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REVIEW 4 major objections 6 minor 61 references

Stochastic Non-Linear Influence in Synchronisation Dynamics

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict The numerical three-regime picture is solid, but the analytic stability section has a concrete algebraic error that invalidates the claimed imaginary-eigenvalue mechanism. read the letter →

arxiv 2608.04749 v1 pith:TBOIPTJU submitted 2026-08-05 math.DS nlin.AOphysics.soc-ph

Hannah Gallant , Dale Roberts , Alexander Kalloniatis This is my paper · ORCID
classification math.DSnlin.AOphysics.soc-ph MSC 34C1560H1005C80
keywords KuramotomodelsynchronisationexternalinfluencestochasticlinkweightsGammanoiseErdos-Renyigraphsentrainmenttransitionphasecoherence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

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.

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 (4)
  1. [Section IV.B, Eqs. (17)–(19)] 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.
  2. [Section IV.B, Eqs. (19) and (22)] 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.
  3. [Section IV.B, Eq. (21)] 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.
  4. [Section IV.B and Appendix C] 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.
minor comments (6)
  1. [Fig. 10 caption] 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.
  2. [Throughout] The spelling of Erdős-Rényi is inconsistent (e.g., 'Erd˝ os-R´ enyi' in section headings), and reference [57] should be Bollobás, not Bollabás.
  3. [Section II, after Eq. (4)] 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.
  4. [Section III.A] 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.
  5. [Eq. (7)] 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.
  6. [Section V] 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').

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the three-regime claim rests on independent numerics and an in-paper linearisation; same-author citations frame the approach but are not load-bearing.

  1. other [Section II (Eqs. 3-4) and Section III.A (Eq. 6), with refs [35], [51]]
    "This formulation can be seen to be consistent with the multiplicative stochastic approach of [51]. ... To understand the external node's ability to entrain the network to its frequency, we define a 'closeness' (Δ) of the average instantaneous frequency to the external node's frequency, similar to [35]."

    References [35], [51] (and [34], [52], [53], [59]) overlap with the present authors, so parts of the paper's framing borrow from the authors' own prior stochastic- and control-Kuramoto work. This is a minor self-citation footprint, not a circular reduction: the model is fully specified in-paper by Eqs. (3)-(4), the closeness metric by Eq. (6), and the three-regime conclusion (captured/stretched/decoupled) is supported by the paper's own simulations (Figs. 1-3) and an in-paper linear analysis (Section IV) with no fitted parameter forcing the target regimes. The cited works supply context and pedigree, not the load-bearing premise; accordingly this step is recorded for transparency and weighs at most 2, not higher.

full rationale

The central claim — a three-regime (captured, stretched, decoupled) response in the stochastic-influence Kuramoto model of Eqs. (3)-(4) — is established by self-contained numerical experiments (Section III, Figs. 1-5) plus an analytic linearisation derived in-paper from the model equations (Section IV, Eqs. 8-23). No parameter is fitted to force the regimes: σ, τ, α and Ω are scanned independently, and the τ=0 baseline is checked against the analytically known Kuramoto critical coupling, an external benchmark. The Section IV.C classification (ϵ̇=δ̇=0 captured; ϵ̇≠0, δ̇=0 stretched; ϵ̇=0, δ̇≠0 decoupled) is a definitional partition of the linearised phase space, but the numerical discovery of three regimes in the full nonlinear model is independent of that partition, so the claim does not reduce to its definition. Same-author citations ([13, 34, 35, 51, 52, 53, 59]) provide context, not load-bearing premises. The paper also states its own scope limit honestly, noting in Appendix C that 'for the linearisation above to be valid, XD must remain small.' The skeptic's algebraic objection to Section IV.B (the discriminant of Eq. (17) equals 4σ² + τ²(Ḡ1−Ḡ2)² ≥ 0, so eigenvalues are always real and the claimed imaginary-eigenvalue kick regime cannot occur, and Eq. (22) appears to drop a factor of τ) is a correctness/internal-consistency issue, not a circularity issue; per the rubric it does not raise the circularity score.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The model rests on the Kuramoto substrate, a postulated Gamma-noise coupling mechanism, and a small-separation linear ansatz. No parameter is fitted to force the target regimes, but sigma, tau, alpha, N, and p are hand-chosen, and the analytic section adds assumptions that are not justified for the stretched regime.

