{"id":"530876f5-5f0d-4816-afcb-a1559bab36b3","arxiv_id":"2607.24615","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Non-Markovian SIR outbreaks on networks reduce to an effective Markovian model via a single per-edge transmissibility, giving a universal curve for the full outbreak-size distribution on weakly heterogeneous networks.","lead":"This paper shows that for non-Markovian epidemics on networks, the spread of waiting-time distributions can be compressed into a single number per link, so the full outbreak-size distribution can be predicted from an effective Markovian model. A smart generalist might read it because it offers a route from measured infection-timing data to the probability of unusually large outbreaks, which matters for hospital capacity and risk planning.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Transmissibility T fixes only marginal transmission probabilities; the joint distribution of transmissions from one infected node depends on the full waiting-time shape, so the claimed full-distribution equivalence to effective Markovian dynamics is not exact.","rationale":"The reader's weakest assumption concerns the universal collapse for weakly heterogeneous networks, with the caveat that it is empirical rather than derived. This stress-test identifies a more specific mechanism: the transmissibility reduction fixes only first-order (marginal) statistics, while the joint distribution of multiple transmissions from one infected node depends on the full waiting-time shape through integrals like E[F_i(τ)^2]. This affects the full outbreak-size distribution, not just the normalized collapse, and it is exactly the quantity the paper claims to predict. The star test is deliberately simple: on a one-generation tree the final-size distribution is just the offspring distribution, and the pgf formula is exact, so the test can settle the concern unambiguously. If the test shows different G(s) for fixed T, the mapping is not exact; the paper's claims would need qualification as approximate or restricted to regimes where the correlations average out. If the test passes, the concern is refuted and the paper's empirical evidence is stronger. Since the reader already conditionally accepts with requests for derivation and error bars, this concern reinforces that conditional verdict without changing it.","tokens_in":11166,"tokens_out":25727,"duration_ms":260096,"concrete_test":"On a star with m leaves and the hub initially infected, let the hub recovery time τ be Exp(1) and let F_i be the infection WT CDF. The pgf of the number of infected leaves is G(s) = E_τ[(1 - F_i(τ) + F_i(τ)s)^m]. Compare G(s) for gamma-distributed infection WTs (Eq. 4) with shape α and for Markovian exponential infection, choosing parameters so that T = E[F_i(τ)] is identical in both cases (e.g., T = 1/2). If G(s) differs — specifically, if P(two leaves infected) ≠ [E F_i(τ)]^2 — then the effective-Markovian mapping at fixed T is not exact even on this tree, and the claim that the full outbreak-size distribution is captured by transmissibility alone is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The per-edge transmissibility T in Eq. (2) determines only the marginal probability that a given S-I pair transmits before recovery. It does not determine the joint law of transmissions from a single infected node: conditional on the node's recovery time τ, the number of transmissions to its k susceptible neighbors is Binomial(k, F_i(τ)), where F_i is the infection waiting-time CDF. Unconditionally, the offspring probability generating function is E_τ[(1 - F_i(τ) + F_i(τ)s)^k], which depends on the whole function F_i, not only on T = E[F_i(τ)]. Two waiting-time distributions with identical T but different shapes therefore lead to different final-size distributions on a star or on a regular network. Consequently, the central claim that an effective Markovian process with the same T (Eq. 9) reproduces the full outbreak-size distribution cannot follow from T alone. The paper's numerical collapse (Fig. 2) covers only a narrow range of shapes, and for the Hamsterster network R_net^0 is fitted to the simulated mean (Appendix D), so the full-distribution comparison is partly circular.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a mapping from non-Markovian SIR dynamics on networks to an effective Markovian description, parameterized by a single per-edge transmissibility T defined in Eq. (2). The authors claim that this mapping reproduces the full outbreak-size distribution: for weakly heterogeneous networks, the distribution collapses onto a universal well-mixed WKB curve governed by R0^eff from Eq. (8), and for highly heterogeneous or empirical networks, Markovian simulations on the same network at R_net^0 from Eq. (9) capture the distribution. The claims are supported by extensive simulations on regular, Erdős–Rényi, gamma-distributed, and Hamsterster networks with gamma and log-normal waiting-time distributions.","tokens_in":11325,"tokens_out":17464,"duration_ms":178288,"significance":"If the central equivalence held, the paper would provide a