{"id":"f07c87dd-9021-4cc1-b589-ab9f461b36cd","arxiv_id":"2507.01640","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"In a two-layer adaptive oscillator network with pairwise and triadic couplings, phase-lag suppresses tiered and explosive synchronization when pairwise coupling is varied, but promotes discontinuous synchronization when triadic coupling is varied.","lead":"This paper studies two groups of coupled oscillators with adaptive coupling strengths and both pairwise and three-body connections, and asks how a phase delay changes the way they synchronize. It reduces the system to two equations and maps where synchronization appears smoothly, in steps, or abruptly.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper never specifies A and B in the adaptation functions; all figures imply A=2, B=1, so the central claim about beta's dual role is only demonstrated for one unstated parameter choice.","rationale":"The reader's weakest_assumption focused on the Ott-Antonsen ansatz requiring a Lorentzian distribution and symmetry assumptions. Those are explicitly stated assumptions of the model, and the paper's own simulations use Lorentzian frequencies, so they do not threaten the internal consistency of the argument. A more load-bearing concern is that the adaptation functions contain constants A and B that are never specified, even though the quantitative and qualitative results depend on them. The pitchfork location in Fig. 1(a) forces A=2,B=1, but the paper never says so, and the text highlights the A=0 case as 'well-known' without acknowledging that A=0 would qualitatively change the bifurcation structure. This is an under-specification of the model, not a mathematical error, and it is addressable by stating the parameter values and checking sensitivity. Therefore the conditional verdict remains appropriate: the results are plausible but not fully reproducible until A and B are reported and the robustness of the qualitative claims to their values is demonstrated. My recommended verdict is UNCHANGED (still CONDITIONAL), matching the reader's assessment but with a different emphasis on the weakest point.","tokens_in":18771,"tokens_out":12670,"duration_ms":126543,"concrete_test":"Recompute the bifurcation diagrams of Eq. (13) for the exact parameter values of Fig. 1 and Fig. 2 (beta=0,0.4,0.8,1.2; K2=5 and K1=1.2, respectively) with A=2,B=1, then repeat with A=1,B=1 and A=0.1,B=1. Record the saddle-node and pitchfork locations, the hysteresis width, and the existence of the weak-synchronization branch for each A. If changing A eliminates the pitchfork bifurcation at K1=1, reverses the direction of hysteresis shrinkage with beta, or removes the dual-role effect, the central claim is contingent on the unspecified A,B. Also ask the authors to report the A and B values used in all runs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The adaptation functions in Eq. (13) are fp,l=(A+B r_other)^(p_l) and fh,l=(A+B r_other)^(h_l) with A,B in R+, but the paper never assigns numerical values to A and B. The threshold formula (17) for the incoherent state, K1* = 2 Delta / (A^p cos beta), depends explicitly on A. The pitchfork PB1 at K1=1 in Fig. 1(a) for beta=0, p=1, Delta=1 implies A=2 (since 2/A=1), and the text says A=0,B=1 is the 'well-known power-law form' while the figures clearly use A>0. If A were 0, the linear coupling near r=0 would vanish (the term is O(r1 r2^p)), so the incoherent state would remain linearly stable for all K1 and the pitchfork bifurcations shown in Figs. 1 and 5 would not exist; synchronization would arise only via saddlenode bifurcations at finite amplitude. Thus the claimed dual role of beta—inhibiting tiered/explosive transitions for K1 variation and promoting discontinuous transitions for K2 variation—is only established for the unstated choice A=2, B=1, and the counterintuitive effects of exponents p and h may be specific to that choice. Because A and B are not reported, the model is not fully defined and the results cannot be reproduced or checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a two-layer adaptive network of Sakaguchi-Kuramoto oscillators with pairwise and triadic interactions, where the coupling strengths are modulated by power-law functions of the other layer's order parameter. Using the Ott-Antonsen ansatz for Lorentzian frequency distributions, the authors derive a two-dimensional reduced system for the order-parameter amplitudes, Eq. (13), and analyze its bifurcations. They report that the phase-frustration parameter β suppresses tiered and explosive transitions when the pairwise coupling K1 is varied with K2 fixed, but promotes discontinuous transitions when the higher-order coupling K2 is varied with K1 fixed. They further identify opposite effects of the adaptation exponents p and h on the width