{"id":"eea97264-1f15-4ff6-b149-49ad52908a07","arxiv_id":"2501.02716","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Unidirectional drive from a middle layer makes two response layers in a multiplex network synchronize with each other, with controllable amplitude amplification.","lead":"This paper studies a three-layer network of oscillators where the middle layer drives the other two through one-way connections. It finds that the two outer layers synchronize with each other, with larger oscillation amplitudes that can be tuned by changing the drive strength or the drive's speed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Single-realization results leave the central control claim untested against multistability; the reported relay-synchronized amplified state may be one of several coexisting attractors.","rationale":"The reader's weakest assumption identifies the lack of parameter and initial-condition dependence testing as the main gap, and I agree that generality is not demonstrated. My concern is more specific: the paper's central 'control' claim is built on single-realization curves, so the possibility of coexisting attractors is not addressed. This is a real soft spot, but it is a conditionality rather than a refutation: the reported states are plausibly real, and the figures are internally consistent. I recommend keeping the CONDITIONAL verdict (hence UNCHANGED relative to the reader), with the added condition that ensemble/multistability checks and quantitative quasi-periodicity diagnostics be supplied before the control claims are stated as general. I do not see an internal inconsistency or a fatal error in the equations as written; the concern is about the robustness and interpretation of the numerical evidence.","tokens_in":11246,"tokens_out":10043,"duration_ms":112207,"concrete_test":"Run an ensemble of at least 50 independent random initial conditions for representative points spanning Fig. 5, e.g., (epsilon,tau) = (3,1), (1.5,1), (3,2.5), and (3,4), and also for a second system size such as N=200, P1=P2=P3=50, K2=5. For each realization compute S_inter between L1 and L3, the amplitude ratio A(epsilon), the phase difference, and a quantitative quasi-periodicity diagnostic (largest Lyapunov exponent or Poincare section). If the distributions of S_inter and A are not sharply peaked around the values shown in Figs. 3 and 6, or if the tau=4 point is chaotic rather than quasi-periodic, then the central control and classification claims fail for those parameters.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The synchronization errors and amplitude ratios in Figs. 2, 3, and 6 are computed from one set of random initial conditions per parameter choice, as stated in Section 2: 'The system of equations in Eq. (1) are integrated for random initial conditions between (-1,1)'. The central claim that tuning epsilon or tau 'controls' the response-layer dynamics presupposes that the observed relay-synchronized, amplified state is the unique, or at least the representative, attractor for each parameter value. This is not established. Coupled Stuart-Landau oscillators with nonlocal coupling are known to be multistable, and in the present setting the response-layer synchronization manifold is independent of K1 and K3 once intralayer synchronization is achieved, so the only nonlinear selection mechanism is the common drive. If different initial conditions relax to non-synchronized response layers, or to synchronized states with different amplitudes at the same (epsilon, tau), then the plotted curves are single-path artifacts and the control claim is not robust. The quasi-periodic region (iv) in Fig. 5 is a related weakness: it is classified by visual inspection of time series and phase portraits, without Lyapunov exponents, Poincare sections, or any quantitative frequency analysis, so the boundary of region (iv) is unsupported. The ensemble/multistability issue is the more load-bearing of the two because it directly affects the paper's core claim of controllable, reproducible dynamics.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a three-layer multiplex network of Stuart-Landau oscillators with unidirectional interlayer coupling from a middle drive layer L2 to two identical response layers L1 and L3. For feedback-type coupling, numerical integration of Eq. (1) shows that the response layers achieve relay synchronization with amplified oscillations that are phase synchronized with the drive, and the response amplitude can be tuned by the interlayer coupling strength epsilon or the time-scale mismatch tau. For diffusive coupling, the identical layers achieve complete synchronization, while a time-scale or parameter mismatch leads to relay synchronization with frequency synchronization and reduced amplitude. A phase diagram in the (epsilon, tau) plane is presented, identifying regions of amplification, equal amplitude, reduced amplitude, and quasi-periodic dynamics.","tokens_in":11510,"tokens_out":10609,"duration_ms":98652,"significance":"If the reported effects are robust, the paper offers a simple mechanism for remotely controlling oscillation amplitudes in multiplex networks via unidirectional drive, which is relevant to applications such as neuronal relay systems and smart grids. The study is purely numerical but uses standard diagnostics (S_intra, S_inter, A(epsilon), phase difference) and clearly specifies the model and parameter values. Its main value is a credible demonstration of a phenomenon in a specific setup, not a general theory. The lack of ensemble statistics and the absence of quantitative measures for quasi-periodicity currently limit the strength of the central control claim.","major_comments":[{"comment":"The