free parameters (5)
  • Influence strength tau = tau in [0,50] in main scans
    Hand-specified control parameter; the captured/stretched/decoupled boundaries and the r and Delta maps depend on it.
  • Gamma shape parameter alpha = alpha = 1 in main results; alpha in {0.25,...,20} in Appendix A
    Controls the variability of the stochastic influence weights; the stochastic-versus-deterministic comparison is a central qualitative conclusion.
  • Internal coupling sigma = sigma = 2
    Chosen so the autonomous ER network synchronises, above the critical coupling; not derived from data or from the influence model.
  • Network size N = N = 300 in main results; N in {40,...,640} for scaling
    Chosen to be near the purposeful-community scale; not derived, and the scaling results show a soft N dependence.
  • ER connection probability p = p = 0.3
    A single hand-chosen graph density is used for all main results, so topology dependence is not established.
assumptions (5)
  • domain assumption The Kuramoto model with degree-normalised internal coupling is an adequate substrate for social influence.
    Eq. (3) adopts the Kuramoto model as the internal dynamics; its applicability to social systems is asserted via cited literature rather than demonstrated in this paper.
  • ad hoc to paper Gamma multiplicative noise with a Lorentzian phase kernel models autonomy-respecting influence.
    Eq. (4) sets B_i = G_i / ((phi_i - theta)^2 + 1); the choice is motivated by qualitative sociology and is not derived or validated against empirical data.
  • ad hoc to paper The two-cluster ansatz with small delta and epsilon is valid for the regimes analysed.
    Section IV.A linearises around small phase separations in Eqs. (15) to (17); the stretched regime by construction has larger separations, so the linear system may not apply there.
  • ad hoc to paper Typical Gamma draws satisfy G1*G2 approximately zero.
    Section IV.B uses this to conclude H is small and sigma-independent; for alpha = 1 the Gamma distribution has mean 1, so the product is not obviously negligible.
  • domain assumption Symmetric equally spaced natural frequencies with mean zero are representative.
    Section III.A sets omega_i evenly spaced on [-1,1]; this removes frequency-distribution effects and may affect the location and sharpness of the transitions.
invented entities (1)
  • External influencer node theta
    purpose: Acts as an outside actor pulling the oscillator network's collective frequency toward its own frequency Omega.
    Introduced as a mathematical construct in Eq. (3); no empirical or experimental handle outside the model is provided.

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Cite this review

Pith. "Pith review of Stochastic Non-Linear Influence in Synchronisation Dynamics." pith.science (2026). https://pith.science/paper/TBOIPTJU

@misc{pith2026260804749,
  author       = {Pith},
  title        = {Pith review of: Stochastic Non-Linear Influence in Synchronisation Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBOIPTJU}},
  note         = {Machine review of arXiv:2608.04749}
}
read the original abstract

We propose a mathematical model that represents the influence of an external actor (influencer) on a network of actors (influenced). The model is an adaptation of control systems within the family of formulations inspired by the Kuramoto Model of synchronisation. Capturing influence as a capacity to affect the character or behaviour of another, we study an external node's ability to influence the synchronisation of the Kuramoto system while its global behaviour is pulled to the external node's frequency and away from its natural mean frequency. In our work, stochastically generated dynamical weights assigned to the links between the network and the external influencer whereby links from one system to the other are assigned via one-sided heavy tail noise, generated by the Gamma distribution. We perform numerical experiments to examine transition points in the ability of the external node to alter the behaviour of the influenced network; either to disrupt synchronisation or to drive it to collective frequencies determined by the external node. We examine the dependence of transition points on the external node's frequency, where too ambitious a driving frequency fails to influence the system while retaining a synchronised state, and reducing achieves a state of synchronisation at a frequency shifted from the mean natural frequency. We also look at the analytic approximation for the system close to synchronisation and fragmentation to understand its behaviour around this limit.