of the hysteresis. Numerical simulations of the full N-dimensional system are superimposed on the analytical bifurcation diagrams and are claimed to agree.","tokens_in":19063,"tokens_out":10056,"duration_ms":105272,"significance":"The theoretical reduction is the main contribution: starting from the microscale equations, the authors obtain a closed low-dimensional model in a transparent way, and the bifurcation analysis is extensive and internally consistent. The reported numerical data do visually track the stable and unstable branches obtained from the reduced model, which strengthens confidence in the reduction. If the results survive a proper specification of all model parameters, they would provide a useful map of synchronization-transition types in adaptive higher-order multilayer networks and extend recent findings on phase frustration in such systems. However, the omission of the adaptation constants A and B, and the lack of a quantitative error assessment, currently limit the reproducibility and generality of the central claims.","major_comments":[{"comment":"The adaptation constants A and B are never assigned numerical values, even though every quantitative result in the manuscript depends on them. Equation (17) gives the incoherence threshold K1* = 2Δ/(A^p cosβ); the reported pitchfork PB1 at K1 = 1 for p = 1, Δ = 1, β = 0 implies A = 2, and comparison with all figures implies B = 1, but these values are stated nowhere. If A were 0, the case the text calls the 'well-known power-law form' in Section II, the linear term in Eq. (13) would vanish and the incoherent state would remain linearly stable for all K1, so the pitchfork scenario that underlies the tiered and explosive transitions in Figs. 1 and 5 would not exist. The central claims about the dual role of β and the effects of p and h are therefore only demonstrated for one unstated parameter choice. Please state the values used and test the robustness of the reported scenarios to variation of A and B.","section":"Section II, Eq. (13), and throughout Section IV"},{"comment":"The numerical validation of the explosive and tiered routes depends on what the text calls 'proper choice of initial condition,' but the manuscript never specifies which initial conditions produce the dashed and dash-dotted curves in Figs. 1, 2, 6, 8, 11, and 13, nor how the forward/backward sweeps were initialized in those cases. Without this information, the numerical results are not reproducible and the claimed agreement between the full N-dimensional system and the reduced model cannot be independently checked.","section":"Section IV, 'Synchronization profile' paragraphs"},{"comment":"The agreement between the N-dimensional simulations and the reduced-order model is presented only through visual superposition of data points on bifurcation curves; no error bars, ensemble statistics, or quantitative mismatch measures are provided. Since the paper's central validation claim is that the reduced model faithfully reproduces the full system, I ask for a quantitative assessment, for example the rms deviation of the numerical order parameter from the analytically stable branch over a range of K1 or K2.","section":"Section IV, first paragraph"},{"comment":"The assumption p1=p2, h1=h2, and β1=β2, stated in Section IV, makes the two layers dynamically identical, so the 'multilayer' aspect enters only through the symmetric cross-adaptation. The concluding claim in Section V that the counterintuitive exponent effects arise 'due to the multilayer configuration' is not supported by a comparison with a single-layer version or with asymmetric layer parameters; a brief numerical test of at least one asymmetric case would substantiate that attribution.","section":"Section IV, symmetry assumption; Section V"}],"minor_comments":[{"comment":"In both the abstract and the conclusion, 'tired' should read 'tiered' in the phrase about inhibiting tiered transitions.","section":"Abstract and Section V"},{"comment":"The panels in Figs. 1, 2, 6, 8, 11, and 13 are not explicitly associated with the parameter values in the captions; please add the parameter values to the panel labels or caption for readability.","section":"Figure captions"},{"comment":"Equation (17) should specify the range of β for which the condition is meaningful; for cosβ_l ≤ 0 the incoherent state is always stable, and the inequality with a negative threshold is trivially satisfied only for negative K1.","section":"Eq. (17)"},{"comment":"The Jacobian derivative ∂G1/∂r2 contains factors B p1 (A+B r2)^{p1-1}; for negative or non-integer exponents, the domain of definition requires A+B r2 > 0. Since A is positive but not specified, the paper should state the admissible parameter ranges used in the bifurcation analysis.","section":"Eq. (16)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid