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Amplification of explosive width in complex networksPitambar Khanra,1 Prosenjit Kundu,1 Pinaki Pal,1 Peng Ji,2 and Chittaranjan Hens31)Department of Mathematics, National Institute of Technology, Durgapur 713209,

India2)The Institute of Science and Technology for Brain-inspired Intelligence, Fudan University, Shanghai,

China3)Physics & Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108,

India

(Dated: 15 April 2021)

We present an adaptive coupling strategy to induce hysteresis/explosive synchronization (ES) in complexnetworks of phase oscillators Sakaguchi-Kuramoto model). The coupling strategy ensures explosive synchro-nization with significant explosive width enhancement. Results show the robustness of the strategy and thestrategy can diminish (by inducing enhanced hysteresis loop) the contrarian impact of phase frustration inthe network, irrespective of network structure or frequency distributions. Additionally, we design a set offrequency for the oscillators which eventually ensure complete in-phase synchronization behavior among theseoscillators (with enhanced explosive width) in the case of adaptive-coupling scheme. Based on a mean-fieldanalysis, we develop a semi-analytical formalism, which can accurately predict the backward transition ofsynchronization order parameter.

PACS numbers: 05.45.Xt, 05.45.Gg, 89.75.FbKeywords: Coupled Oscillators, Synchronization, Adaptive Coupling

The phenomenon of synchronization has fasci-nated researchers for many years due to its ap-pearance in variety of natural as well as man-made systems. However, the study of synchro-nization in adaptively coupled complex networkshas got less attention. Here, we have presentedan adaptive coupling scheme and explored theimpact of that in the enhancement of explosivewidth in Sakaguchi-Kuramoto model on complexnetworks. Numerical investigation shows thatthe proposed scheme ensures explosive synchro-nization (ES) with significant explosive widthenhancement for diverse frequency distributionswith different network realizations. More im-portantly, it is found that the proposed couplingscheme can inhibit the contrarian impact of thephase frustration in the network. We also have es-tablished a semi analytical treatment for investi-gating the system using the Ott-Antonsen ansatz.The results obtained from the semi analytical ap-proach are found to match closely with the nu-merical simulation results. The analysis showsthat the adaptive coupling function is robust andcan enhance the hysteresis width in different com-plex networks having variety of natural frequencydistributions and for a broad range of frustrationparameter.

I. INTRODUCTION

Adaptively coupled oscillators have been found to ex-hibit a wide range of complex behaviors1–4. For in-stance, link weights co-evolving with underlying dy-

namics can enhance the synchronizability in heteroge-neous networks5 and this mechanism can induce spon-taneous synchronization in e.g., certain group of neu-rons of a brain by adjusting the adaptive Hebbian learn-ing process6–9. Recently, it has been reported that theadaptation of the coupling configuration by the local or-der parameter may result in explosive synchronization, aphenomenon characterized by discontinuous (first-order)phase transitions between incoherent and coherent statesaccompanied by hysteresis in networks of coupled oscil-lators10–17.Of particular interest, the explosive synchronization18,19

has been found to emerge among adaptively-coupled os-cillators irrespective of the structure of the networks orintrinsic frequency distributions. We raise here the ques-tion what happens if we use a modified strategy, i.e in-corporating the high power of magnitude of the adaptingparameter?Here, we have used the modified adaptive-couping

scheme proposed by Filatrella et al.18 on Sakaguchi-Kuramoto model in complex network environment, andresults show that the proposed scheme can amplify or in-crease the hysteresis width (difference between the back-ward transition and forward transition points of synchro-nization) for diverse frequency distributions with differ-ent network realizations. For demonstration, we startwith the classical network-coupled Sakaguchi-Kuramotophase oscillators of size N20–23. Dynamics of each oscil-lator in the network is governed by the equation

dθi

dt= ωi + λli

N∑

j=1

Aij sin(θj − θi − α), (1)

where ωi is the intrinsic natural frequency of the ith os-cillator, Aij is an ijth element of the adjacency matrix

2

z=1

z=3

z=2

(a)

(b)

(c)

FIG. 1. Expansion of hyeteresis width in small network

Visualization of the expansion of hysteresis width for ER net-work of network size N = 300, 〈k〉 = 12 with increasing z.Right panel synchronization diagram is constructed using themodel Eqn. (1) with phase frustration value α = 0.5. Dou-ble headed arrow in the last figure of right panel indicatesexplosive width of the synchronization transition.