paper states that the system is 'integrated for random initial conditions between (-1,1)' (Section 2), but every displayed curve is evidently based on a single realization per parameter value. Coupled Stuart-Landau oscillators with nonlocal coupling are known to exhibit multistability, so the plotted S_intra, S_inter, and A(epsilon) values may belong to one of several coexisting attractors. In particular, the control claim in the abstract that the response-layer amplitude 'can be controlled by tuning' epsilon presupposes that the relay-synchronized amplified state is the unique or representative attractor for each (epsilon, tau). Please add an ensemble study: for a grid of (epsilon, tau) values, report the fraction of initial conditions that converge to relay synchronization and the mean plus or minus standard deviation of A(epsilon, tau) and of the synchronization errors. If multiple attractors coexist, characterize their basins and restrict the control claim to the parameter and initial-condition region where it holds.","section":"Section 2, Eq. (1), Figs. 2, 3, 6"},{"comment":"The quasi-periodic region (iv) in Fig. 5 is assigned by visual inspection of time series and phase portraits (Fig. 4(a3,b3)), and no quantitative criterion (largest Lyapunov exponent, power spectrum with incommensurate peaks, or Poincare section) is given. Consequently the boundaries of region (iv) are not reproducible. Moreover, the axes of Fig. 5, as printed, cover tau in [1.0,2.49] and epsilon in [1.5,1.99], which does not include the values tau=4 and epsilon=3 used in Fig. 4 and Fig. 6. Please clarify the axis ranges and the classification algorithm, and report the quantitative measure used to define each of the four regions.","section":"Section 3, Fig. 5"}],"minor_comments":[{"comment":"The notation A(epsilon) is used for a quantity that Fig. 6 plots against tau; rename it to A(tau) for fixed epsilon or use A(epsilon, tau) to avoid confusion. Also clarify that a(0) is the constant drive-layer amplitude, not the epsilon=0 value of the response layer.","section":"Section 2, Eq. (4) and Fig. 6"},{"comment":"Please specify the integration method, step size, transient removal criterion, and whether the results in each figure come from one run or from averaging over runs; this information is necessary for reproducibility.","section":"Section 2"},{"comment":"The epsilon-axis ranges differ between panels (a1) (about 0 to 0.019) and (a2) (about 0 to 0.195), and both differ from the ranges used in Figs. 3 and 6 (up to 2.5 or 3). State the intended epsilon ranges in the captions and explain the different scales.","section":"Fig. 2"},{"comment":"The time axis in the time series panels is labeled only as 'Time'; state whether it is in dimensionless time units or in integration steps.","section":"Fig. 4"},{"comment":"The parameter mismatch results (omega_drive=2, omega_response=1.5) are described without any supporting figure or quantitative measure. Add a figure panel or explicitly present these statements as qualitative observations subject to further study.","section":"Section 3, last paragraph"},{"comment":"The Data Availability section states that computations use publicly available packages but provides no link to the code or data. A persistent link to the integration scripts would substantially improve reproducibility.","section":"Section 5"},{"comment":"The intralayer coupling is only through the x variable, while the y equations do not contain coupling terms; state this explicitly in the text so that readers do not assume the standard complex-field coupling.","section":"Eq. (1)"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of Physica D as a numerical dynamical-systems study. The central phenomenon is plausible, but the control claim is currently stronger than the evidence supports: the single-realization analysis leaves multistability untested, and the quasi-periodic classification in Fig. 5 is not quantitatively grounded. Both issues are fixable with additional computations, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a competent numerical study that adds a genuinely new configuration to the relay-synchronization literature. The core observation—unidirectional feedback from a drive layer makes two response layers synchronize with each other while amplifying their oscillations, with amplification tunable by coupling strength or time-scale mismatch—is clearly demonstrated in the figures. The authors also show that diffusive interlayer coupling produces complete synchronization instead, which is a useful contrast.\n\nWhat I like: the synchronization-error and amplitude measures are standard and well-defined. The paper is honest about what it computes. It does not try to sell approximate analytic theory. The comparison of feedback vs diffusive coupling and of time-scale vs frequency mismatch is the kind of systematic exploration that makes a paper useful to people working on multiplex control.\n\nThe soft spots are real but not fatal. First, every result comes from a single realization of random initial conditions. The stress-test concern is on point: coupled Stuart-Landau oscillators with nonlocal coupling are known to be multistable, and without an ensemble of initial conditions we do not know whether the relay-synchronized amplified state is unique or just one attractor. The control claim in the abstract is therefore stronger than the evidence. Second, the quasi-periodic region in Fig. 5 is classified by eye; no Lyapunov exponents or frequency spectra are shown. Third, only one parameter set is used (N=100, P=25, K2=5, omega=2), so the phase diagram in Fig. 5 may not be representative. The authors should release code and show error bars or at least multiple runs.