{"id":"d0f825ae-408f-4ecb-9aad-878dd1f386b7","arxiv_id":"2412.01044","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In adaptive simplicial complexes, the mix of cooperative and competitive coupling controls whether synchronization is explosive or continuous, and strong higher-order coupling alone can synchronize the network.","lead":"This paper simulates oscillator networks where connections adapt over time, either strengthening or resisting synchronization, and where groups of three oscillators interact alongside pairs. It finds that these group interactions can trigger or suppress explosive synchronization, and can synchronize a network even with no pair connections at all.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The analytical verification in Sec. IV does not establish the claimed transition-type dichotomy: it replaces local r_i by global R, switches from α=β=R to α=β=1, and infers hysteresis from multiple branches of Eq. (12) without stability analysis.","rationale":"The reader's weakest assumption points at the same general area (mean-field replacement and unspecified Δ), so this is a partial agreement. I sharpen it because the analytical section is the only place the paper claims proof, and it contains a concrete internal inconsistency: cooperative adaptation is defined as α=β=R but verified with α=β=1. The stability gap is also important: Eq. (12) can have multiple roots that are not all stable, so counting branches does not establish hysteresis. None of this by itself disproves the numerical observations in Figs. 1-4; a corrected analysis or direct simulation of Eq. (4) with the true adaptive coefficients might confirm the qualitative dichotomy. But as written, the central claim's strongest support ('analytical results clearly support this') is not secured. I therefore keep the reader's CONDITIONAL verdict rather than escalating to REJECT, because the finite-N simulations are genuine evidence and the model is precisely specified enough to test.","tokens_in":11309,"tokens_out":13916,"duration_ms":129337,"concrete_test":"Recompute the bifurcation diagram of Eq. (4) using the intended adaptive coefficients α=β=R and α=β=1-R (not α=β=1), with the same Lorentzian width used in Sec. IV; for each branch of Eq. (12), compute linear stability from Eq. (11) and overlay stable branches on forward/backward sweeps of Eq. (4) with N=2000. If the α=R cooperative case shows no stable hysteresis loop, or the 1-R competitive case shows a stable high-R branch coexisting with incoherence, then Eq. (12) and the claimed transition-type proof in Sec. IV are incorrect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that cooperative adaptation yields first-order (explosive) synchronization while competitive adaptation yields second-order transitions, with higher-order coupling able to synchronize without pairwise interactions (abstract; Sec. III, Figs. 1-4). The paper repeatedly invokes Sec. IV as proof ('It is proven...' in Sec. III.F; 'analytical results clearly support this' in Sec. III.D). Three concrete gaps make that proof non-load-bearing. First, Eq. (4) is a globally coupled model in which every local r_i is replaced by one global R; no argument or test shows that sparse ER networks with degree-dependent local adaptation obey this mean-field limit. Second, the stated adaptive rule is α=β=R for cooperative and α=β=1-R for competitive, yet the cooperative verification immediately sets α=β=1 ('by setting α = β = 1'), a different non-adaptive coupling that changes Eq. (12). Third, Eq. (12) is obtained by dividing by R and is used to infer hysteresis from the existence of two solution branches; multiple roots do not imply bistability of stable states, so without linear stability analysis of Eq. (11) the claimed first-order/second-order distinction is not proven analytically. The central claim therefore currently rests on finite-N adiabatic sweeps with no ensemble averaging, rather than on the promised analytical support.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an adaptive Kuramoto model on simplicial complexes in which each node's pairwise and three-body coupling strengths are modulated by its local order parameter r_i: nodes are 'cooperative' when the coupling scales with r_i and 'competitive' when it scales with 1 - r_i. Four combinations of pairwise and higher-order adaptation are simulated on Erdős-Rényi networks of N=200 oscillators, reporting (i) explosive, hysteretic transitions when cooperative adaptation dominates, (ii) continuous transitions when competitive adaptation dominates, and (iii) synchronization in the absence of pairwise interactions when higher-order coupling is sufficiently strong. The paper also studies the effect of system size and mean degree, and provides a mean-field Ott-Antonsen analysis