Figures

Figures reproduced from arXiv: 2608.04749 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Ensemble-averaged effective frequency [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: (a) [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Mean internal oscillator frequency [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Relative phase [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

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Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    Stochastic vs deterministic influence links For the deterministic case, which corresponds to ¯G1 = ¯G2 = 1 the eigenvalues are simplyλ={τ,2σ+τ}(thus stable). But as shown in Appendix C, the coefficients in the solutions to the linear system reduce to the ordi- nary Kuramoto case; this more as a consequence of their 10 dependence on the noise difference,δG...

    work page

  2. [2]

    Increasingαshifts the stretching through to larger source frequencies: the minimum moves from Ω≈1.63 atα= 0.25 to Ω≈2.45 atα= 1, and to Ω≈3.7–3.9 for α= 10–20. Thus lower-variance influence links extend the captured regime before the source-frame stretching sets in.As discussed in the introduction, this is represen- tative of an interaction between the ex...

    work page

  3. [3]

    Barrat, A., Barth´ elemy, M., Vespignani, A. (2008)Dy- namical Processes on Complex Networks, Cambridge University Press, New York 14 k=16 k=14 k=13 k=10 k=7 = 0 = 0 = 0.25 = 0.5 = 1 = 1.5 = 2 = 2.5 = 3 k=16 k=14 k=13 k=10 k=7 = 1 k=16 k=14 k=13 k=10 k=7 = 5 k=16 k=14 k=13 k=10 k=7 = 10 k=16 k=14 k=13 k=10 k=7 = 15 0 250 t k=16 k=14 k=13 k=10 k=7 = 20 0 2...

    work page 2008

  4. [4]

    (2025) Influence Indicators: Principles and Practices, UniSA, https://socialinfluence.org.au/reports/

    West, B., Quarisa, C, Thomas, E., Spry, D. (2025) Influence Indicators: Principles and Practices, UniSA, https://socialinfluence.org.au/reports/

    work page 2025

  5. [5]

    (2023) Social Influence: Multi-level influence for engagement and resilience, UniSA,https://socialinfluence.org

    West, B., Thomas, E., Quarisa, C, Spry, D. (2023) Social Influence: Multi-level influence for engagement and resilience, UniSA,https://socialinfluence.org. au/reports/

    work page 2023

  6. [6]

    (2017)Model of Social Influence: Towards the Next Frontiers, JASS20(4) 2

    Flache, A., Mas, M., Feliciani, T., Chattoe-Brown, E., Deffuant, G., Huet, S., Lorenz, J. (2017)Model of Social Influence: Towards the Next Frontiers, JASS20(4) 2

    work page 2017

  7. [7]

    (2009)Statistical physics of social dynamics, Rev

    Castellano, C., Fortuato, S., Loreto, V. (2009)Statistical physics of social dynamics, Rev. Mod. Phys.81

    work page 2009

  8. [8]

    (2020)Low-dimensional dynamics of phase oscillators driven by Cauchy noise, Phys

    Tanaka, T. (2020)Low-dimensional dynamics of phase oscillators driven by Cauchy noise, Phys. Rev. E102 042220

    work page 2020

Show all 61 references
  1. [9]

    (2021)Chimera states in coupled Hindmarsh-Rose neu- rons withα-stable noise, Chaos, Fractals and Fractals 148110976

    Wang, Z., Xu, Y., Yongge, L., Kapataniak, T., Kurths, J. (2021)Chimera states in coupled Hindmarsh-Rose neu- rons withα-stable noise, Chaos, Fractals and Fractals 148110976

    2021

  2. [10]

    (2004) Emergence of Dynamical Order: Synchronization Phe- nomena in Complex SystemsWorld Scientific Publishing, 15 Singapore

    Manrubia, S.C., Mikhailov, A.S., Zanette, D.H. (2004) Emergence of Dynamical Order: Synchronization Phe- nomena in Complex SystemsWorld Scientific Publishing, 15 Singapore

    2004

  3. [11]

    (2010) Human-robot ensemble between robot thereminist and human percussionist using coupled oscillator model

    Mizumoto, T., Otsuka, T., Nakadai, K., Takahashi, T., Komatani, K., Ogata, T., Okuno, H.G. (2010) Human-robot ensemble between robot thereminist and human percussionist using coupled oscillator model. In IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Taipei, Taiwan, 19...