practical and valuable route from measured waiting-time distributions to extreme-outbreak risk on networks. The manuscript contains useful ingredients: an explicit transmissibility formula for gamma-distributed infection and recovery, a generalized bond-percolation treatment for degree-correlated networks, and an unusually broad simulation campaign. The paper is also clearly written and carefully describes its simulation methodology. However, the central claim that a single edge transmissibility determines the full outbreak-size distribution is not correct in general, and the empirical evidence for the universal collapse is partly circular. Because the advertised result is the full-distribution equivalence, the error is load-bearing.","major_comments":[{"comment":"The transmissibility T is only the marginal probability that a given S–I edge transmits before recovery. The joint law of transmissions from a single infected node is not determined by T. If an infected node has recovery time ρ and k susceptible neighbors, the number of transmissions has generating function E_ρ[(1 - F_i(ρ) + F_i(ρ) s)^k], which depends on the whole infection CDF F_i, not only on T = E[F_i(ρ)]. A concrete counterexample is a star with a single infected center and k leaves: the final outbreak size is exactly this number of transmissions, so two infection waiting-time distributions with identical T but different F_i give different final-size distributions. Equation (9) constructs an effective Markovian process with the same T, which predicts a Binomial(k, T) offspring distribution; this cannot reproduce both non-Markovian distributions. Thus the abstract claim that arbitrary waiting-time statistics can be incorporated through T alone and that the full outbreak-size statistics are reproduced is not supported.","section":"Eq. (2) and the paragraph following Eq. (9)"},{"comment":"For the Hamsterster network, R_net^0 is defined as the value that reproduces the simulated mean outbreak fraction x*_r in a Markovian simulation on the same network (last paragraph of Appendix D). The agreement shown in Figs. 4(c,d) and S3 is therefore not an independent test of the mapping: the single free parameter is calibrated to the first moment of the simulated distribution, and the comparison only checks whether the remaining shape is captured. To validate the predictive claim, R_net^0 should be computed directly from T and the network structure (e.g., through the correlated percolation equations S1–S2) without reference to the simulated x*_r, or the authors should provide an out-of-sample test on a network not used for calibration.","section":"Appendix D and Fig. S3"},{"comment":"The universal collapse onto the well-mixed WKB action, Eq. (6), is presented as an empirical observation. No derivation shows that the large-deviation rate function for the network SIR final size equals the well-mixed action as a function of R0^eff alone. Since R0^eff is computed from the deterministic final size via Eq. (8), it carries information only about the first moment; the right tail and the fluctuation scale could in principle depend on higher moments of the offspring distribution. The rescaling in Fig. 2(a) by the empirical mode and conditional standard deviation can mask such dependencies. The authors should either derive the claimed universality or test it directly by fixing T and varying the waiting-time shape while holding the network and mean final size fixed; the present simulations vary T and shape simultaneously.","section":"Eq. (8) and Fig. 2"},{"comment":"The numerical evidence does not directly probe the degenerate case that the theory is built on. In all reported simulations, the infection shape parameter α_inf is varied together with T, so T and the higher-order structure of the waiting-time distribution change at the same time. The claim that T alone controls the distribution requires comparing two non-Markovian processes with the same T but different waiting-time shapes. Such a comparison is absent, and the star counterexample above shows that the two processes will generally differ. At minimum, the manuscript needs to clearly state that the equivalence is approximate and holds only within the tested range of shapes and network parameters, and to add the fixed-T comparison.","section":"Figs. 2–4 and the effective-Markovian claim"}],"minor_comments":[{"comment":"The conditioning rule for 'extensive outbreaks' is not fully specified: 'the minimum separating the two modes of the bimodal outbreak-size distribution' is ambiguous for parameter sets where the two modes are not well separated. A precise algorithmic definition and a sensitivity check would be helpful.","section":"Appendix A"},{"comment":"The probability categories 'up to 20% below the mean', 'up to 20% above the mean', and 'more than 20% above the mean' should be defined as fractions of the conditional mean x*_r, and the interval endpoints should be stated explicitly, since the quoted percentages will depend on the conditioning event.","section":"Risk assessment paragraph"},{"comment":"The statement that 'similar trends are