{"id":"b003172e-8c41-45dc-93f8-7bc7b339ed77","arxiv_id":"2606.30069","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A six-parameter interaction model describes the extinction-persistence phase boundary of the Watts-threshold SIS model on ER and BA networks; the boundary is invariant across topologies, transitions are sharp, and adoption threshold dominates.","lead":"The paper maps the extinction-persistence phase boundary for a stochastic Watts-threshold SIS contagion model on Erdos-Renyi and Barabasi-Albert networks using over 180,000 Monte Carlo simulations and logistic regression. A smart generalist might read it to see how reinforcement thresholds affect whether complex contagions with recovery can sustain on networks.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Adaptive Delaunay sampling + weighted logistic regression may bias the reconstructed (β, θ, d) boundary","rationale":"The reader's weakest assumption correctly isolates the methodological step whose failure would invalidate both the six-parameter fit and the invariance claim. No other internal inconsistency is evident from the abstract description of the procedure.","tokens_in":1673,"tokens_out":358,"duration_ms":41581,"concrete_test":"Re-run the boundary reconstruction on a fixed 2-D slice (e.g., d = 5, β ∈ [0.01,0.2], θ ∈ [0.1,0.9]) using both the adaptive Delaunay procedure and a uniform 50×50 grid with the same total trial budget; compare the resulting logistic surface parameters and 10–90 % band width. A shift > 2× the reported uncertainty in the fitted coefficients would indicate bias from adaptivity.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that the six-parameter interaction model accurately and invariantly describes the true continuous extinction-persistence surface. This rests on the assumption that adaptive Delaunay triangulation in the three-dimensional parameter space, followed by weighted logistic regression on the Monte Carlo outcomes, yields an unbiased estimate of the probability surface. Adaptive refinement can preferentially allocate trials near an evolving estimate of the boundary, potentially under-sampling the interior or exterior regions and introducing systematic error in the logistic fit, especially if the initial coarse grid or the adaptation criterion (e.g., uncertainty or gradient) is not validated against uniform or stratified sampling. The reported sharpness (Δθ ≈ 0.005–0.008) and topology invariance could therefore be sensitive to this sampling choice rather than intrinsic to the model.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to reconstruct the extinction-persistence phase boundary of a stochastic Watts-threshold SIS model on Erdős-Rényi and Barabási-Albert networks in the three-dimensional parameter space (β, θ, d) using adaptive Delaunay sampling and weighted logistic regression on 180,000 Monte Carlo trials. It reports that a six-parameter interaction model fits the boundary invariantly across topologies, with sharp transitions (Δθ ≈ 0.005–0.008) dominated by the threshold θ.","tokens_in":1848,"tokens_out":406,"duration_ms":45206,"significance":"Should the numerical reconstruction prove robust, the work would supply a quantitative benchmark for complex-contagion SIS dynamics on networks, filling a gap left by monotone threshold models. The volume of simulations and the cross-topology invariance constitute strengths that could make the six-parameter form a useful reference, provided the adaptive sampling does not introduce systematic bias.","major_comments":[{"comment":"The central claim that the six-parameter model accurately describes the phase boundary rests on the adaptive Delaunay-based sampling and weighted logistic regression producing an unbiased estimate. However, no comparison to uniform sampling or stratified designs is reported to confirm that the adaptive refinement does not bias the reconstructed surface, particularly near the boundary where most trials are allocated.","section":"Methods (adaptive sampling)"},{"comment":"The abstract and results report 180k trials but provide no error bars on the fitted parameters, no validation against known limits (such as the standard SIS threshold when θ=1), and no details on network sizes or stopping criteria; these omissions are load-bearing for assessing the reported sharpness and invariance of the model.","section":"Results (simulation details)"}],"minor_comments":[{"comment":"The definition of the infectious duration d and its relation to the recovery process should be stated explicitly in the model section for clarity.