    2010

  4. [12]

    Strogatz, S.H., Abrams, D.M., McRobie, A., Eckhardt, B., Ott, E. (2005). Theoretical mechanics: Crowd Syn- chrony on the Millenium Bridge, Nature, 438(7064), 43- 44

    2005

  5. [13]

    Neda, Z., Ravasz, E., Vicsek, T., Brechet, Y., Barabasi, A.L. (2000). Physics of the rhythmic applause. Phys.Rev.E 61(6) 6987-6992

    2000

  6. [14]

    (2006) Opinion dynamics and synchronisation in a net- work of scientific collaborations

    Pluchino, A., Boccaletti, S., Latora, V., Rapisarda, A. (2006) Opinion dynamics and synchronisation in a net- work of scientific collaborations. Physica A: Statistical Mechanics and its Applications. 372(2) 316-325

    2006

  7. [15]

    (2020) Modelling distributed decision-making in Com- mand and Control using stochastic network synchronisa- tion

    Kalloniatis, A.C., McLennan-Smith, T.A, Roberts, D.O. (2020) Modelling distributed decision-making in Com- mand and Control using stochastic network synchronisa- tion. European Journal of Operational Research 284(2) 588-603

    2020

  8. [16]

    (1984)Chemical Oscillations, Waves, and Turbulence, Springer, Berlin

    Kuramoto, Y. (1984)Chemical Oscillations, Waves, and Turbulence, Springer, Berlin

    1984

  9. [17]

    (2026)Half a century of the theory of syn- chronization, J

    Kuramoto, Y. (2026)Half a century of the theory of syn- chronization, J. Stat. Mech. 044001

    2026

  10. [18]

    (2000)From Kuramoto to Crawford: ex- ploring the onset of synchronization in populations of coupled oscillators, Physica D143

    Strogatz, S.H. (2000)From Kuramoto to Crawford: ex- ploring the onset of synchronization in populations of coupled oscillators, Physica D143

    2000

  11. [19]

    (2005)The Kuramoto Model: A sim- ple paradigm for synchronization phenomena, Rev

    Acebr´ on, J.A., Bonilla, L.L., P´ erez Vincente, C.J., Ri- tort, F., Spigler, R. (2005)The Kuramoto Model: A sim- ple paradigm for synchronization phenomena, Rev. Mod. Phys.77137

    2005

  12. [20]

    (2016)The Kuramoto model in complex networks, Phys

    Rodrigues, F.A., Peron, T.K.DM., Ji, P., Kurths, J. (2016)The Kuramoto model in complex networks, Phys. Rep.6101-98

    2016

  13. [21]

    (2001)A timely account of the role of duration in decision making, Acta Psychologica108 187

    Ariely, D., Zakay, D. (2001)A timely account of the role of duration in decision making, Acta Psychologica108 187

    2001

  14. [22]

    (2009)Experts and Decision Making: First Steps towards a Unifying Theory of Deci- sion Making in Novices, Intermediates and Experts, MPI Collective Goods Preprint, 2009/2

    Herbig, B., Gl¨ ockner, A. (2009)Experts and Decision Making: First Steps towards a Unifying Theory of Deci- sion Making in Novices, Intermediates and Experts, MPI Collective Goods Preprint, 2009/2

    2009

  15. [23]

    (1988) Chaos in Random Neural NetworksPhys

    Sompolinsky, H., Crisanti, A., Sommers, H.J. (1988) Chaos in Random Neural NetworksPhys. Rev. Lett.61 3

    1988

  16. [24]

    Ghosh, D., Frasca, M., Rizzo, A., Majhi, S., Rakshit, S., Alfaro-Bittner, K., Boccaletti, S.The synchronized dynamics of time-varying networks, Phys. Rep.949

  17. [25]

    (2008)Synchroniza- tion in interacting populations of heterogeneous oscilla- tors with time-varying coupling, Chaos,18037114

    So, P., Cotton, B.C., Barreto, E. (2008)Synchroniza- tion in interacting populations of heterogeneous oscilla- tors with time-varying coupling, Chaos,18037114

    2008

  18. [26]

    (2023)The Kuramoto model on dynamic random graphs, Nonlinearity,366177

    Groisman, P., Huang, R.,Vivas, H. (2023)The Kuramoto model on dynamic random graphs, Nonlinearity,366177

    2023

  19. [27]

    (2019) Synchronisation in Time-Varying Random Networks with Varying Connectivity, Scientific Reports910207

    Faggian, M., Ginelli, F., Rosas, F., Levnajic, Z. (2019) Synchronisation in Time-Varying Random Networks with Varying Connectivity, Scientific Reports910207

    2019

  20. [28]

    (2015)Con- trolling synchrony in a network of Kuramoto oscillators with time-varying coupling, Physica D301–302