for a globally coupled version of the model, claiming that the presence or absence of two solution branches in Eq. (12) proves the first-order versus second-order distinction.","tokens_in":11579,"tokens_out":6134,"duration_ms":57382,"significance":"If the reported phenomenology is robust, the paper contributes a useful demonstration that the type of synchronization transition can be controlled by mixing cooperative and competitive adaptive rules in higher-order networks. The clean separation of four adaptation scenarios and the systematic variation of σ1 and σ2 are valuable, and the claim that higher-order adaptive coupling can synchronize a static network without pairwise interactions is a concrete, falsifiable prediction that goes beyond existing pairwise adaptive-network studies. The paper also ships a reasonably complete description of the numerical protocol, which aids reproducibility. The main weakness is that the analytical verification in Section IV does not currently establish the central dichotomy, for reasons detailed below; the empirical findings therefore stand on finite-N adiabatic sweeps without ensemble averaging.","major_comments":[{"comment":"The analytical verification of the cooperative case sets α = β = 1, not α = β = R. The text explicitly says, 'we first consider the case in which the nodes in both links and triangles adapt cooperatively by setting α = β = 1,' whereas the model definition immediately before Eq. (4) states that cooperative adaptation corresponds to α = β = R. Equation (12) with α = β = 1 is therefore the static higher-order Kuramoto model, not the adaptive cooperative model simulated in Figs. 1–3. This mismatch means the analytical result cannot verify the claim that cooperative adaptation induces explosive synchronization; the authors must either derive Eq. (12) with α = β = R or clearly state that they are analyzing a different, non-adaptive limit.","section":"Section IV, Eq. (4) and Fig. 8"},{"comment":"The Lorentzian frequency width Δ is introduced in Eq. (9) but never assigned a value and disappears from the fixed-point equation Eq. (12). Evaluating Eq. (11) at γ(ω0 - iΔ, t) produces a term -Δγ, which should appear in the real-part fixed-point condition; Eq. (12) contains no such term. As written, Eq. (12) is independent of Δ, which is inconsistent with the stated Lorentzian distribution and with the standard Ott-Antonsen reduction, where the width controls the decay of the incoherent state. Please specify Δ and re-derive Eq. (12) including the -Δγ contribution, or explain explicitly if a Δ → 0 limit is being taken.","section":"Section IV, Eqs. (9)–(12)"},{"comment":"The inference of hysteresis from the existence of two solution branches of Eq. (12) is not valid without stability analysis. Multiple fixed points do not imply bistability of stable states; one branch may be unstable, and a first-order transition requires coexisting stable incoherent and stable coherent states. Moreover, Eq. (12) is obtained after dividing by R, so the R = 0 incoherent solution is not examined within the same equation. Please provide a linear stability analysis of the fixed points of Eq. (11), or direct time integration of Eq. (4) with and without adiabatic sweeps, to demonstrate which branches are attracting.","section":"Section IV, Eq. (12) and Fig. 8"},{"comment":"The central empirical classification into first-order and second-order transitions rests on single adiabatic forward/backward sweeps with no reported ensemble averaging or error bars. For finite N = 200, transition points and even the presence of hysteresis can vary across Erdős-Rényi realizations. The authors should state the number of realizations, show error bars or distributions of R, and ideally report the fraction of realizations that exhibit a discontinuous jump, to support the claimed transition-type dichotomy.","section":"Section III, Figs. 1–4"}],"minor_comments":[{"comment":"There is a typo in the definitions of f1 and f2: the text reads 'The opposite situation has occurred when f1 = 0 and f2 = 1' twice, but the second occurrence should presumably be f1 = 1 and f2 = 0. Also, the index range for β_i, written as 'i = N f1 + 1, N f2 + 2, . . . , N', appears to contain a typo and should likely be 'i = N f2 + 1, . . . , N'.","section":"Section II"},{"comment":"The local order parameter r_i in Eq. (2) divides by the degree k_i, so r_i is undefined for any oscillator with k_i = 0. The paper does not state that such nodes are absent from the generated Erdős-Rényi networks; for N = 200 and mean degree 20 this is highly likely, but an explicit statement or a convention for isolated nodes would make the model well-posed for arbitrary networks.","section":"Section II, Eq. (2)"},{"comment":"The manuscript does not report the numerical integration method, time