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive comments, which highlight important aspects of the methods and results that require clarification. We address each major comment below and will revise the manuscript accordingly.","responses":[{"response":"We agree that the absence of a direct comparison leaves open the possibility of bias in the adaptive sampling procedure. In the revised manuscript we will add a supplementary analysis that repeats the boundary reconstruction on a reduced three-dimensional grid using uniform sampling (approximately 25,000 trials) and compares the resulting six-parameter fits to those obtained with the adaptive Delaunay scheme. Any statistically significant differences will be quantified and discussed; if differences are negligible, this will support the robustness of the original surface.","revision_made":"yes","referee_comment":"[Methods (adaptive sampling)] The central claim that the six-parameter model accurately describes the phase boundary rests on the adaptive Delaunay-based sampling and weighted logistic regression producing an unbiased estimate. However, no comparison to uniform sampling or stratified designs is reported to confirm that the adaptive refinement does not bias the reconstructed surface, particularly near the boundary where most trials are allocated."},{"response":"These omissions are acknowledged. The revised manuscript will include: (i) 95 % confidence intervals on all six fitted interaction-model parameters obtained from the weighted logistic regression; (ii) an explicit validation subsection in which θ is fixed at 1 and the resulting extinction-persistence boundary is compared with the known SIS threshold on both ER and BA networks; (iii) precise statements of the network sizes (N = 5000 nodes for both topologies) and stopping criteria (each realization runs for a maximum of 10 000 time steps or until extinction, with 200 independent Monte Carlo realizations per sampled point). These additions will be placed in the Methods and Results sections.","revision_made":"yes","referee_comment":"[Results (simulation details)] The abstract and results report 180k trials but provide no error bars on the fitted parameters, no validation against known limits (such as the standard SIS threshold when θ=1), and no details on network sizes or stopping criteria; these omissions are load-bearing for assessing the reported sharpness and invariance of the model."}],"tokens_in":1352,"tokens_out":471,"duration_ms":44319,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this paper supplies the first quantitative phase boundary for a stochastic Watts-threshold SIS model on ER and BA networks, using 180k Monte Carlo trials to fit a six-parameter interaction model in (β, θ, d) space.\n\nWhat is new is the joint mapping itself. Monotone threshold models have phase diagrams, but adding SIS-style recovery left the extinction-persistence surface unmapped; the work fills that gap with concrete simulation output and reports that the same functional form works on both topologies, that the threshold dominates, and that the 10-90% band is narrow.\n\nThe scale of the runs and the focus on the dominant parameter are the parts that hold up. Those are useful reference points for anyone modeling complex contagion with recovery.\n\nThe soft spots sit in the reconstruction method. The abstract describes adaptive Delaunay sampling followed by weighted logistic regression but gives no error bars, no comparison to uniform grids, and no checks against known limiting cases. The stress-test point about possible bias from adaptive refinement therefore lands: if the sampling concentrates near an evolving boundary estimate, the fitted surface and its claimed invariance could shift. Without those diagnostics the sharpness numbers (Δ\theta ≈ 0.005-0.008) rest on unverified simulation fidelity.\n\nThis is for network epidemiologists who need a starting quantitative reference for threshold models that include recovery. A reader already working on phase diagrams in contagion processes will extract the parameter ranges and the topology-invariance observation even if they later rerun the fits.