    Leander, R., Lenhart, S., Protopopescu, V. (2015)Con- trolling synchrony in a network of Kuramoto oscillators with time-varying coupling, Physica D301–302

    2015

  21. [29]

    (2006)Dynamical Weights and En- hanced Synchronization in Adaptive Complex Networks Phys

    Zhou, C., Kurths, J. (2006)Dynamical Weights and En- hanced Synchronization in Adaptive Complex Networks Phys. Rev. E96164102

    2006

  22. [30]

    (2016)Concurrent enhancement of percolation and synchronization in adaptive networks, Scientific Reports627111

    Eom, Y., Boccaletti, S. (2016)Concurrent enhancement of percolation and synchronization in adaptive networks, Scientific Reports627111

    2016

  23. [31]

    (1994)Optimal Control and Estimation Dover, New York

    Stengel, R.F. (1994)Optimal Control and Estimation Dover, New York

    1994

  24. [32]

    (2023)Con- trolling complex networks with complex nodes, Nat

    D’Souza, R.M., di Bernardo, M., Liu, Y. (2023)Con- trolling complex networks with complex nodes, Nat. Rev. Phys.5250-262

    2023

  25. [33]

    (2016)Control principles of complex systems, Rev

    Liu, Y., Barabasi, A. (2016)Control principles of complex systems, Rev. Mod. Phys.88

    2016

  26. [34]

    (2011)Controllability of complex networksNature473167-173

    Liu, Y., Slotine, J., Barabasi, A. (2011)Controllability of complex networksNature473167-173

    2011

  27. [35]

    (2017) Using Hamiltonian control to desynchronize Kuramoto, Phys

    Gjata, O., Asllani, M., Barletti, L., Carletti, T. (2017) Using Hamiltonian control to desynchronize Kuramoto, Phys. Rev. E.,95022209

    2017

  28. [36]

    (2016)Frustration tun- ing and perfect phase synchronization in the Kuramoto- Sakaguchi modelPhys

    Brede, M., Kalloniatis, A.C. (2016)Frustration tun- ing and perfect phase synchronization in the Kuramoto- Sakaguchi modelPhys. Rev. E.93062315

    2016

  29. [37]

    (2019)Controlling and enhancing synchronization through adaptive phase lags Phys

    Kalloniatis, A.C., Brede, M. (2019)Controlling and enhancing synchronization through adaptive phase lags Phys. Rev. E.99032303

    2019

  30. [38]

    (2015)Control of coupled oscil- lator networks with application to microgrid technologies Sci

    Skardal, P.S., Arenas, A. (2015)Control of coupled oscil- lator networks with application to microgrid technologies Sci. Adv.17

    2015

  31. [39]

    (2007)Controllability of complex networks via pinning, Phys

    Sorrentino, F., di Bernardo, M., Garofalo, F., Chen, g. (2007)Controllability of complex networks via pinning, Phys. Rev. E75046103

    2007

  32. [40]

    Porfiri, M., di Bernardo (2008)Criteria for global pinning-controllability of complex networks, Automomat- ica44

    2008

  33. [41]

    (2009)Statisti- cal physics of social dynamics, Rev

    Castellano, C., Fortunato, S., Loreto, V. (2009)Statisti- cal physics of social dynamics, Rev. Mod. Phys81591

    2009

  34. [42]

    (2018)The Anatomy of Friendship, Trends in Cognitive Sciences221

    Dunbar, R.I.M. (2018)The Anatomy of Friendship, Trends in Cognitive Sciences221

    2018

  35. [43]

    Dunbar, R.I.M (2015)Do online social media cut through the constraints that limit the size of offline social net- works?R. Soc. open sci.5150292

    2015

  36. [44]

    (2014)Fractal multi- level organisation of human groups in a virtual world, Sci

    Fuchs, B., Sornette, D., Thurner, S. (2014)Fractal multi- level organisation of human groups in a virtual world, Sci. Rep.46526

    2014

  37. [45]

    (2023)A spectrum of complexity uncovers Dunbar’s num- ber and other leaps in social structureChaos, Solitons and Fractals170113389

    Saavedra, M., Mira, J., Mu˜ nuzuri, A.P., Seoane, L. (2023)A spectrum of complexity uncovers Dunbar’s num- ber and other leaps in social structureChaos, Solitons and Fractals170113389

    2023

  38. [46]

