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arxiv: 2604.01724 · v2 · pith:IUBN43GKnew · submitted 2026-04-02 · 🌊 nlin.AO

Complete Synchronization and its Transition in Higher Harmonic Sakaguchi--Kuramoto Oscillators

Subhasish Chowdhury , Sangita Dutta , Pitambar Khanra , Swarup Kumar Laha , Prosenjit Kundu This is my paper

Pith reviewed 2026-05-22 10:14 UTC · model grok-4.3

classification 🌊 nlin.AO
keywords complete synchronizationSakaguchi-Kuramoto modelphase frustrationbi-harmonic couplingheterogeneous networksscale-free networksmean-field approximationhysteresis
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The pith

Selecting natural frequencies that scale with node degree and tuning bi-harmonic coupling parameters achieves complete synchronization at small coupling strengths in phase-frustrated Sakaguchi-Kuramoto oscillators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that phase frustration in Sakaguchi-Kuramoto oscillators on heterogeneous networks can be overcome by choosing natural frequencies proportional to node degree together with suitable bi-harmonic coupling terms. This choice produces complete synchronization, where the global order parameter reaches exactly one, at a designated small value of the coupling strength. A sympathetic reader would care because many real systems rely on collective locking of oscillators yet are hindered by frustration; removing that barrier at low cost matters for power grids, neural circuits, and engineered networks. The work further shows that the synchronization transition is discontinuous with hysteresis on scale-free networks but continuous on Erdős-Rényi networks, and derives an analytic critical coupling for the pure second-harmonic case via mean-field theory, yielding a perfectly ordered two-cluster state.

Core claim

By choosing natural frequencies that scale linearly with node degree and appropriate parameters in the bi-harmonic coupling function, the phase-frustrated Sakaguchi-Kuramoto model reaches complete synchronization (r = 1) at a specified small coupling value on heterogeneous networks. The transition is first-order with hysteresis in scale-free networks and second-order in Erdős-Rényi networks. For the pure second-harmonic case the mean-field approximation supplies an analytic expression for the critical coupling from the self-consistent equations and produces a perfectly ordered two-cluster synchronized state.

What carries the argument

Optimal natural frequencies that scale linearly with node degree, paired with bi-harmonic coupling parameters, which permit derivation of the critical coupling from mean-field self-consistent equations.

If this is right

  • Complete synchronization remains robust when the same frequency scaling and coupling choice are applied to empirical networks such as the C. elegans neural network and the Zachary Karate Club.
  • The same construction extends to higher-order harmonic coupling functions while preserving complete synchronization.
  • The transition order depends on network topology: discontinuous with hysteresis on scale-free graphs and continuous on Erdős-Rényi graphs.
  • In the pure second-harmonic regime the system settles into a perfectly ordered two-cluster state once the critical coupling is crossed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the frequency scaling works on real networks, it suggests a design rule for assigning intrinsic frequencies according to measured node degrees to promote synchronization with minimal coupling strength.
  • The two-cluster state found in the second-harmonic case may appear in other frustrated oscillator models when similar degree-proportional frequencies are imposed.
  • Testing the same construction on networks that evolve in time or include transmission delays would clarify how robust the complete-synchronization regime is outside the static undirected case.

Load-bearing premise

The mean-field approximation accurately captures the dynamics when natural frequencies are set exactly proportional to node degree and the network is large and sufficiently heterogeneous.

What would settle it

Numerical integration on a large scale-free network with the proposed degree-proportional frequencies and bi-harmonic parameters, checking whether the order parameter reaches and stays at exactly 1 above the analytically predicted small coupling value.

Figures

Figures reproduced from arXiv: 2604.01724 by Pitambar Khanra, Prosenjit Kundu, Sangita Dutta, Subhasish Chowdhury, Swarup Kumar Laha.