\n\nOverall: this is a modest but legitimate contribution. It deserves peer review, because the phenomenon is plausible and the numerics are clean; a referee can ask for the missing robustness checks. I would not cite it in the next twelve months myself, but I would point a student to it if they were working on relay synchronization with directional coupling.","headline":"Useful numerical study of relay synchronization with unidirectional coupling; the control claim needs robustness checks before being taken as general.","tokens_in":11964,"tokens_out":1856,"would_cite":false,"duration_ms":18807,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","34D06"],"pacs":["05.45.Xt"],"model":"deepseek-v4-flash","headline":"Unidirectional feedback from a middle drive layer makes two identical outer layers relay-synchronize with amplified, phase-locked oscillations whose amplitude can be set by the coupling strength or the time-scale mismatch.","keywords":["multiplex network","relay synchronization","unidirectional coupling","time scale mismatch","Stuart-Landau oscillator","amplitude control","interlayer coupling","quasi-periodic dynamics"],"falsifier":"Integrate Eq. (1) for the same feedback coupling with a different network size and coupling range, e.g. $N=200$, $P=50$, and with initial conditions drawn from outside $(-1,1)$; if the interlayer synchronization error $S_{\\text{inter}}$ between L1 and L3 does not fall to zero as $\\epsilon$ grows, or the response amplitude does not increase with $\\epsilon$ and decrease with $\\tau$, the claimed mechanism is specific to the chosen ring parameters rather than generic.","tokens_in":11077,"feed_emoji":"🔄","tokens_out":10734,"duration_ms":96654,"temperature":0.7,"pith_summary":"This paper sets out to establish that in a three-layer multiplex network of Stuart-Landau oscillators, unidirectional feedback from a middle drive layer forces the two outer response layers into relay synchronization with each other. The response oscillations are amplified relative to the drive and are phase-locked to it, and their amplitude can be tuned continuously by adjusting the interlayer coupling strength $\\epsilon$ or the time-scale mismatch parameter $\\tau$. With diffusive rather than feedback coupling, the identical-layer setup instead yields complete synchronization across all three layers. The paper also shows that sufficiently large $\\tau$, or a mismatch in the oscillators' intrinsic frequencies, replaces phase locking with frequency synchronization and eventually quasi-periodic dynamics in the responses. If correct, the results provide a mechanism for remotely controlling the collective state of response layers by tuning only the drive layer.","feed_headline":"One-way coupling yields relay synchronization with tunable amplitude","feed_subtitle":"A middle drive layer synchronizes the two outer layers and sets their oscillation amplitude.","key_machinery":"The load-bearing object is the unidirectional feedback term in Eq. (1): each node in L1 and L3 receives $\\epsilon(x_{i2}, y_{i2})$ from its counterpart in L2, while L2 receives nothing back. Because the two response layers are forced by the identical drive signal, they inherit a common input and synchronize with each other even though no direct L1-L3 link exists; the same term also injects energy, which enlarges the response oscillations. The time-scale parameter $\\tau$ multiplies the derivatives of L2, effectively making the drive faster or slower than the responses, and the paper shows this changes the balance between amplification and phase locking. The quantitative diagnostics are the intra- and interlayer synchronization errors and the average amplitude ratio $A(\\epsilon)$ measured relative to the drive.","core_discovery":"Starting from the system in Eq. (1), with $N=100$ Stuart-Landau oscillators per layer on a ring with nonlocal coupling range $P=25$, the authors find that when the middle layer L2 drives L1 and L3 through unidirectional feedback terms $\\epsilon(x_{i2}, y_{i2})$, the two response layers synchronize completely with each other, while their oscillation amplitude grows above that of the drive and their average phase difference to the drive falls to zero. The synchronization errors $S_{\\text{intra}}$ and $S_{\\text{inter}}$ confirm that intralayer order in the responses and relay synchronization between them both set in as $\\epsilon$ is raised. The same feedback with a time-scale mismatch $\\tau$ inserted in the drive layer's equations tunes the response amplitude down to equality with the drive at $\\tau=2.5$ and then into smaller-amplitude and quasi-periodic regimes at larger $\\tau$, where relay synchronization is replaced by frequency synchronization between the responses. With unidirectional diffusive coupling, the identical-layer case yields complete synchronization across all three layers, while parameter mismatch (for example $\\omega_{\\text{response}}=1.5$ versus $\\omega_{\\text{drive}}=2$) preserves relay synchronization but gives only frequency synchronization with the drive.","pith_inferences":["A direct extension the paper does not run: with three or more identical response layers all driven by the same L2 signal, relay synchronization should hold pairwise among all of them, with $\\epsilon$ and $\\tau$ controlling the common amplitude.","The arrangement is structurally an auxiliary-system experiment: L1 and L3 are identical systems forced by the same drive, so their synchronization with each other is equivalent to each being in generalized synchronization with L2, a connection that suggests master-stability-function conditions for when relay synchronization should