    (2021)Why do people partic- ipate in small online communitiesProc

    Hwang, S., Foote, J.D. (2021)Why do people partic- ipate in small online communitiesProc. ACM Hum.- Comput. Interact., Vol.5, No.CSCW2, Article 462. https://doi.org/10.1145/3479606

    2021 doi

  39. [47]

    (2006)Evidence for time-variant decision making, Eur

    Ditterich, J. (2006)Evidence for time-variant decision making, Eur. J. Neurosci24, 3628

    2006

  40. [48]

    (2014)Hand movementsreflect com- petitive processing in numerical cognition, Can

    Faulkenberry, T.J. (2014)Hand movementsreflect com- petitive processing in numerical cognition, Can. J. Exp. Psychol.68(3)147

    2014

  41. [49]

    (2009)The Analysis of Cognitive Processes with Stochastic Diffusion Models, Habilitation, Heidel- berg University

    Voss, A. (2009)The Analysis of Cognitive Processes with Stochastic Diffusion Models, Habilitation, Heidel- berg University

    2009

  42. [50]

    (2013)Diffusion Mod- els in Experimental Psychology: A Practical Introduction, Exp

    Voss, A., Nagler, M., Lerche, V. (2013)Diffusion Mod- els in Experimental Psychology: A Practical Introduction, Exp. Psych.606

    2013

  43. [51]

    (2007)Influence of noise on the synchronisation of the stochastic Kuramoto model

    Bag, B.C., Petrosyan, K., Hu, C.-K. (2007)Influence of noise on the synchronisation of the stochastic Kuramoto model. Phys.Rev.E 76(5) 056210

    2007

  44. [52]

    (2008).Phase synchronization on scale-free networks in the presence of 16 noise

    Khoshbakht, H., Shabazi, F., Samani, K.A. (2008).Phase synchronization on scale-free networks in the presence of 16 noise. J.Stat.Mech.Theory Exp. 10, 10020

    2008

  45. [53]

    (2013)Stochastic (in)stability of synchronisation of oscillators on net- works

    Zuparic, M.L., Kalloniatis, A.C. (2013)Stochastic (in)stability of synchronisation of oscillators on net- works. Phys.D.255, 13-51

    2013

  46. [54]

    (2017)Synchronisa- tion of networked Kuramoto oscillators under L´ evy noise

    Kalloniatis, A.C., Roberts, D.O. (2017)Synchronisa- tion of networked Kuramoto oscillators under L´ evy noise. Phys.A.466, 476-491

    2017

  47. [55]

    (2018).Synchronisation under shocks: the L´ evy Kuramoto model

    Roberts, D., Kalloniatis, A.C. (2018).Synchronisation under shocks: the L´ evy Kuramoto model. Phys.D. 368, 10-21

    2018

  48. [56]

    Kuramoto networks with infinitely many sta- ble equilibria (2022)

    Sclosa, D. Kuramoto networks with infinitely many sta- ble equilibria (2022). SIAM J. Appl. Dyn. Syst. 22, 3267

    2022

  49. [57]

    (2021)Global Stochastic Synchronization of Kuramoto-Oscillator Networks With Distributed Control IEEE Transactions on Cybernetics5112

    Wu, J., Li, X. (2021)Global Stochastic Synchronization of Kuramoto-Oscillator Networks With Distributed Control IEEE Transactions on Cybernetics5112

    2021

  50. [58]

    (2024)Stochas- tic synchronization of Kuramoto-oscillator network with pinning controlPhys

    Zhu, J., Li, S., Sun, Y., Shi, H., Ma, C. (2024)Stochas- tic synchronization of Kuramoto-oscillator network with pinning controlPhys. Scr.995

    2024

  51. [59]

    (1998)Modern Graph Theory, Springer, New York

    Bollab´ as, B. (1998)Modern Graph Theory, Springer, New York

    1998

  52. [60]

    Gao, J., Liu,Y., D’Souza, R.M., Barab´ asi (2014)Target control of complex networks, Nature Comms

    2014

  53. [61]

    (2017) Gaussian noise and the two-network frustrated Kuramoto model, Physica D: Nonlinear Phenomena341, 10–32

    Holder, A.B., Zuparic, M.L., Kalloniatis, A.C. (2017) Gaussian noise and the two-network frustrated Kuramoto model, Physica D: Nonlinear Phenomena341, 10–32

    2017

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