{"id":"b83f4c71-00d6-4abd-ab5d-8ced99b8f7fe","arxiv_id":"2608.04749","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"An external node with random Gamma-distributed influence links can capture, stretch, or decouple a synchronised oscillator network, with stochastic links preserving internal cohesion better than deterministic ones.","lead":"A new model puts an outside influencer into a network of coupled oscillators whose link strengths fluctuate randomly, and maps when the group locks to the influencer, stretches, or ignores it. It matters because it gives a testable mathematical picture of how gentle, intermittent influence can steer a group without breaking its internal cohesion.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section IV.B's stability analysis is internally inconsistent: the eigenvalues of Eq. (17) are always real, so the claimed imaginary-eigenvalue regime and the approximate eigenvalues in Eqs. (19)/(22) as printed cannot be correct.","rationale":"The paper's central claim is primarily empirical: the stochastic influence model on ER graphs exhibits captured, stretched, and decoupled regimes, with a soft transition and no fracture of an already synchronised system. The numerical phase diagrams, finite-size scaling to N=640, and deterministic/stochastic comparisons provide independent, reasonably documented support for that claim. However, the analytic section is not a minor typo: the eigenvalues of the linearised system are always real, the factorisation in Eq. (19) and the approximate eigenvalues in Eq. (22) are algebraically wrong, and the claim that cluster-averaged Gamma draws make Ḡ1Ḡ2 vanishingly small is statistically incorrect. Because the paper explicitly uses Section IV.B to 'understand' and 'confirm' the numerically observed regimes, the analytic support for the central claim is invalid as printed. This does not by itself overturn the numerical three-regime conclusion, but it does mean the paper's own stated analytic confirmation is unsupported and needs correction. The reader's CONDITIONAL verdict is therefore appropriate: the numerical contribution may stand, but the analytic claims and ideally the code/data should be corrected and released before the paper is accepted in its current form.","tokens_in":21148,"tokens_out":12489,"duration_ms":151309,"concrete_test":"Symbolically recompute the eigenvalues of Eq. (17), correct Eq. (19) to λ±=(tr/2)(1±√(1−4τH/tr)) and Eq. (22) to λ≈{τH, tr−τH}, then evaluate these corrected eigenvalues using cluster-averaged Ḡ1, Ḡ2 measured from the actual N=300 simulations in the stretched regime. If the corrected eigenvalues are real and positive, the imaginary-kick mechanism and the 'H>1/(4τ)' regime in Section IV.B are refuted as printed; the remaining question is whether the numerically observed dip-and-recovery is explained by the corrected linear theory or requires a genuinely nonlinear cluster analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IV.B's linear stability analysis is the paper's analytic confirmation of the three-regime picture, but it is algebraically inconsistent. The discriminant of the linearisation matrix in Eq. (17) simplifies exactly to 4σ² + τ²(Ḡ1−Ḡ2)², which is nonnegative for every σ, τ, Ḡ1, Ḡ2 ≥ 0. Therefore the eigenvalues in Eq. (18) are always real, and the claimed regime 'H > 1/(4τ)' with imaginary eigenvalues and noise-induced oscillations cannot occur. The factorisation in Eq. (19) is also missing a factor of the trace: writing tr = 2σ + τ(Ḡ1+Ḡ2) and det = τH·tr, the correct form is λ± = (tr/2)(1 ± √(1 − 4τH/tr)), not (tr/2)(1 ± √(1 − 4τH)). Consequently Eq. (22) drops a factor of τ: in the small-noise limit λ₋ ≈ τH, not tr·H. A separate statistical slip compounds this: Ḡ1 and Ḡ2 are cluster averages of O(N) Gamma draws, so for large clusters they concentrate near E[G]=1 with O(N^{−1/2}) fluctuations; they are not typically 'vanishingly small', so the small-H approximation in Eq. (21) is not the typical case. The numerical three-regime observation may still be correct, but the analytic confirmation as printed does not hold, and the specific explanation of the stretched state via imaginary kicks is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a stochastic influence model in which N Kuramoto oscillators on an Erdős-Rényi graph are coupled to a single external node θ whose frequency Ω is fixed. The influence links are directed and carry Gamma-distributed multiplicative weights modulated by a Lorentzian phase kernel. The paper reports extensive numerical phase diagrams in the (τ, Ω) plane, identifies three regimes (captured, stretched, decoupled), compares stochastic with deterministic influence links, and studies finite-size scaling of the order parameter. It then proposes a two-cluster linear