Figure 1
Figure 1. Figure 1: Synchronization dynamics in a scale-free network (𝑁 = 500, 𝛾 = 2.8) under phase frustration 𝛼 = 𝛽 = 0.5. Purple diamond and green square markers denote the synchronization levels of the order parameters (𝑟1 and 𝑟2 ) obtained from normal and homogeneous natural frequency distributions, respectively. The blue solid and red dashed curves correspond to forward and backward adiabatic continuation of the couplin… view at source ↗
Figure 2
Figure 2. Figure 2: Order parameters 𝑟1 and 𝑟2 vs. coupling strength 𝐾2 . Numerical and analytical results for a scale-free network (𝑁 = 500, 𝛾 = 2.8, ⟨𝑞⟩ ≈ 9, 𝛽 = 0.5). Blue and red lines denote simulation results via forward and backward integration, respectively. (a) Behavior of 𝑟1 showing only a backward phase transition. (b) Bifurcation diagram of 𝑟2 exhibiting a first-order transition and a distinct hysteresis area. Cya… view at source ↗
Figure 3
Figure 3. Figure 3: The effect of frequency deviation on system synchronization. The graph compares the simulated error 𝜂 (red symbols) against the theoretical scaling limit 𝜂 ∼ 𝜎 2 (blue line) under varying noise strengths 𝜎. The system exhibits two distinct regimes: a quadratic growth of error at low 𝜎, followed by a saturation phase (𝜂 ⟶ 1) indicating total loss of synchronization at high 𝜎. Results are shown for a scale-f… view at source ↗
Figure 4
Figure 4. Figure 4: Synchronization transitions in empirical networks. Variation of order parameters 𝑟1 and 𝑟2 as functions of the coupling strength 𝐾2 for the Zachary Karate Club network (𝑁 = 34) (a,b) and the C. elegans network (𝑁 = 131) (c,d) with 𝛽 = 0.2. Blue solid and red dashed curves denote forward and backward numerical simulations, respectively. In both networks, 𝑟1 displays a backward transition, whereas 𝑟2 exhibit… view at source ↗
Figure 5
Figure 5. Figure 5: Third-order harmonic extension of the coupled oscillator model. Order parameters 𝑟1 , 𝑟2 , and 𝑟3 as functions of the coupling strength for a scale-free network with degree exponent 𝛾 = 2.8 and size 𝑁 = 500, where phase frustration 0.5. Blue and red curves represent forward and backward simulations obtained using the proposed optimal frequency distribution. The maximum values (complete synchronization) of … view at source ↗
Figure 6
Figure 6. Figure 6: Synchronization dynamics in a Erdős–Rényi (ER) network with 𝑁 = 500 and phase frustration 𝛼 = 𝛽 = 0.5. The blue solid and red dashed curves correspond to forward and backward adiabatic continuation of the coupling strength, computed using the proposed optimal frequency assignment. (a)–(b) Variation of order parameters with the second-harmonic coupling 𝐾2 at fixed 𝐾1 = 0.3. (c)–(d) Variation of order parame… view at source ↗
read the original abstract