Ott-Antonsen reduction study, but the unstated values of A and B are a serious reproducibility issue; the authors should also clarify how this work differs from Ref. [69], which already treats adaptive higher-order multilayer networks without phase frustration. If the authors supply the missing parameter values, specify the initial conditions for the different numerical routes, and provide a sensitivity check, the paper could become acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is a solid extension of the same group's earlier work on adaptive multilayer networks with higher-order interactions, using the Ott-Antonsen reduction to get a two-dimensional order-parameter model. The central claim is that phase-lag inhibits tiered and explosive transitions when K2 is fixed and K1 varies, but promotes discontinuous transitions when K1 is fixed and K2 varies. That claim is plausible and is backed by systematic bifurcation analysis of the reduced model, with finite-N simulations that visually track the stable branches. The two-parameter diagrams in K1-p, K1-h, K2-p, and K2-h are new and do a clean job of organizing the regimes. I also credit them for being honest about initial-condition dependence and for showing basins of attraction.\n\nThe soft spots are real but fixable. Most importantly, the paper never gives numerical values for A and B in the adaptation functions; all figures imply A=2, B=1, but that is never stated. This matters more than a typo: the incoherent-state threshold in Eq. (17) depends on A, and if A were 0 the incoherent state would remain linearly stable for all K1, so the pitchfork bifurcations and the claimed dual role of beta are only demonstrated for that one unstated parameter choice. As written, the model is not fully defined and the results are not reproducible. Second, the novelty is modest: the dual role of beta already appears in refs. [60,62], and the new contribution is mostly the exponent effects and the parameter-space maps. The authors should demarcate what is genuinely new. Third, the finite-N simulations come with no error bars or ensemble statistics; the agreement is convincing visually, but a few repeated-run averages would make it quantitative.\n\nTo be clear, the circularity concern does not land: the reduced model is derived from the original equations under explicit assumptions (Lorentzian frequencies, N to infinity, OA ansatz), and the transitions are read off from bifurcation analysis, not fitted to target data. The central argument holds up under the stated assumptions.\n\nThis paper deserves a serious referee. The missing A and B values, the parameter table, and a sharper separation from prior work are all addressable in revision. I would send it out, and if the authors fix the reproducibility issue, it becomes a useful contribution for the synchronization subfield.","headline":"A competent OA-based extension of this group's adaptive multilayer program: the beta dual-role claim is plausible, but the paper never states the A,B values it uses, and the whole bifurcation story is only shown for A=2, B=1.","tokens_in":19612,"tokens_out":1930,"would_cite":false,"duration_ms":22071,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","34D06","37G10","05C82"],"pacs":["05.45.Xt","89.75.Fb"],"model":"deepseek-v4-flash","headline":"The phase-frustration parameter $\\beta$ plays opposite roles in a two-layer adaptive oscillator network: it suppresses abrupt synchronization transitions when pairwise coupling is scanned, and promotes them when triadic coupling is scanned.","keywords":["synchronization transition","multilayer networks","higher-order interactions","phase frustration","Ott-Antonsen ansatz","explosive synchronization","adaptive coupling","bifurcation analysis"],"falsifier":"Simulate the full $N$-oscillator system with a Gaussian or bimodal frequency distribution and compare the hysteresis width and the locations of the saddle-node bifurcations with the predictions of Eq. (13); if the dual role of $\\beta$ disappears or the reduced model fails to track the simulations, the central claim is falsified. A second check is to measure order-parameter distributions over random initial conditions in the multistable regime and compare the observed fractions with the basin fractions of the reduced model.","tokens_in":18542,"feed_emoji":"⚡","tokens_out":5370,"duration_ms":56150,"temperature":0.7,"pith_summary":"The paper studies how phase frustration (a phase lag in the coupling) reshapes the routes to synchronization in an adaptive two-layer network of phase oscillators with both pairwise and three-body interactions. It claims that the phase-lag parameter $\\beta$ has a dual, direction-dependent effect: for fixed triadic coupling and varying pairwise coupling $K_1$, increasing $\\beta$ suppresses tiered and