# Onset of synchronization of Kuramoto oscillators in scale-free networks.

TL;DR: It is provided compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent 23, and it is shown that thecritical coupling remains finite, in agreement with HMF calculations.

Abstract: Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent 2 3, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators.

## Summary (2 min read)

### I. INTRODUCTION

- Synchronization processes are pervasively observed in a wide range of physical, chemical, technological, and biological systems [1] .
- Arguably, one of the most studied models in this context is the one proposed by Kuramoto [2] , which in the last decade was extensively investigated when the oscillators are placed on complex networks (see [3, 4] and references therein).
- One of these problems is whether the critical coupling strength for the onset of synchronization remains finite in the thermodynamic limit for scale-free (SF) networks characterized by a power-law degree distribution with an exponent 2 < γ.
- In Sec. III, the authors compare the estimations by mean-field theories with numerical simulations.

### II. MEAN-FIELD THEORIES FOR PHASE OSCILLATORS IN HETEROGENEOUS NETWORKS

- The authors provide a brief review of the main analytical approximations used to deal with ensembles of phase oscillators in heterogeneous networks.
- In contrast to the case of globally connected populations, the original analytical treatment via a self-consistent analysis by Kuramoto [2] cannot be directly extended to the network case.
- Instead, an exact decoupling is only achieved by defining local order parameters as EQUATION which leads to EQUATION.
- Therefore, in this paper, the authors evaluate the onset synchronization numerically using the standard order parameter R in Eq. ( 2).
- The authors goal is to systematically investigate the behavior of the onset of synchronization as the size of SF networks increases, comparing the theoretical predictions provided by the current mean-field approaches.

### IV. COUPLING NORMALIZATION

- The size dependence of the onset of synchronization on the system's size brings back to attention a topic intensively debated in early studies of network synchronization [3, 4] , namely, the choice for the normalization of the coupling function.
- The authors compare the impact of different prescriptions for the coupling function in large heterogeneous networks in the light of the latter points.
- Reasonable choices for N i would then be quantities that are related to the network topology.
- Curiously, this conclusion is not evident from early works [3, 4] . if the problem of finding the appropriate normalization has been solved: Nevertheless, while this choice leads to a finite onset of synchronization in the thermodynamic limit-and moreover sets the same K c for all heterogeneous networks-it imposes a vanishing coupling strength to low connected nodes.

### V. CONCLUSION AND DISCUSSION

- The authors have analyzed the onset of synchronization of Kuramoto phase oscillators in scale-free networks.
- Specifically, both theories predicted a vanishing critical value for the onset of synchronization.
- The authors pointed out that this is a noticeable difference between the critical properties of synchronization of phase oscillators and the SIS dynamics.
- Nevertheless, in this regime of γ , the latter approximation estimates correctly secondary peaks in susceptibility curves associated with localization effects due to the epidemic activation of the largest hub in the network-a phenomenon for which the authors have not observed a counterpart in the synchronization dynamics of large SF networks.
- While this prescription could be appropriate in cases in which the focus of the analysis is not on the role played by the network topology in the dynamics (e.g., [36] ), it seems counterintuitive that large heterogeneous networks should synchronize similarly as homogeneous ones.

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8,690 citations

### "Onset of synchronization of Kuramot..." refers background in this paper

...Another possible source of disagreement between mean-field theories and numerical simulations in the estimation of the critical coupling strength is the consideration of different definitions of order parameters [3, 8]....

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...[8] Stefano Boccaletti, Vito Latora, Yamir Moreno, Martin Chavez, and D-U Hwang, “Complex networks: Structure and dynamics,” Physics Reports 424, 175–308 (2006)....

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...Subsequent theoretical approaches [6, 7] estimated via mean-field approximations that, in the absence of degree-degree correlations, the critical coupling should converge to zero as the number of oscillators tends to infinity – similarly to what happens for other dynamical processes on networks [8]....

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6,169 citations

### "Onset of synchronization of Kuramot..." refers background in this paper

...[1] Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization: a universal concept in nonlin-...

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...Synchronization processes are pervasively observed in a wide range of physical, chemical, technological and biological systems [1]....

[...]

2,552 citations

### "Onset of synchronization of Kuramot..." refers background or methods in this paper

...The size dependence of the onset of synchronization on the system’s size brings back to attention a topic intensively debated in early studies of network synchronization [3, 4], namely, the choice for the normalization of the coupling function....

[...]

...Curiously, this conclusion is not evident from early works [3, 4]....

[...]

...The relationship between structure and synchronous dynamics has been studied in many scenarios: from homogeneous and unclustered networks to heterogeneous and modular ones, in addition to variations of phase oscillator models including correlations between intrinsic dynamics and local topology [3, 4]....

[...]

...Arguably, one of the most studied models in this context is the one proposed by Kuramoto [2], which in the last decade was extensively investigated when the oscillators are placed on complex networks (see [3, 4] and references therein)....

[...]

...One of these prescriptions discussed in previous works is Ni = kmax ∀i [3, 4]....

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^{1}, Charles III University of Madrid

^{2}, University of Barcelona

^{3}, Leonardo

^{4}

2,463 citations

### "Onset of synchronization of Kuramot..." refers background or methods in this paper

...In contrast to the case of globally connected populations, the original analytical treatment via a selfconsistent analysis by Kuramoto [2] cannot be directly extended to the network case....

[...]

...[2] Juan A Acebrón, Luis L Bonilla, Conrad J Pérez Vicente, Félix Ritort, and Renato Spigler, “The Kuramoto model: A simple paradigm for synchronization phenomena,” Reviews of modern physics 77, 137 (2005)....

[...]

...In order to assess the overall synchrony of an ensemble of oscillators, Kuramoto [2] introduced the order parameter...

[...]

...Arguably, one of the most studied models in this context is the one proposed by Kuramoto [2], which in the last decade was extensively investigated when the oscillators are placed on complex networks (see [3, 4] and references therein)....

[...]

1,017 citations

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##### Frequently Asked Questions (2)

###### Q2. What are the future works in "Onset of synchronization of kuramoto oscillators in scale-free networks" ?

Future research should, however, investigate the limits of the self-consistent approach in predicting the critical points of the second-order model in light of the recent results present in [ 42 ]. A systematic study about the dependence of the nature of the synchronization transition on N as well as on the distribution of connection weights for SF networks made up of secondorder Kuramoto oscillators is also an interesting topic for future research.