observed for Weibull and log-normal WT distributions' is not shown; the authors should either include the figure or label the statement as a remark based on simulations.","section":"Fig. 1 caption and text after Eq. (5)"},{"comment":"The validation of the first-reaction method against the modified next-reaction method of Ref. [49] is mentioned without quantitative results. A brief description of the validation (e.g., maximum relative error in the quantities reported later) would strengthen confidence in the numerics.","section":"Appendix A, simulation validation"}],"recommendation":"reject","confidential_remarks":"The central issue is that the claimed equivalence is false at the level of exact full-distribution statistics, as shown by the mixed-binomial offspring distribution, and the paper's own Hamsterster calibration is circular. The paper could potentially be reframed as an approximate collapse observed in a restricted parameter regime, but that would be a substantially weaker claim than the one advertised in the abstract and title. I would not block publication of a more cautious version, but the present manuscript's central claim is not correct as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take on Taitelbaum & Assaf, arXiv:2607.24615. The paper goes after a real gap: the full outbreak-size distribution, including the extreme right tail, for non-Markovian SIR on networks. Previous work had well-mixed large deviations and network thresholds, but not this combination. The practical recipe is simple and apparently useful: compute the per-edge transmissibility T, then run either the well-mixed WKB formula at an effective R0 for weakly heterogeneous networks, or Markovian simulations on the same network at an effective R0 for strongly heterogeneous ones. If that mapping holds, it gives a direct route from generation-interval data to extreme-outbreak risk.\n\nThe simulations are extensive and honestly described: 10^6 realizations per parameter set, several network families, gamma and log-normal waiting times, and the conditioning on extensive outbreaks is stated. The collapse in Fig. 2 is striking, and the WKB curve tracks the right tail well. For the weakly heterogeneous case, the mapping via R_eff^0 = -ln(1-x_r*)/x_r* is not fitted—it comes from the bond-percolation mean, so the distribution comparison is a real test, and it passes.\n\nThe soft spots are the usual suspect. The stress-test note is right: the transmissibility T only fixes the marginal probability of transmission along each edge. It does not fix the joint distribution of transmissions from one node, which depends on the whole infection waiting-time CDF through E_τ[Binomial(k, F_i(τ))]. Two waiting-time shapes with identical T can have different offspring distributions, so the claim that arbitrary WT statistics reduce to a single T for the full distribution is not exact. The paper's numerical evidence covers a modest range of gamma shapes plus one log-normal case; the collapse is likely approximate, not universal. The authors should either provide a derivation or soften the claim and test the collapse with distributions that share T but differ in shape.\n\nThe Hamsterster analysis is the weaker leg. There, R_net^0 is defined as the value that reproduces the simulated mean outbreak in a Markovian simulation, and then the full distribution is compared. That's less circular than fitting the whole distribution, but it is not an independent prediction. The right tail comparison still means something, but the reader should know the mean is used as input.\n\nOverall, the paper deserves a serious referee. The central idea is plausible, the simulations are reproducible, and the practical recipe could be widely used. But the theoretical overstatement about exact equivalence is a load-bearing flaw in the argument as written, and the rebuttal needs to either prove or significantly weaken the claim. I'd send it to review, and ask for the same-T/different-shape test plus a derivation or an explicit approximate-error statement.","headline":"A useful empirical mapping for non-Markovian outbreak tails, but the 'single transmissibility' equivalence is oversold; the paper's own simulations only show approximate collapse.","tokens_in":11916,"tokens_out":2514,"would_cite":true,"duration_ms":31794,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single per-edge transmission probability governs the full outbreak-size distribution of non-Markovian SIR epidemics on networks, including rare large outbreaks.","keywords":["non-Markovian epidemics","outbreak-size distribution","extreme outbreak risk","edge transmissibility","bond percolation","large deviations WKB","complex networks","waiting-time distributions"],"falsifier":"Run stochastic SIR simulations on the same weakly heterogeneous network with two different waiting-time distributions—for example gamma with $\\alpha_\\mathrm{inf}<1$ and a log-normal distribution—tuned so that Eq. (2) gives the same $T$ and hence the same deterministic final size $x_r^*$; if the normalized outbreak-size distributions do not collapse onto the WKB curve