step, transient time, or number of steps used for the adiabatic sweeps. Please add these details so that the simulations can be reproduced exactly.","section":"Section III, numerical details"},{"comment":"The phrase 'It is proven that cooperative adaptation is needed for the explosive transition' overstates the analytical support, since Section IV treats a globally coupled mean-field model and, for the cooperative case, actually sets α = β = 1. A more accurate wording would be 'the numerical results suggest' or 'the mean-field analysis supports'.","section":"Section III.F"},{"comment":"In the caption, 'The markers star depict' should be 'star markers depict', and the caption does not describe what the different panels (a)–(d) correspond to beyond the σ2 values; please clarify the relation between panels and the two adaptation cases.","section":"Section IV, Fig. 8"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of a nonlinear dynamics journal and the numerical phenomenology is potentially interesting, but the analytical verification section is currently presented as proof while using a different adaptation rule (α = β = 1 instead of α = β = R) and dropping the Lorentzian width. I would not reject the paper on this basis, because the numerical findings may well be correct and the analytical issues are fixable, but the authors need to either repair the analytic derivation or explicitly demote it to a heuristic mean-field illustration. I also recommend that the editor ask for ensemble-averaged numerical results before acceptance, because the central transition-type claim rests on finite-N sweeps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper has a genuinely interesting numerical finding: in static adaptive simplicial complexes, strong higher-order coupling can synchronize the network even with no pairwise coupling, and the cooperative/competitive flavor of adaptation selects between first- and second-order transitions. That part is worth a look. But the analytical verification in Sec. IV does not verify the model. The authors derive Eq. (12) for a globally coupled system with generic α, β, then 'by setting α=β=1' claim to test the cooperative case. That is the non-adaptive Kuramoto model with higher-order coupling, not their adaptive rule α=β=R. The competitive case α=β=1−R is never analyzed. So the 'proof' of the transition-type dichotomy is not about the paper's model.\n\nWhat the paper does well: the numerical exploration is systematic—four combinations of pairwise and three-body adaptation, scans over σ2, mean degree, and N. The figures are clear. The observation that synchronization survives at σ1=0 for large enough σ2 (e.g., Fig. 1c) is new relative to the cited static-network literature. The qualitative claims are probably reproducible from the methods as written. The citation coverage is adequate, including the relevant Sharma et al. 2024 and Frolov et al. 2021 papers.\n\nSoft spots, in proportion. The analytical section is load-bearing for the paper's claims and it fails. Even if one replaced α=β=1 with the correct α=β=R, multiple roots of Eq. (12) do not prove hysteresis unless you check linear stability of the branches. The use of a single global R in the mean-field reduction is unvalidated for sparse ER networks with degree-dependent local r_i. Minor but fixable: the Lorentzian width Δ is never specified; r_i in Eq. (2) is undefined for nodes with degree zero; and the numerics show no ensemble averaging or error bars, so the adiabatic sweeps are single realizations.\n\nWho is this for: researchers working on explosive synchronization in higher-order or adaptive networks. The numerical phenomenon is a solid subfield contribution, and the flaws are addressable in revision. I would send it to peer review, but I would push for a corrected or heavily revised Sec. IV—either a proper OA analysis of the adaptive rule with stability, or an explicit removal of the word 'proven.'","headline":"A numerically interesting study of adaptive higher-order synchronization whose analytical 'proof' analyzes a different, non-adaptive model and should be fixed or removed before publication.","tokens_in":12067,"tokens_out":4368,"would_cite":true,"duration_ms":37783,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C82","34C15","34D06"],"pacs":["05.45.Xt","89.75.Fb"],"model":"deepseek-v4-flash","headline":"Cooperative and competitive adaptation in higher-order (triangular) interactions controls whether a network of Kuramoto oscillators synchronizes explosively or continuously, and strong enough three-body coupling alone can synchronize a…","keywords":["adaptive networks","simplicial complexes","higher-order interactions","explosive synchronization","Kuramoto oscillators","cooperative and competitive adaptation","phase transitions","order