explosive transitions and shrinks hysteresis, whereas for fixed pairwise coupling and varying triadic coupling $K_2$, increasing $\\beta$ promotes discontinuous transitions and bistability. The adaptation exponents act in opposite directions: the pairwise exponent $p$ narrows hysteresis while the higher-order exponent $h$ widens it. These conclusions rest on a two-dimensional order-parameter model derived from the Ott-Antonsen ansatz, whose bifurcation analysis matches direct simulations of the full $N$-oscillator system. If correct, the results give controllability principles: frustration and adaptation exponents are knobs that can suppress or encourage abrupt synchronization.","feed_headline":"Phase lag flips explosive sync on and off in two-layer nets","feed_subtitle":"Frustration shrinks hysteresis when pairwise coupling is scanned, but creates it when triadic coupling is scanned.","key_machinery":"The load-bearing object is the two-dimensional reduced order-parameter system for $\\dot{r}_1$ and $\\dot{r}_2$ in Eq. (13), obtained by applying the Ott-Antonsen ansatz to the continuity equation in the thermodynamic limit. The system encodes the global order parameters of the two layers with power-law cross-adaptation functions $(A+B r_{\\text{other}})^{p}$ and $(A+B r_{\\text{other}})^{h}$ multiplying the pairwise and triadic coupling terms. Its fixed points and their saddle-node and pitchfork bifurcations, traced by numerical continuation, explain the full system's synchronized, weakly synchronized, and incoherent states, and yield the stability condition for incoherence $K_1<\\min\\{2\\Delta/(A^p \\cos\\beta)\\}$.","core_discovery":"The central discovery is that phase frustration reverses its qualitative effect depending on which coupling is varied. When $K_1$ is varied at fixed $K_2$, increasing $\\beta$ shifts all bifurcations to larger $K_1$, shrinks the interval where a weak partially synchronized branch exists between two saddle-node and pitchfork bifurcations, and eventually eliminates the tiered transition, leaving only explosive synchronization. When $K_2$ is varied at fixed $K_1$, the opposite happens: for small $\\beta$ the transition is continuous, but past a critical $\\beta$ the incoherent state regains stability and saddle-node bifurcations create coexisting coherent and incoherent branches, so frustration induces explosive transitions and multistability. The paper further establishes that the pairwise adaptation exponent $p$ reduces hysteresis width while the higher-order exponent $h$ promotes bistability, even though increasing these exponents reduces the corresponding effective coupling strengths, a counterintuitive effect attributed to the multilayer cross-adaptation structure. All these transitions are reproduced quantitatively by the Ott-Antonsen-derived system of two ODEs for the layer order parameters $r_1$ and $r_2$, validated against simulations with $N=10000$ oscillators per layer.","pith_inferences":["If the reduced model is structurally stable, its predictions should survive moderate deviations from the Lorentzian frequency distribution, but the bifurcation points will shift quantitatively; a systematic comparison with Gaussian or bimodal distributions would test the robustness of the dual role of $\\beta$.","The counterintuitive exponent effects suggest a design principle for engineered oscillator networks: adaptation exponents and phase lag can tune hysteresis without changing nominal coupling strengths, which may matter for power-grid or neuronal models where abrupt synchronization is undesirable.","Because the two layers are coupled only through order-parameter adaptation, varying the baseline $A$ and feedback strength $B$ could produce transition types not classified in the current four-regime diagrams, a directly testable extension of the parameter-space maps.","A concrete experimental test: in the multistable regime, measuring the order parameters after many random initial conditions should yield a trimodal distribution matching the basin fractions from the reduced model, which could be checked with electronic or optoelectronic oscillator arrays."],"forward_implications":["The reduced two-dimensional model in Eq. (13) can locate synchronization-transition boundaries and hysteresis widths at negligible computational cost, replacing direct simulation of the full $N$-oscillator system.","Phase frustration can act as a control parameter: in the $K_1$-scan regime it suppresses abrupt transitions, while in the $K_2$-scan regime it creates them, so the same network can be pushed between continuous and explosive synchronization by choosing which coupling is tuned.","The pairwise adaptation exponent $p$ is a hysteresis-reducing knob even when higher-order coupling dominates, and the higher-order exponent $h$ is a bistability-promoting