of N used here. Since an incorrect ε0 would shift the entire delay-time and width statistics, the authors should at least verify ⟨S(ε0)⟩ ≈ 0 for representative (N, ξ) points across the three graph models, and report the uncertainty in the fitted coefficients if available.","section":"Section IV, Eq. (18)"},{"comment":"The statements in the abstract and conclusions that the distributions are invariant under ξ for all three graph models, and that mostly connected graphs match the RMT forms, are too broad given the manuscript's own caveat that BRGGs(4N/5) never reaches the RMT limit because of the asymmetry of attaching the lead to the larger set. The authors should explicitly separate the scaling universality claim, which is intended to hold for all three models, from the RMT-approach claim, which is restricted to ERGs and RGGs (and to BRGGs only in other bipartitions).","section":"Abstract and Conclusions"}],"minor_comments":[{"comment":"There is a typo in the caption text: 'BRRGs' should read 'BRGGs'.","section":"Section IV, Fig. 2 caption"},{"comment":"The caption says 'Right, middle, and left columns show the results', while Figs. 2–7 use 'first, second, and third columns'. Please make the caption consistent with the other figures.","section":"Section IV, Fig. 8 caption"},{"comment":"The word 'analitycal' appears in the Conclusions; it should be 'analytical'.","section":"Section V"},{"comment":"The phrase 'deviations betweem numerical data' contains a typo; 'betweem' should be 'between'.","section":"Section IV"},{"comment":"The column header 'BRGGs(4 N/5)' is formatted inconsistently; consider using 'BRGGs(4N/5)' as in the text. Also, the tables would benefit from a statement of the statistical uncertainty in the quoted values of Δ, ⟨τ^{-1}⟩, τ_typ, and Γ_typ.","section":"Tables I and II"}],"recommendation":"major_revision","confidential_remarks":"The paper's main novelty is the empirical ξ-scaling of delay-time and width distributions for three graph families. The reliance on prior fits for α and ε0, combined with the absence of quantitative collapse metrics, makes the central claim currently under-supported. However, the issues are addressable within the scope of a revision, provided the authors add collapse statistics, recompute RMT parameters at multiple N, and qualify the claims for BRGGs. I would also note that the abstract slightly overstates the scope given the BRGG(4N/5) caveat."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the distributions themselves are new and the numerics are careful, but the central scaling claim is demonstrated by eye, not by measure, and that is exactly the part that needs to carry the paper.\n\nWhat is actually new: the Wigner delay-time and resonance-width distributions for ERGs, RGGs, and BRGGs in the one-channel setup, and their apparent collapse under xi = <k> N^{-alpha}. The complete-graph limit reproduces the known RMT forms, which is a good sanity check. The authors are also straight about the BRGG(4N/5) case: they say it does not reach the perfect-coupling/RMT regime because of the asymmetric setup and the graph's limited complexity. That kind of honesty is rare and worth crediting.\n\nThe soft spots are the usual ones for a numerical scaling study. The collapse is read off log-linear histograms; there is no KS statistic, no residual-versus-N plot, no statement about how the spread between N=100, 200, 400 shrinks. For a claim whose content is “these distributions are N-independent at fixed xi,” visual superposition is not enough. Second, alpha comes from earlier spectral fits in work by the same group, not from the delay-time/width data. That is not circular in a disqualifying sense—using an independently determined exponent is a legitimate test—but it is a fitted input, and the conclusion inherits its uncertainty. A refit or a sensitivity scan over alpha would settle it. Third, the RMT comparisons in Tables I and II use N=100 only, so the stated trend of improving agreement with N is not actually tested. And the BRGG limitation means the word “universal” in the abstract covers all three models, but the RMT leg only really applies to ERGs and RGGs.\n\nNone of this sinks the paper. The methodology is transparent, the statistics are large, and the limitations are mostly acknowledged. I would send it to an expert referee and ask for quantitative collapse metrics, an alpha sensitivity analysis, and N-dependent RMT comparisons. After that, it would be a solid reference for the random-graph scattering community and a useful pointer for microwave experiments.","headline":"A careful numerical extension that likely gives real xi-scaling for delay times and resonance widths, but the collapse is only shown visually and deserves a quantitative pass before publication.","tokens_in":17390,"tokens_out":3074,"would_cite":false,"duration_ms":36035,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["46.65.