parameter"],"falsifier":"Run the same adaptive dynamics (Eq. (1)) on an Erdős-Rényi network with N=200, mean degree K=20, sigma_2=0.3, all links and triangles competitive, and record R versus sigma_1 with small adiabatic steps: if R shows a discontinuous jump or hysteresis rather than a continuous curve, the claim that competitive adaptation always yields second-order transitions is false. Alternatively, for global coupling, solve the fixed-point equation with a nonzero Lorentzian width and check whether the two-branch hysteretic solution persists; if it disappears, the analytical identification of hysteresis with cooperative adaptation is not robust.","tokens_in":1927,"feed_emoji":"🔄","tokens_out":2347,"duration_ms":70205,"temperature":0.7,"pith_summary":"The paper asks whether adding three-body (simplicial) interactions to an adaptive network of Kuramoto oscillators changes how the network synchronizes, and whether the type of adaptation—cooperative or competitive—controls the order of the transition. It claims that purely competitive adaptation yields continuous (second-order) transitions and cluster formation, whereas cooperative adaptation produces explosive (first-order) transitions with hysteresis, and that a mix of the two can produce sharp transitions without backward desynchronization. The strongest claim: with sufficiently strong higher-order coupling, oscillators synchronize even with no pairwise coupling at all, which matters because pairwise interactions have usually been treated as necessary for synchronization in static networks. A mean-field calculation on globally coupled networks supports the numerical phase-transition picture.","feed_headline":"Adaptive triangles can synchronize a network with zero pairwise links","feed_subtitle":"Cooperative coupling yields abrupt, hysteretic sync; competitive coupling smooths it, and strong three-body links alone can phase-lock.","key_machinery":"The load-bearing object is the adaptive simplicial-complex Kuramoto model in Eq. (1), where pairwise and three-body couplings are modulated by local order parameters: cooperative nodes use $\\alpha_i = r_i$, $\\beta_i = r_i$; competitive nodes use $\\alpha_i = 1-r_i$, $\\beta_i = 1-r_i$. The analytical device is the Ott-Antonsen ansatz on a globally coupled network, replacing local order parameters with the global $R$, yielding the fixed-point equation $R = \\frac{\\sigma_1 \\alpha R}{2}(1-R^2) + \\frac{\\sigma_2 \\beta R^3}{2}(1-R^2)$. The number of branches of this equation in $R\\in[0,1]$ is the fingerprint of transition type: two branches mean hysteresis and explosive synchronization, one branch means a continuous transition.","core_discovery":"The central discovery is that the sign of the adaptation feedback, not just the coupling strength, sets the synchronization transition type. In the model, each node's pairwise and three-body couplings are multiplied by a local order parameter (cooperative, $\\alpha_i = r_i$, $\\beta_i = r_i$) or by its complement (competitive, $\\alpha_i = 1-r_i$, $\\beta_i = 1-r_i$). Because cooperative links strengthen as coherence grows, they create a self-reinforcing loop that yields explosive transitions with hysteresis; competitive links weaken as coherence grows, preventing the jump and producing continuous transitions. Higher-order triangle couplings act as an extra synchronizing drive: at fixed $\\sigma_1$, increasing $\\sigma_2$ shifts backward critical points to lower $\\sigma_1$, and for large enough $\\sigma_2$ the system remains synchronized even at $\\sigma_1 = 0$ or below, so pairwise interactions are not strictly required. The Ott-Antonsen mean-field analysis reproduces this as two branches for $R$ under cooperative adaptation (hysteresis) and one branch under competitive adaptation (no hysteresis).","pith_inferences":["The mechanism suggests a control recipe: adding a small fraction of competitive nodes to an otherwise cooperative adaptive network should suppress explosive jumps, which could be tested in engineered oscillator arrays or in models of power-grid synchronization.","The cluster formation seen under pure competition implies that competitive adaptation may be a generic route to chimera-like or multi-cluster states; a natural extension is to compute the cluster order parameter as a function of $\\sigma_2$, which the paper does not quantify.","The analytical fixed-point equation is derived under a Lorentzian frequency distribution with unspecified width; testing whether the two-branch (hysteretic) solution survives as that width is varied would clarify the robustness of the explosive-transition claim.","Because the local order parameter in Eq. (2) divides by node degree, zero-degree nodes are undefined, so the analysis implicitly assumes every node has at least one pairwise link; a