knob even when pairwise coupling dominates.","The coexistence of incoherent, weakly coherent, and strongly coherent states means initial conditions can select the final synchronization level, and the paper's basin diagrams map which initializations reach which state.","As the authors state, the framework generalizes to more than two layers and to interactions beyond triadic simplices."],"supporting_citations":[{"why":"Supplies the Ott-Antonsen ansatz that reduces the continuity equation to the low-dimensional order-parameter dynamics.","marker":"[70]"},{"why":"Provides the Sakaguchi-Kuramoto phase-frustrated oscillator model that the network equations generalize.","marker":"[59]"},{"why":"Introduces order-parameter-based adaptive coupling, the cross-adaptation scheme used here.","marker":"[52]"},{"why":"Demonstrates tiered synchronization in adaptive higher-order systems, a phenomenon the paper reproduces and modifies with phase lag.","marker":"[56]"},{"why":"Shows double explosive synchronization in adaptive higher-order networks, giving the explosive-transition context.","marker":"[57]"},{"why":"Establishes that phase lag has opposite effects on transitions with variation of pairwise versus higher-order coupling, the result this paper extends to multilayer adaptive networks.","marker":"[62]"},{"why":"Studies adaptation in higher-order multilayer networks and provides the baseline for the synchronization scenarios explored here.","marker":"[69]"}],"fun_headline_variants":["Phase frustration: a toggle for explosive sync in multilayer nets","Phase lag decides if sync jumps or slides in adaptive networks","Frustration flips sync transition: from tiered to explosive","Higher-order coupling changes how phase lag shapes sync","Phase lag: one parameter, two faces for multilayer sync"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes the intrinsic frequencies are exactly Lorentzian and the phase density stays on the Ott-Antonsen manifold, with identical parameters across the two layers; if the frequency distribution departs from Lorentzian, the reduced two-ODE model is no longer exact and the reported bifurcation scenarios are not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Phase frustration: a toggle for explosive sync in multilayer nets","Phase lag decides if sync jumps or slides in adaptive networks","Frustration flips sync transition: from tiered to explosive","Higher-order coupling changes how phase lag shapes sync","Phase lag: one parameter, two faces for multilayer sync"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000514,"raw_usage":{"total_tokens":2573,"prompt_tokens":1095,"completion_tokens":1478,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":711,"completion_tokens_details":{"reasoning_tokens":1397}},"tokens_in":711,"tokens_out":1478,"duration_ms":12027,"temperature":1.0,"reasoning_tokens":1397,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:47:06.528517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the full $N$-oscillator system with a Gaussian or bimodal frequency distribution and compare the hysteresis width and the locations of the saddle-node bifurcations with the predictions of Eq. (13); if the dual role of $\\beta$ disappears or the reduced model fails to track the simulations, the central claim is falsified. A second check is to measure order-parameter distributions over random initial conditions in the multistable regime and compare the observed fractions with the basin fractions of the reduced model.","supporting_citations":[{"cited_title":"Dutta, P","cited_arxiv_id":null,"evidence_quote":"Supplies the Ott-Antonsen ansatz that reduces the continuity equation to the low-dimensional order-parameter dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Sakaguchi-Kuramoto phase-frustrated oscillator model that the network equations generalize."},{"cited_title":"Boccaletti, A","cited_arxiv_id":null,"evidence_quote":"Introduces order-parameter-based adaptive coupling, the cross-adaptation scheme used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates tiered synchronization in adaptive higher-order systems, a phenomenon the paper reproduces and modifies with phase lag."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows double explosive synchronization in adaptive higher-order networks, giving the explosive-transition context."},{"cited_title":"G´ omez-Gardenes, Y","cited_arxiv_id":null,"evidence_quote":"Establishes that phase lag has opposite effects on transitions with variation of pairwise versus higher-order coupling, the result this paper extends to multilayer adaptive networks."},{"cited_title":"Rajwani, A","cited_arxiv_id":null,"evidence_quote":"Studies adaptation in higher-order multilayer networks and provides the baseline for the synchronization scenarios explored here."}],"review_version":1}