In heterogeneous networks of coupled oscillators, phase frustration typically prevents the emergence of synchronization in the Sakaguchi--Kuramoto (SK) model. In this study, we propose an analytical framework to overcome this barrier and induce complete synchronization at a specified small coupling value in oscillators governed by phase-frustrated bi-harmonic coupling. We derive an optimal set of natural frequencies that is robust against added noise and correlated with the network degree heterogeneity, along with the parameters involved in the bi-harmonic coupling function that lead to complete synchronization ($r = 1$). In addition, we find complete synchronization transitions accompanied by hysteresis in scale-free networks, indicating a first-order (discontinuous) phase transition, whereas Erd\H{o}s--R\'enyi networks exhibit complete synchronization through a second-order (continuous) phase transition. Furthermore, we use the mean-field approximation in the presence of optimal frequencies to determine the critical coupling strength associated with the synchronization transition in the pure second-harmonic Sakaguchi--Kuramoto model. Here, the obtained optimal natural frequencies scale linearly with the node degree, and the critical coupling strength for the onset of synchronization is derived analytically from the self-consistent equations. In this specific regime, we observe a perfectly ordered two-cluster synchronized state. These findings remain robust for higher-order harmonic coupling schemes, as well as across a diverse range of synthetic and empirical networks, including scale-free, Erd\H{o}s--R\'enyi, Zachary Karate Club, and the \textit{C.~elegans} neural network, demonstrating their general applicability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes an analytical framework to achieve complete synchronization (r=1) in phase-frustrated bi-harmonic Sakaguchi-Kuramoto oscillators on heterogeneous networks. Optimal natural frequencies are chosen to scale linearly with node degree, together with tuned bi-harmonic coupling coefficients, to reach complete synchronization at a small specified coupling strength. The work reports first-order transitions with hysteresis in scale-free networks and second-order transitions in Erdős-Rényi networks. For the pure second-harmonic case, mean-field approximation is applied to self-consistent equations to derive the critical coupling analytically, yielding a perfectly ordered two-cluster state. Results are claimed to be robust across higher-harmonic schemes and both synthetic (scale-free, ER) and empirical networks (Zachary Karate Club, C. elegans).

Significance. If the mean-field closure and analytical critical-coupling derivation hold under the degree-frequency correlation, the result would provide a concrete route to overcome phase frustration in Sakaguchi-Kuramoto models on heterogeneous networks. The explicit analytical expression for the critical coupling, the demonstration of both continuous and discontinuous transitions, and the numerical robustness across multiple network topologies constitute a substantive contribution to the study of synchronization in complex systems.

major comments (3)
  1. [Abstract / mean-field section] Abstract and § on mean-field derivation: the claim that the critical coupling is derived analytically from self-consistent equations under the optimal-frequency choice lacks visible explicit steps, error estimates, or checks against finite-size effects. Without these, it is unclear whether the resulting expression is an independent prediction or is effectively conditioned on the linear scaling constant between ω_i and k_i.
  2. [Mean-field approximation] Mean-field closure under ω_i ∝ k_i: in scale-free networks the correlation between the frequency distribution and the degree distribution means that high-degree nodes (which dominate the global mean field) carry systematically shifted natural frequencies. The standard single-order-parameter mean-field ansatz invoked to close the self-consistent equations for the two-cluster state therefore requires additional justification or degree-resolved corrections to remain consistent with the microscopic dynamics.
  3. [Results on complete synchronization] Table or figure reporting the synchronization transition: the reported achievement of r=1 at a small coupling value depends on the specific choice of the linear scaling constant and the bi-harmonic coefficients; these appear as free parameters whose selection must be shown to be independent of the target r=1 outcome rather than tuned to produce it.
minor comments (2)
  1. [Notation] Notation for the order parameter r and the coupling strength K should be used consistently between the analytic expressions and the numerical figures.
  2. [Figures] Figure captions should explicitly list the network size, degree exponent (for scale-free cases), and noise amplitude used in each panel to facilitate reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and have revised the manuscript to enhance clarity and provide additional justification where needed.

read point-by-point responses

  1. Referee: [Abstract / mean-field section] Abstract and § on mean-field derivation: the claim that the critical coupling is derived analytically from self-consistent equations under the optimal-frequency choice lacks visible explicit steps, error estimates, or checks against finite-size effects. Without these, it is unclear whether the resulting expression is an independent prediction or is effectively conditioned on the linear scaling constant between ω_i and k_i.

    Authors: We agree that the original presentation could benefit from expanded detail. The revised manuscript now includes the full algebraic steps deriving the critical coupling from the self-consistent equations under the optimal linear frequency scaling. We have added an error estimate for the mean-field closure and new numerical checks of finite-size effects across network sizes from N=500 to N=5000. The expression is derived analytically once the scaling constant is fixed by the requirement that the bi-harmonic terms produce r=1 at the target coupling; we explicitly separate this choice from the subsequent analytic solution. revision: yes

  2. Referee: [Mean-field approximation] Mean-field closure under ω_i ∝ k_i: in scale-free networks the correlation between the frequency distribution and the degree distribution means that high-degree nodes (which dominate the global mean field) carry systematically shifted natural frequencies. The standard single-order-parameter mean-field ansatz invoked to close the self-consistent equations for the two-cluster state therefore requires additional justification or degree-resolved corrections to remain consistent with the microscopic dynamics.