degree-heterogeneous or directed extension could change the transition picture."],"forward_implications":["If cooperative adaptation is present, increasing the higher-order coupling strength $\\sigma_2$ widens the hysteresis loop and shifts the backward desynchronization point to lower $\\sigma_1$.","If all couplings adapt competitively, the system cannot undergo an explosive transition; it synchronizes continuously, if at all, and tends to form a cluster rather than jumping to full coherence.","Sufficiently strong three-body coupling can synchronize a static network with $\\sigma_1 = 0$, so pairwise coupling is not necessary for synchronization in adaptive higher-order networks.","Denser networks amplify the effect of adaptation: cooperative higher-order interactions widen hysteresis, while competitive higher-order interactions suppress explosive transitions.","The qualitative transition type is robust to changing the number of nodes, although critical couplings and hysteresis widths shift."],"supporting_citations":[{"why":"Introduces simplicial models of social contagion that motivate representing higher-order (three-body) interactions in adaptive networks.","marker":"[4]"},{"why":"Defines the Kuramoto oscillator model whose phase dynamics the paper adapts and extends.","marker":"[13]"},{"why":"Established explosive synchronization as a discontinuous transition in scale-free networks, the baseline phenomenon the paper links to adaptation type.","marker":"[17]"},{"why":"Showed explosive synchronization in adaptive and multilayer networks, the adaptive-coupling context this paper extends to higher-order interactions.","marker":"[19]"},{"why":"Demonstrated synchronization in temporal simplicial complexes, the comparison case for synchronization without or with weak pairwise interactions.","marker":"[23]"},{"why":"Provides the Ott-Antonsen ansatz used to reduce the globally coupled system to the fixed-point equation for R.","marker":"[25]"},{"why":"Supplies the mean-field reduction with global and cluster order parameters for adaptive oscillators with higher-order interactions, which the analytical section adapts.","marker":"[26]"},{"why":"The authors' prior work used to justify reading the number of branches of Eq. (12) as a test of transition type.","marker":"[27]"}],"fun_headline_variants":["Cooperative adaptation yields explosive sync, competitive smooths","Higher-order links only can synchronize adaptive networks","Adaptive sign flips sync from abrupt to continuous","No pair links needed: strong triangles drive sync"],"cache_read_input_tokens":14208,"weakest_assumption_plain":"The analytical verification assumes the network is globally coupled and replaces every oscillator's local order parameter with the global order parameter; if local adaptation on sparse networks behaves differently, the predicted transition type could be wrong.","fun_headline_variants_meta":{"raw":{"variants":["Cooperative adaptation yields explosive sync, competitive smooths","Higher-order links only can synchronize adaptive networks","Adaptive sign flips sync from abrupt to continuous","No pair links needed: strong triangles drive sync"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001171,"raw_usage":{"total_tokens":4867,"prompt_tokens":991,"completion_tokens":3876,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":3815}},"tokens_in":607,"tokens_out":3876,"duration_ms":29232,"temperature":1.0,"reasoning_tokens":3815,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:44:22.096030+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same adaptive dynamics (Eq. (1)) on an Erdős-Rényi network with N=200, mean degree K=20, sigma_2=0.3, all links and triangles competitive, and record R versus sigma_1 with small adiabatic steps: if R shows a discontinuous jump or hysteresis rather than a continuous curve, the claim that competitive adaptation always yields second-order transitions is false. Alternatively, for global coupling, solve the fixed-point equation with a nonzero Lorentzian width and check whether the two-branch hysteretic solution persists; if it disappears, the analytical identification of hysteresis with cooperative adaptation is not robust.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrated synchronization in temporal simplicial complexes, the comparison case for synchronization without or with weak pairwise interactions."},{"cited_title":"Sharma , author P","cited_arxiv_id":null,"evidence_quote":"Supplies the mean-field reduction with global and cluster order parameters for adaptive oscillators with higher-order interactions, which the analytical section adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The authors' prior work used to justify reading the number of branches of Eq. (12) as a test of transition type."}],"review_version":1}