\n\nIt deserves a serious referee. The gap is real and the data volume is substantial; the methodological gaps are the sort that review can tighten rather than fatal flaws.","headline":"This gives the first simulated phase diagram for a Watts-threshold SIS model with recovery, but the adaptive sampling and fit need validation before the sharpness and invariance claims can be trusted.","tokens_in":2295,"tokens_out":433,"would_cite":false,"duration_ms":37320,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A six-parameter interaction model captures the phase boundary of the stochastic Watts-threshold SIS model on random networks, invariant across topologies.","keywords":["complex contagion","Watts threshold","SIS model","phase boundary","random networks","epidemic threshold","Monte Carlo simulation","logistic regression"],"falsifier":"New Monte Carlo simulations on held-out parameter combinations or a third network topology that deviate substantially from the predictions of the fitted six-parameter model would show the description does not hold.","tokens_in":2580,"feed_emoji":"","tokens_out":712,"duration_ms":43496,"temperature":0.7,"pith_summary":"This paper reconstructs the extinction-persistence phase boundary for a stochastic Watts-threshold SIS model of complex contagion with recovery, in the joint space of transmission rate, adoption threshold and infectious duration. It shows that the boundary is well described by a six-parameter interaction model whose form does not change between Erdos-Renyi and Barabasi-Albert networks. The adoption threshold dominates feasibility, the transition is sharp, and the work supplies a quantitative reference for the complex-contagion analogue of the classical SIS threshold. A sympathetic reader would care because the result gives concrete conditions under which reinforced adoption persists or dies out on networks.","feed_headline":"Six-parameter model maps complex contagion phase boundary","feed_subtitle":"The extinction-persistence boundary for a Watts-threshold SIS model stays invariant across Erdos-Renyi and Barabasi-Albert networks and is d","key_machinery":"The six-parameter interaction model for the phase boundary in the joint parameter space of transmission rate β, adoption threshold θ and infectious duration d, obtained from adaptive Delaunay-based sampling and weighted logistic regression on Monte Carlo trials.","core_discovery":"The extinction-persistence phase boundary of the stochastic Watts-threshold SIS model is well described by a six-parameter interaction model whose structure is invariant across Erdos-Renyi and Barabasi-Albert networks. The transition is sharp, with the 10-90% extinction-probability band spanning only Δθ ≈ 0.005-0.008, and the adoption threshold is the dominant parameter governing epidemic feasibility, with transmission rate and infectious duration playing secondary and asymmetric roles.","pith_inferences":["If the invariance extends, the same functional form could be tested for predicting boundaries on other random or empirical networks.","The dominance of θ implies that small shifts in the number of required reinforcing neighbors can switch an outbreak from persistence to extinction.","The reported asymmetry between β and d suggests that control measures shortening infectious periods may have different leverage than measures reducing transmission probability."],"forward_implications":["The adoption threshold θ dominates the feasibility of epidemic persistence or extinction.","Transmission rate β and infectious duration d play secondary and asymmetric roles in setting the boundary location.","The phase transition is sharp, with the extinction probability changing rapidly over a small interval Δθ ≈ 0.005-0.008.","The structure of the six-parameter model is the same for both Erdos-Renyi and Barabasi-Albert networks."],"fun_headline_variants":["Six-parameter model maps SIS threshold phase boundary","Adoption threshold dominates complex contagion boundary","Sharp transition in stochastic Watts-threshold SIS","Six-param model invariant on ER and BA networks"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The adaptive Delaunay-based sampling combined with weighted logistic regression on the Monte Carlo trials produces an unbiased reconstruction of the true continuous phase boundary in the (β, θ, d) space.","fun_headline_variants_meta":{"raw":{"variants":["Six-parameter model maps SIS threshold phase boundary","Adoption threshold dominates complex contagion boundary","Sharp transition in stochastic Watts-threshold SIS","Six-param model invariant on ER and BA networks"]},"model":"grok-4.3","cost_usd":0.007244,"raw_usage":{"total_tokens":3340,"prompt_tokens":670,"num_sources_used":0,"completion_tokens":52,"cost_in_usd_ticks":72437000,"prompt_tokens_details":{"text_tokens":670,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2618,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":670,"tokens_out":52,"duration_ms":39129,"temperature":1.0,"reasoning_tokens":2618,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T03:47:29.022241+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"New Monte Carlo simulations on held-out parameter combinations or a third network topology that deviate substantially from the predictions of the fitted six-parameter model would show the description does not hold.","supporting_citations":[],"review_version":1}