stability analysis to explain these regimes analytically, concluding that an already synchronised system is not fractured by the external node.","tokens_in":21487,"tokens_out":7972,"duration_ms":89495,"significance":"If the numerical three-regime picture is correct, the paper makes a useful contribution to stochastic influence and control in oscillator networks, with a socially motivated model and substantial ensemble statistics. The numerical results are internally coherent: the order-parameter and closeness heat maps, the oscillator-frequency distributions, the scaling exponent, and the stochastic-versus-deterministic comparison all support the three-regime interpretation. The analytic Section IV is intended as confirmation of this picture, but in its current form it contains algebraically incorrect eigenvalue expressions and an unsupported statistical approximation. The strengths of the paper are the breadth and reproducibility of the numerical experiments and the clear use of two complementary diagnostics, r and Δ; the weaknesses are concentrated in the analytic stability analysis, which needs to be corrected before the paper's claims can be accepted as stated.","major_comments":[{"comment":"The claimed imaginary-eigenvalue regime is algebraically impossible. For the matrix L in Eq. (17), tr L = 2σ + τ(Ḡ1+Ḡ2) and det L = τ[(Ḡ1+Ḡ2)σ + τḠ1Ḡ2], so the discriminant appearing in Eq. (18) is tr² − 4 det = 4σ² + τ²(Ḡ1−Ḡ2)², which is nonnegative for all σ, τ, Ḡ1, Ḡ2 ≥ 0. Therefore λ± are always real, and the statements in Section IV.B that for H > 1/(4τ) the system has imaginary components and that 'an imaginary part generates an oscillation' are not supported by the linearisation. The stretched regime cannot be explained by noise-induced imaginary eigenvalues within this analysis.","section":"Section IV.B, Eqs. (17)–(19)"},{"comment":"The factorisation in Eq. (19) is missing a factor of the trace. With tr = 2σ + τ(Ḡ1+Ḡ2) and det = τH·tr, the correct expression is λ± = (tr/2)(1 ± √(1 − 4τH/tr)), not (tr/2)(1 ± √(1 − 4τH)). Consequently, the small-H limit in Eq. (22) should read λ− ≈ τH and λ+ ≈ tr − τH, whereas the printed λ = tr{H, 1−H} gives λ− ≈ tr·H and is therefore wrong by a factor of τ/tr. This affects the subsequent qualitative discussion of damping and of the τ-dependence of the stability.","section":"Section IV.B, Eqs. (19) and (22)"},{"comment":"The statistical claim that 'typically Ḡ1Ḡ2 will be vanishingly small' is incorrect for the quantities as defined. Ḡ1 and Ḡ2 are normalised sums over O(N) Gamma(α, 1/α) variables with E[G] = 1; for the cluster sizes used in this paper they concentrate near 1 with fluctuations of order N^(−1/2), and the product Ḡ1Ḡ2 has expectation near 1, not near 0. The 'typical' small-H approximation in Eq. (21) is therefore not the typical case, and the stability classification built on Eqs. (21)–(22) needs to be rederived using the actual typical values of Ḡ1 and Ḡ2. This is load-bearing because those equations form the analytic basis for the claimed suppression of fracture.","section":"Section IV.B, Eq. (21)"},{"comment":"The stability analysis is performed on a time-dependent linear system, since Ḡ1 and Ḡ2 are resampled continuously, yet the regime classification is based on pointwise eigenvalues of the frozen matrix in Eq. (17). For a stochastic linear system Ẋ = ξ − L(t)X, the signs of the real parts of instantaneous eigenvalues do not by themselves determine almost-sure stability; an analysis via the fundamental matrix or Lyapunov exponents is required. The time-integrals in Appendix C are written down, but they are not used to justify the regime boundaries, so the analytic three-regime claim is not yet established even after the algebraic errors are corrected.","section":"Section IV.B and Appendix C"}],"minor_comments":[{"comment":"The caption lists the Ω values as {0, 0.25, 0, 0.25, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0}, with 0 and 0.25 each appearing twice; please correct the intended set of values and label the columns consistently.","section":"Fig. 10 caption"},{"comment":"The spelling of Erdős-Rényi is inconsistent (e.g., 'Erd˝ os-R´ enyi' in section headings), and reference [57] should be Bollobás, not Bollabás.","section":"Throughout"},{"comment":"The sentence beginning 'Notes that when they are close in phase...' is ungrammatical and should be rewritten; additionally, please state explicitly the sign convention of the influence term in Eq. (3) so that a reader can see that the term pulls φi toward θ when φi is behind θ.","section":"Section II, after Eq. (4)"},{"comment":"The description 'Runge-Kutta method 4 (10 substeps)' is vague; please specify the integrator and the exact convention for when the noise Gi is resampled relative to the