at $R_0^{\\mathrm{eff}}$ beyond finite-size fluctuations, the universal-collapse claim is falsified.","tokens_in":10900,"feed_emoji":"🦠","tokens_out":9758,"duration_ms":80623,"temperature":0.7,"pith_summary":"Extreme epidemic risk lives in the right tail of the outbreak-size distribution, but that tail is hard to compute when infection and recovery are not memoryless. The paper's central claim is that for SIR dynamics on networks, arbitrary infection and recovery waiting-time distributions can be folded into a single per-edge transmissibility $T$, which turns the non-Markovian process into an effective Markovian one with the same full outbreak-size statistics. On weakly heterogeneous networks this gives a universal well-mixed semiclassical curve parameterized by the effective reproduction number $R_0^{\\mathrm{eff}} = -\\ln(1-x_r^*)/x_r^*$; on strongly heterogeneous and empirical networks it gives Markovian dynamics on the same network at $R_0^{\\mathrm{net}} = T/(1-T)\\,\\lambda_c^{-1}$. If the claim is right, a measured generation interval and contact pattern are enough to predict the probability of an unusually large outbreak, without simulating the full non-Markovian dynamics.","feed_headline":"A single transmissibility predicts extreme outbreak risk on networks","feed_subtitle":"Measured infection and recovery timings fold into one curve, making extreme-outbreak tails predictable.","key_machinery":"The load-bearing object is the per-edge transmissibility $T = \\int_0^\\infty f_i(t)\\Phi_r(t)\\,dt$, the probability that infection along one susceptible–infected edge fires before recovery. For gamma-distributed infection and recovery it takes the closed form $T = I_z(\\alpha_\\mathrm{inf}, \\alpha_\\mathrm{rec})$ with $z = \\alpha_\\mathrm{inf}\\beta/(\\alpha_\\mathrm{inf}\\beta + \\alpha_\\mathrm{rec}\\gamma)$. This one number converts the final state into a bond-percolation configuration, and the mapping then works in two directions: invert the percolation final size $x_r^*$ to get $R_0^{\\mathrm{eff}}$ for the well-mixed WKB large-deviation action, or invert the Markovian relation to get $R_0^{\\mathrm{net}}$ for Markovian dynamics on the same network. The WKB action and its Gaussian width, taken from the well-mixed theory, supply the shape of the universal curve.","core_discovery":"On its own terms, the paper demonstrates that the final state of a non-Markovian SIR epidemic is a bond-percolation configuration with per-edge occupation probability $T$, and that this single number controls not just the mean outbreak fraction but the whole distribution of outcomes. For weakly heterogeneous degree distributions, inverting the deterministic final-size relation gives $R_0^{\\mathrm{eff}}$, and the rescaled outbreak-size distributions measured across regular, Erdős–Rényi, and gamma-distributed networks with gamma and log-normal waiting times collapse onto the well-mixed WKB prediction. For strongly heterogeneous or correlated networks, the same transmissibility defines $R_0^{\\mathrm{net}}$, and Markovian simulations on the actual network reproduce the non-Markovian conditional mean, standard deviation, and full distribution, including on the empirical Hamsterster network with assortative mixing. The qualitative asymmetry between infection- and recovery-time shapes follows directly from $T$: broader-than-exponential infections enlarge $T$ and drive larger outbreaks, while broader-than-exponential recoveries shrink it.","pith_inferences":["Inference: the reduction predicts that any two waiting-time distributions producing the same transmissibility $T$ and the same deterministic final size should yield identical normalized outbreak-size distributions; running this matched-transmissibility comparison for Weibull versus gamma or log-normal timings is a direct test the paper does not perform.","Inference: the breakdown of the well-mixed collapse around degree coefficient of variation 0.5 suggests a crossover regime rather than a sharp threshold; a natural extension would be to add a degree-dispersion correction to the WKB action and check whether it interpolates between the two regimes.","Inference: for real-time epidemic risk assessment, the mapping implies that empirically measured contact patterns and inter-event-time distributions can be converted directly into tail-risk probabilities, such as the probability of exceeding hospital capacity, which is the practical payoff the paper points to without developing a forecasting procedure."],"forward_implications":["Given measured generation intervals and infectious-period distributions, practitioners can compute $T$ and then $R_0^{\\mathrm{eff}}$ or $R_0^{\\mathrm{net}}$ to obtain the full outbreak-size distribution without simulating non-Markovian network dynamics.","On weakly heterogeneous networks the right tail is analytic: the well-mixed WKB action gives the probability of an outbreak of any fractional size, including events far above the mean.","On strongly heterogeneous or correlated networks, the equivalence reduces the problem to Markovian simulation on the existing network, which is