hold.","The quasi-periodic region in Fig. 5 is identified by visual inspection of time series and phase portraits; a Lyapunov-exponent or power-spectrum scan would sharpen the region boundaries and determine whether part of region (iv) is chaotic rather than quasi-periodic.","Because the feedback term injects drive-layer amplitude into both responses symmetrically, the mechanism may work for a wider class of limit-cycle and chaotic oscillators, not just Stuart-Landau systems, provided the driven response remains stable."],"forward_implications":["A three-layer multiplex with one-way feedback can act as a remote synchronizer: nodes in two disconnected layers become fully synchronized through the middle layer alone.","The response amplitude can be set by choosing $\\epsilon$, and can be matched to the drive amplitude by choosing $\\tau \\approx 2.5$, giving a two-knob control scheme for remote layers.","Changing the interlayer coupling from feedback to diffusive switches the outcome from amplified relay synchronization to complete synchronization when the layers are identical, and to reduced-amplitude relay synchronization when they are mismatched.","Increasing $\\tau$ beyond the equal-amplitude point destroys phase locking and eventually relay synchronization, leaving only frequency-synchronized quasi-periodic responses; applications needing exact phase locking must keep $\\tau$ small.","With a mismatch in intrinsic frequencies, relay synchronization between the responses survives but phase synchronization with the drive does not, so phase locking is not a generic feature of the unidirectional scheme.","For experimental relay systems, the prediction that response amplitude grows with $\\epsilon$ and shrinks as $\\tau$ moves away from 1 can be tested directly by measuring the oscillation envelope of the remote layers while sweeping the drive amplitude or drive time scale."],"supporting_citations":[{"why":"Introduces relay synchronization as synchronization of distant networks through an intermediate relay layer.","marker":"[32, 33, 34, 35]"},{"why":"Prior demonstration of relay synchronization in multiplex networks that this study extends to unidirectional interlayer coupling.","marker":"[44]"},{"why":"Shows that the nature of the relay layer's coupling changes relay synchronization, motivating the paper's feedback-versus-diffusive comparison.","marker":"[45]"},{"why":"Reports unidirectional interlayer links in two-layer neuronal networks, providing the directionality motivation for the drive-response setup.","marker":"[47]"},{"why":"Shows that tuning dynamical time scales can revive synchronized oscillations in multiplex networks, the control mechanism the paper generalizes.","marker":"[12]"},{"why":"Reports frequency-synchronized states induced by differing dynamical time scales, supporting the use of $\\tau$ to shift responses from phase to frequency synchronization.","marker":"[56]"}],"fun_headline_variants":["Relay sync with amplified responses via one-way drive","Unidirectional coupling lets one layer set two layers' amplitude","Middle layer remotely synchronizes and amplifies outer layers","One-way relay sync, amplitude remote-controlled"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results assume that the behavior observed for the single parameter set $N=100$, $P_1=P_2=P_3=25$, $K_2=5$, $\\omega=2$, with quasi-periodicity judged by visual inspection, carries over to other network sizes, topologies, parameter values, and initial conditions.","fun_headline_variants_meta":{"raw":{"variants":["Relay sync with amplified responses via one-way drive","Unidirectional coupling lets one layer set two layers' amplitude","Middle layer remotely synchronizes and amplifies outer layers","One-way relay sync, amplitude remote-controlled"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001114,"raw_usage":{"total_tokens":4647,"prompt_tokens":957,"completion_tokens":3690,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":3628}},"tokens_in":573,"tokens_out":3690,"duration_ms":26231,"temperature":1.0,"reasoning_tokens":3628,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:06:20.084288+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Integrate Eq. (1) for the same feedback coupling with a different network size and coupling range, e.g. $N=200$, $P=50$, and with initial conditions drawn from outside $(-1,1)$; if the interlayer synchronization error $S_{\\text{inter}}$ between L1 and L3 does not fall to zero as $\\epsilon$ grows, or the response amplitude does not increase with $\\epsilon$ and decrease with $\\tau$, the claimed mechanism is specific to the chosen ring parameters rather than generic.","supporting_citations":[{"cited_title":"Leyva, I","cited_arxiv_id":null,"evidence_quote":"Prior demonstration of relay synchronization in multiplex networks that this study extends to unidirectional interlayer coupling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that the nature of the relay layer's coupling changes relay synchronization, motivating the paper's feedback-versus-diffusive comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports unidirectional interlayer links in two-layer neuronal networks, providing the directionality motivation for the drive-response setup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that tuning dynamical time scales can revive synchronized oscillations in multiplex networks, the control mechanism the paper generalizes."},{"cited_title":"Kachhara, G","cited_arxiv_id":null,"evidence_quote":"Reports frequency-synchronized states induced by differing dynamical time scales, supporting the use of $\\tau$ to shift responses from phase to frequency synchronization."}],"review_version":1}