integration substeps.","section":"Section III.A"},{"comment":"The sum rule in Eq. (7) appears to omit the internal coupling factor σ multiplying the double sum over the adjacency matrix; if this is intentional, please clarify, otherwise add the missing factor.","section":"Eq. (7)"},{"comment":"The phrase 'it’s natural ability' should be 'its natural ability', and the sentence 'The different regimes of behaviour and be seen as dependent...' appears to have a missing word ('can' or 'may').","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":"The numerical study is substantial and likely salvageable, but Section IV.B needs a genuine reworking rather than a cosmetic fix. The authors should correct the eigenvalue algebra, replace the incorrect statistical approximation with a valid estimate of the typical Ḡ1, Ḡ2 values, and either provide a proper stochastic stability argument or explicitly downgrade the analytic claims to a heuristic. On the current version, major revision seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The numerical phase diagram is real and useful: an external oscillator with Gamma-distributed, phase-dependent coupling produces a captured/stretched/decoupled landscape that is coherent across graph realizations and system sizes, and the paper documents it carefully. The scaling analysis and the stochastic-vs-deterministic comparison are solid. But Section IV.B is wrong as printed, and the stress-test note is correct: the eigenvalues of Eq. (17) are always real, so the claimed imaginary-kick mechanism for the stretched state does not exist.\n\nWhat is new: the Gamma-distributed multiplicative influence links with a Lorentzian phase kernel, and the systematic mapping of the (Ω, τ) plane. The numerical work is well specified—ensemble sizes, integration details, diagnostics r and Δ—so it is reproducible in principle. The authors are also candid that the transition is soft and that the system recovers after decoupling.\n\nThe soft spot is the analytic section, and it is not minor. The discriminant in Eq. (18) simplifies to 4σ² + τ²(Ḡ₁−Ḡ₂)², never negative. Eq. (19) drops a factor of the trace in the square root, and Eq. (22) drops a factor of τ. On top of that, the claim that Ḡ₁Ḡ₂ is 'vanishingly small' is statistically wrong for cluster averages: with N=300, those averages concentrate near 1, not near 0. The paper leans on these analytics to 'qualitatively reflect' the numerics, so this needs fixing. The three-regime picture does not collapse—the numerics support it—but the analytic explanation of the stretched state as noise-induced oscillations is unsupported.\n\nThe citation pattern looks fine; self-citations are to their prior stochastic and adaptive Kuramoto work, which is relevant. I would send this to peer review: the numerical contribution is worth referee time, and the analytic section can either be corrected or demoted to a heuristic that does not rely on imaginary eigenvalues. If the authors fix the algebra, the paper would be a reasonable applied contribution to stochastic synchronization and influence.","headline":"The numerical three-regime picture is solid, but the analytic stability section has a concrete algebraic error that invalidates the claimed imaginary-eigenvalue mechanism.","tokens_in":22002,"tokens_out":5307,"would_cite":false,"duration_ms":57154,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","60H10","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes and tests a Kuramoto-based model in which a single external node with Gamma-random influence links can capture, stretch, or decouple a synchronised network depending on the driving frequency and influence strength.","keywords":["Kuramoto model","synchronisation","external influence","stochastic link weights","Gamma noise","Erdos-Renyi graphs","entrainment transition","phase coherence"],"falsifier":"In the stretched regime, record the instantaneous phase difference between the two clusters across many graph realisations; if the typical separation is not small compared with one radian, the linear approximation fails exactly where it is used. A second check is to rerun the numerics with Gamma draws conditioned to be large, so the product of the cluster noise averages is not negligible, and see whether the transition-onset curves predicted by the $\\sigma$-independent parameter $H$ of Eq. (21) shift.","tokens_in":20887,"feed_emoji":"📡","tokens_out":10166,"duration_ms":105799,"temperature":0.7,"pith_summary":"This paper tries to establish that a single external actor can steer a synchronised Kuramoto network through random, intermittent influence links, and that the outcome depends on where the actor's driving frequency sits relative to the group's natural frequencies. Using Erdős-Rényi random graphs with $N=300$ oscillators and Gamma-distributed link weights, the paper finds three regimes as the driving frequency $\\Omega$ and influence strength $\\tau$ vary: captured, where the network synchronises with the external node; stretched, where internal coherence dips while the group is pulled toward $\\Omega$; and decoupled, where the group ignores the influencer and returns to its own synchronised state. The significance is that the model gives a quantitative account of when influence works without destroying group cohesion, and it says that an already synchronised system is stretched but not fractured. The paper also shows that stochastic soft links preserve internal synchronisation better than constant deterministic links, matching the idea that autonomy-respecting influence is more effective.","feed_headline":"One external node can capture, stretch, or decouple a network","feed_subtitle":"In simulations, stochastic soft links let a driven synchronised group keep its internal coherence more than deterministic forcing does.","key_machinery":"The central object is the stochastic influence link of Eq. (4), $B_i = G_i/((\\varphi_i-\\theta)^2+1)$, with $G_i \\sim \\Gamma(\\alpha, 1/\\alpha)$; it makes the external node's pull strongest when phases are close, and its heavy tail makes most draws act as weak nudges punctuated by rare strong kicks. The argument is carried by a two-cluster ansatz $\\Phi_1=\\theta+\\delta$, $\\Phi_2=\\Phi_1+\\epsilon$, whose linearised equations for $\\epsilon$ and $\\delta$ yield the stability matrix and eigenvalues of Eqs. (17)-(19), controlled by the parameter $H$ of Eq. (20). Under the assumption that typical Gamma draws make the product $\\bar G_1\\bar G_2$ negligible, $H$ becomes small and approximately $\\sigma$-independent, predicting that fluctuations in inter-cluster phase separation and source-population deviation are damped more strongly than in the uncoupled Kuramoto system, with rare large draws producing transient oscillatory kicks.","core_discovery":"The paper's central claim is that the stochastic influence model of Eqs. (3)-(4) produces three regime classes on Erdős-Rényi networks: captured, stretched, and decoupled. In the captured regime, the external node's frequency $\\Omega$ lies inside or near the natural frequency band and the influence strength $\\tau$ is adequate, so the internal oscillators synchronise with each other and are entrained to the external node. In the stretched regime, $\\Omega$ approaches or passes the edge of the band, the internal system remains coherent but is pulled toward $\\Omega$, and the Kuramoto order parameter dips to its minimum; this is the region of maximum influence. In the decoupled regime, $\\Omega$ is too ambitious, the external node drops the system, and the oscillators resynchronise at their natural mean frequency. The paper further claims that an already synchronised system is not fractured: the two-cluster stability analysis shows inter-cluster phase fluctuations are damped, so the influence manifests as stretching rather than splitting, and stochastic Gamma links with small shape parameter yield higher internal synchronisation than deterministic links.","pith_inferences":["An extension the paper leaves implicit is that the three-regime structure should survive for other heavy-tailed, non-negative link distributions; the Gamma choice mainly supplies frequent small values plus rare large kicks, so distributions with the same skewness should reproduce captured, stretched, and decoupled behaviour.","The pair of diagnostics, order parameter $r$ and closeness $\\Delta$, measure different failures; an operational extension would be to use $\\Delta/\\Omega$ or the centre-of-mass frequency as a real-time criterion for when an influencer is inside versus outside the group.","A testable social-system hypothesis follows: intermittent weak influence should shift a group's collective behaviour without fracturing it more effectively than constant strong influence, which could be checked in experiments on synchronised movement or consensus formation.","The measured scaling exponent for the fluctuation peak, $N^{-0.38}$, is shallower than the $N^{-1/2}$ an independent-oscillator picture would give; testing how this exponent depends on the Gamma shape $\\alpha$ would separate the graph contribution from the noise contribution to the softness of the transition."],"forward_implications":["If the central claim is right, an influencer whose frequency lies inside the natural frequency band can entrain the whole network to $\\Omega$ without destroying internal synchronisation, provided $\\tau$ is not too large.","For driving frequencies near or beyond the edge of the band, the system enters a stretched regime in which the order parameter dips and oscillators