{"id":"27c43004-1207-41fd-90ed-9ddf96b5f4bf","arxiv_id":"2607.10171","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"On temporal higher-order networks, immunization shows bistability and discontinuous transitions so effectiveness depends on initial prevalence; High Infection Contribution and prevalence-adaptive egocentric strategies beat standard heuristics.","lead":"Immunizing people on networks where groups form and dissolve over time can produce sudden jumps in disease levels and two possible outcomes that depend on how many people are already infected. The authors give a targeting rule based on each person’s contribution to new infections and show that the best local sampling rule changes as the outbreak grows.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"HIC ranking and HA–PA/EPS–EHS crossovers rest on a homogeneous pre-immunization infection profile that the nonlinear higher-order dynamics themselves make least plausible at high ρ₀.","rationale":"The Reader correctly isolates the homogeneous infection-profile closure as the weakest modeling step supporting the quantitative strategy rankings. That closure is load-bearing for the claim that HIC is superior and for the IC explanation of the HA–PA / EPS–EHS crossovers; the discontinuous transitions and bistability themselves follow from the mean-field fixed-point structure without needing the same approximation and are already corroborated by direct simulation and the SocioPatterns check. Because the paper never reports a non-homogeneous re-ranking, the concern remains open, so the verdict stays CONDITIONAL rather than moving to ACCEPT or REJECT. The proposed test is a single, well-defined numerical check on the paper’s own ensemble that either confirms the ranking is robust or quantifies how much the headline strategy claim softens. No stronger internal inconsistency is evident; the issue is the security of the ranking derivation under the dynamics the paper itself studies.","tokens_in":18810,"tokens_out":968,"duration_ms":9516,"concrete_test":"On the same HOAD ensemble used for Fig. 2, record the true pre-immunization activity-resolved densities i^{t⁻₀}_a from free dynamics at several ρ₀ (e.g. 0.05, 0.13, 0.60). Recompute IC scores with the exact post-immunization moments Θ_m = ∫ a^{(m)}(1−q_a)i^{t⁻₀}_a da (no homogeneous closure), re-rank nodes, and re-measure ω_c for HIC vs TA/HA/PA. If HIC’s advantage over HA/PA shrinks by >10% or the HA–PA crossover ρ₀ moves by more than ~0.1, the load-bearing ranking claim weakens.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper’s strongest claim is that prevalence-dependent discontinuous immunization thresholds appear on temporal higher-order networks and that HIC, with IC(a)=2β₁a¹+3β₂\rho₀(1−ω)a², yields the lowest eradication threshold among the global heuristics (Abstract; §2.2–2.3; Figs. 1–2). The ranking and the equal-threshold comparison that explain the HA–PA and EPS–EHS crossovers are derived by closing the one-step growth Δ\rho⁺₀ under the homogeneous pre-immunization profile i^{t⁻₀}_a ≈ ρ₀ n_a and the post-immunization moment closure Θ_m ≈ (ρ⁺₀/χ) B_m(q) (Methods §4.4–4.5, Eqs. 5, 25–26, 28–30, 32–37). Under the same nonlinear higher-order infection rule that produces bistability, high-activity classes are preferentially infected before t₀, so the true i^{t⁻₀}_a is concentrated on large a rather than proportional to n_a. That concentration is strongest precisely in the high-ρ₀ regime where the paper claims HA (and EHS) overtake PA (and EPS). If the homogeneous closure is inaccurate there, both the absolute HIC ranking and the IC-based explanation of the crossovers can shift, even while the qualitative discontinuous/bistable phenomenology remains intact. Instantaneous thinning and independent a¹, a² draws compound the same closure but are secondary to the profile assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies immunization of a nonlinear higher-order contagion process on temporal hypergraphs generated by the higher-order activity-driven (HOAD) model. Using a mean-field description (Eqs. 1–3), it shows that as the immunized fraction ω varies, the steady-state prevalence exhibits discontinuous (hybrid) transitions and bistability, so the eradication threshold ω_c depends on the pre-immunization prevalence ρ₀—unlike temporal pairwise networks. Motivated by that dependence, the authors derive a High Infection Contribution (HIC) ranking IC(a)=2β₁a⁽¹⁾+3β₂ρ₀(1−ω)a⁽²⁾ that minimizes one-step post-immunization growth under a homogeneous-profile closure, and show that HIC yields the lowest ω_c among the global heuristics tested (TA, HA, PA, R). They further introduce egocentric sampling strategies (EPS, EHS, EBS) based on local pairwise/triadic counts and a two-stage rule (TES) that switches with ρ₀, and reproduce the qualitative phenomenology on an augmented SocioPatterns contact sequence.","tokens_in":19286,"tokens_out":1234,"duration_ms":13029,"significance":"If the results hold, the work supplies a concrete, prevalence-aware immunization theory for temporal higher-order systems and a practical ranking (HIC) that systematically outperforms standard activity heuristics. The early-stage threshold (Eq. 4), the fixed-point stability procedure for finite ρ₀, and the closed-form egocentric nomination intensities (Eqs. 11–12, 49) are carefully derived and track Monte Carlo simulations; the SocioPatterns validation shows the same discontinuous/bistable structure and strategy crossovers. These elements are useful for epidemic and misinformation control when group interactions are both