A = (Aij)N×N . Here α accounts for a frustrating termwithin the range of 0 ≤ α < π

224–28 and λ is the cou-

pling strength. li is a time-dependent coupling term andcontributes adiabatically to the coupling function.As a general choice, we consider li as a function of the

Kuramoto order parameter R which quantifies the degreeof synchronization of the entire population. The globalorder parameter is defined as

Rei(ψ−α) =1

N

N∑

j=1

ei(θj−α), (2)

where 0 ≤ R ≤ 1 quantifies the magnitude of coher-ence and ψ denotes the average phase. Prior researchhave shown that the adaptive function li = R can resultin explosive synchronization10,19. However, this strategygenerates a constant hysteresis or explosive width. Weseek here a suitable choice of li which can enhance thehysteresis loop of phase-coupled networks and choose theadaptive function as li = Rz−1, where z (≥ 1) is a posi-tive real number18 .To illustrate the effectiveness of our choice, we con-

sider a heterogeneous Erdos-Renyi (ER) network withN = 3× 102 nodes. We increase (decrease) the couplingstrength λ adiabatically with an increment (decrement)δλ = 0.01 and compute the stationary value of R for eachλ during the forward (backward) transition from the in-coherent to the phase synchronized state. In this simu-lation we take the natural frequencies from a Lorentziandistribution and set the frustration parameter (α) equalto 0.5. In Fig. 1 (a), the order parameter undergoes acontinuous transition for the choice of z = 1 where thenetwork is a purely diffusive i.e. the coupling function isnot controlled by any adaptive coupling. However, set-

ting the adaptive parameter to higher power (z = 2),one can generate discontinuous (explosive) transition inthe synchronization order parameter (see Fig. 1 (b)). In-terestingly, the explosive width is further enhanced if weincrease the value of z from 2 to 3 (Fig. 1 (c)). Thedouble headed arrow in Fig. 1(c) indicates the explosivewidth for the explosive synchronization transition whichis the difference between the backward transition and for-ward transition points of synchronization. In this paper,we have explored the impact of z on transition to syn-chronization in different complex networks of Sakaguchi-Kuramoto oscillators for a broad range of frequency dis-tributions, namely Lorentzian or uniform distributions.We have also considered the degree-frequency correlatedenvironment and a special form of degree-frequency cor-relation which eventually gives perfect synchronization.Our results show that the adaptive coupling function isrobust and can induce the enhanced explosive width fora broad range of frustration parameter over diverse net-work settings. In this work, we validate our detailed nu-merical results by solving (semi-analytically) order pa-rameter equation on the basis of Ott-Antonsen ansatz29.The forward transition curve is perfectly fitted with oursemi-analytical findings.

II. ANALYTICAL APPROACH AND NUMERICAL

VERIFICATION

We start with the annealed network approximationproposed in30–34, which gives critical behavior of phasetransitions (including synchronization transition) in com-plex networks. For the case of a sparse uncorrelated com-plex network with a degree distribution P (k) in the ther-modynamic limit (N → ∞) we may write,

dθi

dt= ωi +

λRz−1ki

N 〈k〉

N∑

j=1

kj sin(θj − θi − α), (3)

where ki is the degree of ith node, and 〈k〉 is the aver-age degree across nodes. To obtain the explicit set ofequations for the time evolution of the complex orderparameter C = Rei(ψ−α), we use here the Ott-Antonsenansatz29 where the density of the oscillators at time twith phase θ for a given degree k and frequency ω begiven by the function f(k, ω; θ, t), and we normalize it as

∫ 2π

0

f(k, ω; θ, t)dθ = h(k, ω) = P (k)g(ω). (4)

To maintain the conservation of the oscillators20,31 in anetwork, the density function f satisfies the continuityequation

∂

∂tf(k, ω; θ, t) +

∂

∂θ[vf(k, ω; θ, t)) = 0, (5)

where v is the velocity field on the circle that drives thedynamics of f , which comes from the right hand side of

3

the Eqn. (3) as follows

v(k, ω; θ, t) =dθ

dt= ω +

λk

2iRz−1

[

Ce−iθ − C∗eiθ]

.(6)

Expanding f(k, ω; θ, t) in a Fourier series in θ, we have

f =h(k, ω)

2π

1 +

[

∞∑

n=1

fn(k, ω, t)einθ + c.c.

]

, (7)

where c.c. stands for complex conjugate. We now con-sider the Ott-Antonsen ansatz

fn = βn(k, ω, t) (8)

to obtain an equation for the function β(k, ω, t) and findthe following equation

∂β

∂t+ iωβ +

λk

2Rz−1(Cβ2 − C∗) = 0, (9)

where |β(k, ω, t)| ≤ 1 to avoid divergence of the series.The above equation is not yet in a closed form. So tomake the above equation in a closed form, we get thecomplex global order parameter as

C(t) =e−iα

〈k〉

∫ ∞

kmin

∫ ∞

−∞

kP (k)g(ω)β∗dωdk. (10)

Now to study a steady state solution we averageC(t) =Reiψ−iα+iΩt for the constant order parameter R, a phaseψ, and a group angular velocity Ω. By suitably varyingthe reference frame, ω 7→ ω−Ω and putting ψ = 0, with-out loss of generality, we have C = Re−iα. Therefore,C∗ = Reiα and for stationary points of Eqn. (9) we find

the solution of β = 0.We obtain the solution as follows,

β(k, ω) =

−ixeiα + eiα√1− x2, |x| ≤ 1.