are pulled toward $\\Omega$ while remaining internally coherent; at higher $\\Omega$ it decouples and returns to its autonomous synchronised state.","The external node does not fracture an already synchronised system: fluctuations in the inter-cluster phase separation decay, so the influence shows up as temporary stretching rather than as two persistent clusters.","Stochastic influence links with small Gamma shape parameter $\\alpha$ preserve more internal synchronisation than deterministic links, because frequent weak nudges let the system re-synchronise after being dropped, while deterministic links delay that recovery.","The entrainment transition is soft rather than a sharp phase transition, with peak fluctuations scaling roughly as $N^{-0.38}$, so the three-regime behaviour is meaningful at finite community sizes rather than only in the thermodynamic limit."],"supporting_citations":[{"why":"It defines the Kuramoto model of coupled oscillators that the influenced network in Eq. (3) is built on, including natural frequencies and coupling strength.","marker":"[14, 15]"},{"why":"It supplies the multiplicative stochastic coupling formulation that the influence-link construction in Eqs. (3)-(4) is described as being consistent with.","marker":"[51]"},{"why":"It provides the adaptive and stochastic pinning-control mechanisms that the influence model extends, and the closeness diagnostic $\\Delta$ is taken from this line of work.","marker":"[34, 35]"},{"why":"It introduced heavy-tailed noise into Kuramoto synchronisation, the precedent for using Gamma-distributed influence weights instead of Gaussian or bounded noise.","marker":"[52, 53]"},{"why":"It states the social-influence principles of autonomy-respecting, identity-based influence that motivate the weak-random-nudge design and the interpretation of the regimes.","marker":"[2]"},{"why":"It argues that randomness is a necessary ingredient of social interactions, justifying the stochastic representation of the influence links.","marker":"[39]"},{"why":"They are recent stochastic pinning-control results that the model is compared with in terms of synchronisation and entrainment behaviour.","marker":"[55, 56]"}],"fun_headline_variants":["Stochastic links let one node capture, stretch, or drop a network","Three fates for a network under stochastic influence","External node's Gamma noise: capture, stretch, or decouple","One node's stochastic pull decides a network's sync fate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analytic argument assumes the two oscillator clusters stay close in phase and that ordinary draws of the random link strengths have almost no effect on each other; in the stretched regime, where phases spread out, those assumptions can fail.","fun_headline_variants_meta":{"raw":{"variants":["Stochastic links let one node capture, stretch, or drop a network","Three fates for a network under stochastic influence","External node's Gamma noise: capture, stretch, or decouple","One node's stochastic pull decides a network's sync fate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00034,"raw_usage":{"total_tokens":1903,"prompt_tokens":1002,"completion_tokens":901,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":831}},"tokens_in":618,"tokens_out":901,"duration_ms":8805,"temperature":1.0,"reasoning_tokens":831,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:28:24.224121+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the stretched regime, record the instantaneous phase difference between the two clusters across many graph realisations; if the typical separation is not small compared with one radian, the linear approximation fails exactly where it is used. A second check is to rerun the numerics with Gamma draws conditioned to be large, so the product of the cluster noise averages is not negligible, and see whether the transition-onset curves predicted by the $\\sigma$-independent parameter $H$ of Eq. (21) shift.","supporting_citations":[{"cited_title":"(2007)Influence of noise on the synchronisation of the stochastic Kuramoto model","cited_arxiv_id":null,"evidence_quote":"It supplies the multiplicative stochastic coupling formulation that the influence-link construction in Eqs. (3)-(4) is described as being consistent with."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It states the social-influence principles of autonomy-respecting, identity-based influence that motivate the weak-random-nudge design and the interpretation of the regimes."},{"cited_title":"(2007)Controllability of complex networks via pinning, Phys","cited_arxiv_id":null,"evidence_quote":"It argues that randomness is a necessary ingredient of social interactions, justifying the stochastic representation of the influence links."}],"review_version":1}