higher-order and time-varying, and they cleanly separate the dynamical novelty (ρ₀-dependent thresholds) from the design of deployable local strategies.","major_comments":[{"comment":"Methods §4.4–4.5 (Eqs. 5, 25–26, 28–30, 32–37): HIC and the equal-threshold IC comparison that explain the HA–PA and EPS–EHS crossovers are derived by closing one-step growth under the homogeneous pre-immunization profile i^{t⁻₀}_a ≈ ρ₀ n_a. Under the same nonlinear higher-order infection rule that produces bistability, high-activity classes are preferentially infected before t₀, so the true profile is concentrated on large a—most strongly at high ρ₀, precisely where HA/EHS are claimed to overtake PA/EPS. The paper should either (i) recompute IC and the crossover loci with the actual pre-immunization fixed-point profile i^{t⁻₀}_a obtained from Eq. (20), or (ii) quantify the ranking error of the homogeneous closure against that profile across the ρ₀ range of Figs. 2–3. Without this check the absolute optimality of HIC and the IC-based explanation of the crossovers remain incompletely supp","section":null},{"comment":"§2.3 and Methods §4.3: Instantaneous thinning of the infected density at t₀ (Eq. 19) is used both for the theoretical thresholds and for the HIC derivation. Real immunization (vaccination, isolation) acts with a delay and does not instantly remove already-infected individuals from the infectious pool. The paper should report at least one delayed-immunization or gradual-rollout protocol (e.g., continuous removal of a fraction of S and I over a finite window after t₀) and show whether the discontinuous transitions, the ρ₀-dependence of ω_c, and the HIC ranking order survive. If they do not, the practical claim that HIC is the preferred strategy needs to be qualified.","section":null}],"minor_comments":[{"comment":"Fig. 1b–c: the hybrid critical scalings |ρ*−ρ*_ωL|∝|ω−ω_L| and |ρ*−ρ*_ωU|∝|ω−ω_U|^{0.5} are stated without error bars or fit ranges; a short table of fitted exponents and residual norms would make the hybrid claim more transparent.","section":null},{"comment":"§2.4 / Methods §4.6: the probe fraction ϕ and window ΔT are free parameters of the egocentric strategies but are not systematically varied in the main figures; a brief sensitivity panel (or SI note) would clarify robustness of the EPS–EHS crossover.","section":null},{"comment":"Methods §4.7: the empirical network is expanded by a factor of 100 to N=15500. The text should state whether the activity-rate ranks (and therefore HIC order) are preserved under this augmentation, or report the same strategy comparison on the original N.","section":null},{"comment":"Notation: ρ₀, ρ^{+}_{0}, ρ̃_M and ω_L / ω_U / ω_c appear with slightly different subscripts across the abstract, §2.1 and Methods; a single consistent glossary would help.","section":null},{"comment":"Discussion: the independence of a⁽¹⁾ and a⁽²⁾ is listed as a limitation; a one-sentence remark on how a modest positive correlation would shift IC weights would be useful for readers applying the method to empirical data.","section":null}],"recommendation":"major_revision","confidential_remarks":"The core dynamical claim (discontinuous/bistable immunization thresholds on temporal higher-order networks) is solid and well supported by theory and simulation. The load-bearing issue is the homogeneous-profile closure used to justify HIC and the strategy crossovers; if the authors supply the requested profile check (or a delayed-immunization robustness test), the paper is close to acceptance. Scope and novelty fit a physics-of-complex-systems / network-science journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing: this paper shows that immunizing on temporal higher-order (HOAD) networks produces hybrid discontinuous transitions and bistability in prevalence as a function of the immunized fraction, so the eradication threshold depends on pre-immunization prevalence ρ₀—unlike the pairwise activity-driven case. That qualitative claim is solid inside their model and is the actual novelty relative to Iacopini-style higher-order contagion and Liu et al. 2014-style activity-driven immunization.\n\nWhat they do well: the mean-field setup (Eqs. 1–3), the early-stage threshold (Eq. 4), and the numerical fixed-point stability analysis for finite ρ₀ are written carefully and track Monte Carlo in Figs. 1–3. The IC score IC(a)=2β₁a⁽¹⁾+3β₂ρ₀(1−ω)a⁽²⁾ is a clean one-step growth argument that consistently ranks HIC above TA/HA/PA/R, and the EPS–EHS crossover with a simple two-stage TES is a practical local-information idea. SocioPatterns (augmented) reproduces the same qualitative picture. Citations are appropriate; circularity is low—thresholds come from the equations and are checked in independent runs.\n\nSoft spots, in proportion: the stress-test lands. HIC and the IC explanation of the HA–PA / EPS–EHS crossovers close under a homogeneous pre-immunization profile i^{t⁻₀}_a ≈ ρ₀ n_a. Under the same nonlinear higher-order rule that creates bistability, infection concentrates on high-activity classes before t₀, most strongly at high ρ₀ where they claim HA/EHS overtake PA/EPS. So the absolute ranking and the IC story of the crossover can shift even if the discontinuous/bistable phenomenology holds. Instantaneous thinning and independent a⁽¹⁾, a⁽²⁾ draws are secondary but real. No code; empirical network is size-augmented. These are modeling closures, not internal contradictions.