+g","89.75.Hc","05.60.Gg"],"model":"deepseek-v4-flash","headline":"The paper claims that Wigner-delay-time and resonance-width distributions of sparse random graphs collapse onto universal curves under the single scaling parameter $\\xi = \\langle k \\rangle N^{-\\alpha}$, matching random-matrix theory in…","keywords":["Wigner delay time","resonance widths","random graphs","Erdos-Renyi graphs","random geometric graphs","tight-binding model","random matrix theory","scaling universality"],"falsifier":"Compute the same $\\xi$-fixed histograms for Erdős–Rényi graphs with $N=1000$ and $N=2000$ using Eq. (18) and $\\alpha=0.075$; if the curves separate by size, the power-law scaling is not the full story. Alternatively, measure the ensemble-averaged scattering matrix at the predicted perfect-coupling strength for these larger sizes; if $\\langle S\\rangle$ is not close to zero, the comparison with the RMT reference distributions loses its justification.","tokens_in":16317,"feed_emoji":"🕒","tokens_out":11752,"duration_ms":104617,"temperature":0.7,"pith_summary":"Using the equivalence between a random graph's adjacency matrix and a tight-binding Hamiltonian, the authors open Erdős–Rényi graphs, random geometric graphs, and bipartite random geometric graphs by a single-channel lead and numerically compute Wigner delay times and resonance widths. Their central claim is that the distributions of both quantities become functions of a single scaling parameter, $\\xi = \\langle k \\rangle N^{-\\alpha}$, so that curves for different graph sizes, connectivities, and graph models fall on top of each other. For mostly connected Erdős–Rényi and random geometric graphs the collapsed distributions agree with the analytic random-matrix-theory expressions for one-channel scattering, Eqs. (11), (12), and (17). The paper also shows that the mean delay time divided by $N$ follows the mean eigenvalue density of the closed graph, $\\langle \\tau \\rangle/N = 2\\pi\\rho(E)$. The value of the claim is practical: delay times and resonance widths are experimentally accessible, so a single-parameter description predicts what microwave photonic realizations of these graphs should measure.","feed_headline":"One parameter collapses random-graph delay-time statistics","feed_subtitle":"Wigner delay times and resonance widths follow one universal curve after rescaling by degree and graph size.","key_machinery":"The load-bearing object is the scaling parameter $\\xi = \\langle k \\rangle N^{-\\alpha}$, built from the average degree and the graph size, with model-dependent exponents taken from prior spectral studies. It fixes the horizontal scale on which delay-time and width distributions are compared and, through the empirical perfect-coupling formula Eq. (18), sets the lead-to-graph coupling strength $\\varepsilon_0$ used in every calculation. The scattering formalism reduces the single-channel $S$-matrix to a phase $S(E)=e^{i\\phi(E)}$; the Wigner delay time is $\\tau=d\\phi/dE$ at $E=0$, and resonance widths $\\Gamma_n$ come from the complex eigenvalues of the effective non-Hermitian Hamiltonian $H_{\\rm eff}=H-e^{ik}WW^T$. The reference universal curves are the one-channel RMT delay-time distributions (11) and (12), derived from the Porter-Thomas distribution of eigenfunction intensities, and the resonance-width distribution (17) for non-isolated resonances.","core_discovery":"The central discovery is the invariance of the distributions of Wigner delay times and resonance widths under the scaling $\\xi = \\langle k \\rangle N^{-\\alpha}$, with $\\alpha$ depending on the graph family ($0.075$ for Erdős–Rényi graphs, $0.26$ for random geometric graphs, $0.3429$ for bipartite random geometric graphs with $s=4N/5$). For graph sizes $N=100,200,400$ and fixed $\\xi$, histograms of $\\ln(\\tau^{-1}/\\langle\\tau^{-1}\\rangle)$, $\\ln(\\tau/\\tau_{\\rm typ})$, and $\\ln(\\Gamma/\\Gamma_{\\rm typ})$ coincide for different sizes and connectivities, and the same collapsed curves are shared by the three graph models. In the mostly connected regime the collapsed curves coincide with the random-matrix-theory distributions (11), (12), and (17), showing a smooth crossover from insulating to metallic scattering as $\\xi$ grows. The exception stated in the paper is the bipartite random geometric graph with the lead attached to the larger set: the scattering setup is too asymmetric to reach the RMT reference, although its $\\xi$-scaled distributions still collapse onto the common curves.","pith_inferences":["Because the exponents $\\alpha$ differ between graph families, the scaling variable is not a single universal quantity; one could test whether $\\alpha$ is controlled by the spectral dimension or by the bipartite and geometric structure of the model.","The collapse is demonstrated for $N\\le 400$; an unambiguous test would push to $N=1000$ or beyond and ask whether the same fixed $\\alpha$ still collapses the curves, or whether the power-law form is only an effective low-size description.","The same $\\xi$-scaling logic might extend to other one-channel scattering observables, such as conductance or shot-noise distributions, and to multi-channel leads, where the RMT reference distributions differ but the scaling parameter may still organize the crossover.","The empirical perfect-coupling formula is fitted numerically; a direct measurement of $\\langle S\\rangle$ at the predicted $\\varepsilon_0$ for larger $N$ would either confirm the universality or reveal the parameter range where the formula breaks down."],"forward_implications":["For a random graph of either family, the full single-channel delay-time and resonance-width statistics can be predicted from $\\langle k\\rangle$, $N$, and the family-specific $\\alpha$, without solving the scattering problem.","Microwave photonic arrays that realize tight-binding random geometric graphs should observe the same $\\xi$-collapse and the same RMT distributions in the dense limit, providing a direct experimental test.","The mean Wigner delay time is tied to the closed graph's eigenvalue density through $\\langle\\tau\\rangle/N = 2\\pi\\rho(E)$, so spectral information of the closed system controls the open-system delay.","For the bipartite graph with an asymmetric lead attachment, the RMT reference is not attainable, but the $\\xi$-scaling still organizes the data, indicating a universality distinct from the GOE limit.","The crossover from insulator to metal as $\\xi$ increases is smooth rather than critical, so the delay time can serve as a convenient transport probe of the crossover without access to eigenfunctions."],"supporting_citations":[{"why":"Supplies the empirical perfect-coupling formula for Erdős–Rényi graphs and the isolated-to-metallic crossover analysis on which the coupling-strength calibration rests.","marker":"[25]"},{"why":"Provides the analogous coupling-strength formulas for random geometric and bipartite graph models and the alpha exponents used in xi, and states that the asymmetric bipartite setup does not reach the RMT limit.","marker":"[26]"},{"why":"Introduced the scaling parameter xi proportional to <k> N^{-alpha} for spectral properties of directed random geometric graphs, which this paper extends to scattering observables.","marker":"[65]"},{"why":"Derives the one-channel Wigner delay time distributions (11) and (12) from the Porter-Thomas distribution of eigenfunction intensities, the RMT reference for the delay-time collapse.","marker":"[56]"},{"why":"Derives the non-isolated resonance-width distribution (16) and its logarithmic form (17), the RMT reference for the width distributions.","marker":"[61]"},{"why":"Establishes the relation <tau>/N = 2 pi rho(E) connecting the mean Wigner delay time to the closed-system eigenvalue density, used to explain the N-scaling of the delay time.","marker":"[35]"}],"fun_headline_variants":["One parameter collapses random-graph delay statistics","Universal scaling for Wigner delay times in random graphs","Single scaling law for resonance widths and delay times","Degree-size rescaling unifies graph scattering universality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the empirical formula Eq. (18) for the coupling strength that achieves perfect coupling, together with the fitted exponents $\\alpha$; if that formula or the exponents are not transferable to larger graphs or to other lead attachments, the apparent universality could be an artifact of the studied parameter range.","fun_headline_variants_meta":{"raw":{"variants":["One parameter collapses random-graph delay statistics","Universal scaling for Wigner delay times in random graphs","Single scaling law for resonance widths and delay times","Degree-size rescaling unifies graph scattering universality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000181,"raw_usage":{"total_tokens":1319,"prompt_tokens":971,"completion_tokens":348,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":300}},"tokens_in":587,"tokens_out":348,"duration_ms":4405,"temperature":1.0,"reasoning_tokens":300,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:19:46.418868+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same $\\xi$-fixed histograms for Erdős–Rényi graphs with $N=1000$ and $N=2000$ using Eq. (18) and $\\alpha=0.075$; if the curves separate by size, the power-law scaling is not the full story. Alternatively, measure the ensemble-averaged scattering matrix at the predicted perfect-coupling strength for these larger sizes; if $\\langle S\\rangle$ is not close to zero, the comparison with the RMT reference distributions loses its justification.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the empirical perfect-coupling formula for Erdős–Rényi graphs and the isolated-to-metallic crossover analysis on which the coupling-strength calibration rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the analogous coupling-strength formulas for random geometric and bipartite graph models and the alpha exponents used in xi, and states that the asymmetric bipartite setup does not reach the RMT limit."},{"cited_title":"Peralta-Mart ´ ınez and J","cited_arxiv_id":null,"evidence_quote":"Introduced the scaling parameter xi proportional to <k> N^{-alpha} for spectral properties of directed random geometric graphs, which this paper extends to scattering observables."},{"cited_title":"Ossipov and Y","cited_arxiv_id":null,"evidence_quote":"Derives the one-channel Wigner delay time distributions (11) and (12) from the Porter-Thomas distribution of eigenfunction intensities, the RMT reference for the delay-time collapse."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the non-isolated resonance-width distribution (16) and its logarithmic form (17), the RMT reference for the width distributions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the relation <tau>/N = 2 pi rho(E) connecting the mean Wigner delay time to the closed-system eigenvalue density, used to explain the N-scaling of the delay time."}],"review_version":1}