substantially cheaper than simulating arbitrary waiting-time clocks.","Changing the shape of the infection-time distribution has opposite and stronger effects than changing the recovery-time distribution, and both are captured quantitatively by the transmissibility formula.","Extreme-outbreak risk quantified this way is sizable: in the paper's examples, about one-fifth of the probability mass of extensive outbreaks lies more than 20% above the mean, on both an Erdős–Rényi and the empirical Hamsterster network."],"supporting_citations":[{"why":"Defines the per-edge transmissibility and the bond-percolation mapping of SIR final states on networks; this is the object the entire reduction rests on.","marker":"[16]"},{"why":"Supplies the well-mixed WKB large-deviation action and Gaussian width that form the universal curve used for weakly heterogeneous networks.","marker":"[10]"},{"why":"Establishes the general epidemic process as dynamical percolation, justifying the treatment of the final outbreak as a connected cluster of occupied bonds.","marker":"[15]"},{"why":"Provides the bond-percolation final-size self-consistency equations from which the paper extracts $x_r^*$ and $R_0^{\\mathrm{eff}}$.","marker":"[17]"},{"why":"Gives the degree-distribution generating functions used in the percolation equations for random networks.","marker":"[41]"},{"why":"Defines the Erdős–Rényi network ensemble used as one of the weakly heterogeneous test cases for the universal collapse.","marker":"[42]"},{"why":"Provides the empirical Hamsterster network data used to test the mapping on a strongly correlated real-world topology.","marker":"[38]"}],"fun_headline_variants":["One transmissibility captures extreme epidemic risk","Non-Markovian outbreaks shrink to one percolation parameter","Extreme outbreak tails collapse onto a single curve","Predicting extreme outbreaks from one measured number","Universal curve predicts extreme outbreak size"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For weakly heterogeneous networks (degree scatter up to about half the mean degree), the paper assumes that the whole rescaled outbreak-size distribution is fixed by one effective reproduction number $R_0^{\\mathrm{eff}}$ and is insensitive to the detailed shape of the waiting-time distributions and the degree distribution; this is supported by simulation collapse rather than derived from the underlying dynamics.","fun_headline_variants_meta":{"raw":{"variants":["One transmissibility captures extreme epidemic risk","Non-Markovian outbreaks shrink to one percolation parameter","Extreme outbreak tails collapse onto a single curve","Predicting extreme outbreaks from one measured number","Universal curve predicts extreme outbreak size"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000297,"raw_usage":{"total_tokens":1700,"prompt_tokens":903,"completion_tokens":797,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":728}},"tokens_in":519,"tokens_out":797,"duration_ms":6885,"temperature":1.0,"reasoning_tokens":728,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:25:52.041652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run stochastic SIR simulations on the same weakly heterogeneous network with two different waiting-time distributions—for example gamma with $\\alpha_\\mathrm{inf}<1$ and a log-normal distribution—tuned so that Eq. (2) gives the same $T$ and hence the same deterministic final size $x_r^*$; if the normalized outbreak-size distributions do not collapse onto the WKB curve at $R_0^{\\mathrm{eff}}$ beyond finite-size fluctuations, the universal-collapse claim is falsified.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the per-edge transmissibility and the bond-percolation mapping of SIR final states on networks; this is the object the entire reduction rests on."},{"cited_title":"Hindes, M","cited_arxiv_id":null,"evidence_quote":"Supplies the well-mixed WKB large-deviation action and Gaussian width that form the universal curve used for weakly heterogeneous networks."},{"cited_title":"Grassberger, On the critical behavior of the gen- eral epidemic process and dynamical percolation, Math","cited_arxiv_id":null,"evidence_quote":"Establishes the general epidemic process as dynamical percolation, justifying the treatment of the final outbreak as a connected cluster of occupied bonds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the bond-percolation final-size self-consistency equations from which the paper extracts $x_r^*$ and $R_0^{\\mathrm{eff}}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the degree-distribution generating functions used in the percolation equations for random networks."},{"cited_title":"Erd˝ os and A","cited_arxiv_id":null,"evidence_quote":"Defines the Erdős–Rényi network ensemble used as one of the weakly heterogeneous test cases for the universal collapse."},{"cited_title":"Kunegis, KONECT – The Koblenz Network Collec- tion, inProc","cited_arxiv_id":null,"evidence_quote":"Provides the empirical Hamsterster network data used to test the mapping on a strongly correlated real-world topology."}],"review_version":2}