\n\nWho it is for: people working on higher-order contagion control and temporal hypergraphs. A serious referee should see it. I would engage, cite the bistability/threshold-dependence result, and treat HIC as a useful heuristic pending a check without the homogeneous closure.","headline":"Real contribution on immunization under temporal higher-order contagion; HIC ranking is useful but rests on a homogeneous-infection closure that is weakest exactly where higher-order effects dominate.","tokens_in":19890,"tokens_out":579,"would_cite":true,"duration_ms":5057,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"On temporal higher-order networks, immunization success depends on how large the outbreak already is, and a prevalence-weighted ranking of nodes clears infection more efficiently than standard activity heuristics.","keywords":["higher-order immunization","infection contribution","higher-order contagion","discontinuous phase transitions","egocentric sampling strategies","temporal hypergraphs","activity-driven model","bistability"],"falsifier":"On the same higher-order activity-driven networks, replace the homogeneous pre-immunization infection profile with a strongly activity-biased one (or measure the actual profile from full simulations) and check whether HIC still yields the lowest eradication threshold and whether the HA–PA and EPS–EHS crossovers still occur at the predicted prevalence.","tokens_in":19668,"feed_emoji":"💉","tokens_out":738,"duration_ms":6330,"temperature":0.7,"pith_summary":"This paper studies how to stop contagion when contacts are both group-based and constantly changing. On such temporal higher-order networks the fraction of people you need to immunize is not a single fixed number: as that fraction rises, prevalence can jump discontinuously to extinction, and for a range of fractions both extinction and a large endemic state are possible, so the outcome hinges on the prevalence at the moment of intervention. That dependence is absent from ordinary pairwise temporal networks. From the same mean-field description the authors derive a High Infection Contribution score that weights each person's pairwise and group activity by the current prevalence, and show that ranking people by this score yields the lowest eradication threshold among the global strategies they test. When only local samples of contacts are available, pairwise-based sampling works better early in an outbreak while group-based sampling works better later; a simple two-stage rule that switches between them therefore improves on random immunization. The same qualitative picture appears on a real face-to-face contact dataset.","feed_headline":"Outbreak size decides who to immunize in group networks","feed_subtitle":"A prevalence-weighted ranking beats activity heuristics; local sampling must switch mid-outbreak.","key_machinery":"The High Infection Contribution (HIC) score IC(a) = 2β₁a⁽¹⁾ + 3β₂ρ₀(1−ω)a⁽²⁾, obtained by minimizing the one-step post-immunization rise in prevalence under a homogeneous-infection closure of the higher-order activity-driven mean-field equations; it supplies both the global ranking rule and the diagnostic that explains when pairwise versus higher-order targeting is superior.","core_discovery":"Immunization on temporal higher-order networks produces bistability and discontinuous (hybrid) transitions in steady-state prevalence as the immunized fraction varies, so the immunization threshold itself depends on pre-immunization prevalence. Ranking nodes by the infection-contribution score IC(a) = 2β₁a⁽¹⁾ + 3β₂ρ₀(1−ω)a⁽²⁾ therefore outperforms total-, pairwise- and higher-order-activity heuristics, while among local (egocentric) strategies the better choice itself switches from pairwise to higher-order sampling as prevalence rises.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Outbreak size sets who to immunize in temporal group networks","Bistability makes immunization threshold prevalence-dependent","Infection-contribution ranking beats activity heuristics","Local sampling strategy flips as contagion prevalence rises","Hybrid transitions emerge under immunization of temporal hypergraphs"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The derivation treats infection as evenly spread across activity classes just before immunization; if high-activity people are already far more infected than average, the ranking and the claimed crossovers can change.","fun_headline_variants_meta":{"raw":{"variants":["Outbreak size sets who to immunize in temporal group networks","Bistability makes immunization threshold prevalence-dependent","Infection-contribution ranking beats activity heuristics","Local sampling strategy flips as contagion prevalence rises","Hybrid transitions emerge under immunization of temporal hypergraphs"]},"model":"grok-4.5","effort":"low","cost_usd":0.004378,"raw_usage":{"total_tokens":1283,"prompt_tokens":734,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":43780000,"prompt_tokens_details":{"text_tokens":734,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":477,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":734,"tokens_out":72,"duration_ms":4157,"temperature":1.0,"reasoning_tokens":477,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T13:45:21.573349+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"On the same higher-order activity-driven networks, replace the homogeneous pre-immunization infection profile with a strongly activity-biased one (or measure the actual profile from full simulations) and check whether HIC still yields the lowest eradication threshold and whether the HA–PA and EPS–EHS crossovers still occur at the predicted prevalence.","supporting_citations":[],"review_version":1}