    Authors: The optimal scaling ω_i = c k_i is chosen precisely so that the frequency offset is canceled by the phase shift induced by the bi-harmonic coupling, yielding a uniform effective field. In the revision we derive this cancellation explicitly and add degree-binned order-parameter plots for the scale-free case in the supplementary material. These confirm that the single global order parameter remains an accurate closure for the two-cluster state; a full degree-resolved mean-field treatment is not required under the present correlation. revision: partial

  3. Referee: [Results on complete synchronization] Table or figure reporting the synchronization transition: the reported achievement of r=1 at a small coupling value depends on the specific choice of the linear scaling constant and the bi-harmonic coefficients; these appear as free parameters whose selection must be shown to be independent of the target r=1 outcome rather than tuned to produce it.

    Authors: The scaling constant and bi-harmonic coefficients are obtained by solving the self-consistent equations for the exact condition r=1 at the prescribed small coupling strength; they are therefore uniquely determined by the network degree sequence and the target synchronization state rather than adjusted numerically after the fact. The revised results section now presents the explicit algebraic procedure used to compute these values and demonstrates that modest deviations from optimality still permit r=1, albeit at slightly larger coupling. revision: yes

Circularity Check

1 steps flagged

Optimal frequencies linear in degree used to close self-consistent equations for critical coupling

specific steps
  1. fitted input called prediction [Abstract and mean-field section for pure second-harmonic Sakaguchi-Kuramoto model]
    "Here, the obtained optimal natural frequencies scale linearly with the node degree, and the critical coupling strength for the onset of synchronization is derived analytically from the self-consistent equations. In this specific regime, we observe a perfectly ordered two-cluster synchronized state."

    The linear scaling of natural frequencies with degree is introduced as the 'optimal' choice that enables complete synchronization at small coupling; this same scaling is then inserted into the mean-field self-consistent equations to extract the critical K analytically. The resulting critical value is therefore conditioned on the frequency-degree correlation that was selected to produce the desired two-cluster state, reducing the 'prediction' to a direct consequence of the input choice rather than an independent derivation.

full rationale

The paper derives an optimal set of natural frequencies that scale linearly with node degree and then applies the standard mean-field closure to obtain an analytic expression for the critical coupling in the pure second-harmonic case. While the mean-field ansatz itself is standard and the linear scaling is presented as a derived result, the abstract and description indicate that this specific choice of frequencies is what permits the self-consistent equations to close analytically and yield a perfectly ordered two-cluster state. This creates a partial dependence where the reported critical value is conditioned on the very frequency-degree correlation introduced to achieve the desired synchronization, rather than emerging as a fully independent prediction from the microscopic dynamics. No explicit self-citation chain or redefinition of the order parameter is evident from the provided text, so the circularity remains moderate.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mean-field closure for the order parameter, the assumption that the chosen frequencies remain optimal under added noise, and the validity of the self-consistent equation for the second-harmonic case. No new particles or forces are postulated.

free parameters (2)
  • linear scaling constant between natural frequency and node degree
    The abstract states that optimal frequencies scale linearly with degree; the proportionality constant is chosen to achieve r=1 and is therefore a fitted or tuned parameter.
  • bi-harmonic coupling coefficients
    The parameters in the bi-harmonic coupling function are tuned to produce complete synchronization.
axioms (1)
  • domain assumption Mean-field approximation holds for the order-parameter dynamics when frequencies are degree-correlated.
    Invoked to obtain the critical coupling from self-consistent equations in the pure second-harmonic regime.

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