−ixeiα[

1−√

1−(

1x

)2]

, |x| > 1.(11)

where x = ω−ΩλkRz and this solution satisfies the require-

ment |β| ≤ 1 for the convergence of the geometrical series(Eqn. (8)). The first solution corresponds to the syn-chronous state and the second solution is due to desyn-chronous state. In this case, the order parameter can berewritten as

R =1

〈k〉

[∫ ∫ ∞

kmin

kP (k)g(ω)β∗(k, ω)H (1− |x|) dkdω

+

∫ ∫ ∞

kmin

kP (k)g(ω)β∗(k, ω)H (|x| − 1) dkdω

]

,(12)

where H is Heaviside functions. The first part of right-hand side of Eqn. (12) encompasses the contribution oflocked oscillators and the second part accounts for thecontribution of drift oscillators to the order parameter R.Splitting the contributions of real and imaginary parts,we can eventually obtain the coupled self consistent equa-tions of R and Ω (see in the appendix).

1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

1 1.5 2 2.5 3

(a) (b)

(III)

(II)

(III)

(II)

(I) (I)

FIG. 2. Phase diagram on z − λ plane showing am-

plification of hysteresis width. Magenta (I), yellow (II)and cyan (III) islands respectively represent asynchronous(R ∼ 0), hysteresis and synchronous (R ∼ 1) regions. Theseregions are separated by solid black line indicating the crit-ical coupling strength for the transition to synchrony duringthe forward (λf ) and backward (λb) continuation. Both thefigures are constructed with a scale free (SF) network of sizeN = 2000, 〈k〉 = 12 and γ = 2.8. Panel (a) indicates zerophase frustration i.e., α = 0 and panel (b) indicates α = 0.5.The solid black boundary separating regions (II) and (III)and solid line separating the regions (I) and (II) has beenobtained from numerical simulation. The black diamonds onthis line are obtained from semi-analytical approach (Eqn. 16-17) showing very close match.

To test the impact of z on transition to synchro-nization, we use a heterogeneous scale-free network(N = 2000, 〈k〉 = 12 and γ = 2.8) generated fromBarabasi-Albert model, the frequencies are drawn from aLorentzian distribution with mean 0 and standard devi-ation 1. The bi-stable state (hysteresis regime) is shownby yellow color (regime:II) in the Fig. 2 (a)-(b) for thefrustration parameters α = 0.0 and α = 0.5 respectively.For lower z, the hysteresis width is found to be negligi-bly small and if we increase this adaptive parameter tohigher values, the width is significantly enhanced irre-spective of the frustration parameter. Here the magentaisland (regime: I) shows the de-synchronized regime andcyan island (regime: III) is the locked part of coupledphase oscillators. It is observed from the figure that asz is increased, the region of the hysteresis is amplifiedsignificantly. Interesting to note here that the higheradaptive coupling parameter (z > 1) not only induces ex-plosive synchronization in the network but also enhancesthe explosive width even for high phase frustration (e.g.α = 0.5 in Fig. 2 (b)). Typically in this kind of network,higher phase frustration (α > 0.3) destroys the hyster-isis behaviour in a degree-frequency correlated networkwhen z = 134. However, for higher value of z, it can over-come the situation and explosive synchronization (ES)re-emerges.

The boundary of the yellow and cyan regions whichsignifies the forward critical point (λf (z)) of the hystere-sis transition is computed numerically from the modelEqn. 1. The backward transition points derived from the

4

0 1 2 3 4

0

0.5

1

0 1 2 3 4-15

-10

-5

(a)

(b) (d)

(c)=0.5z=2

z=3

FIG. 3. Order parameter R and group angular velocity

Ω as a coupling strength λ for two values of z. Bluecurve indicates the synchronization diagram and red curve in-dicates the group angular velocity calculated from the modelEqn. (1) for SF network of network size N = 2000, 〈k〉 = 12and γ = 2.8. Black dotted verticle line indicates the backwardtransition point of R and Ω calculated from the semi-analyticEqn. (16) and (17). All the data are simulated with phasefrustration α = 0.5.

self-consistent coupled equations of R and Ω (see the ap-pendix) are shown with black diamonds in Fig. 2 whichshow close match with numerical simulation.

We have also numerically calculated the order param-eter and the global frequency Ω for z = 2 and z = 3(see Fig. 3). The order parameter (R) and global fre-quency (Ω) are shown in the Fig. 3(a,c) with blue linesand Fig. 3(b,d) with red lines respectively as a functionof λ. The backward transition points determined fromthe self-consistent equations are shown with dashed ver-tical lines in the figure which shows a very close matchwith the backwards points obtained through numericalsimulation of the whole system.

For z = 2, the hysteresis width (see inset of Fig. 3(a,b)) occurs due to the presence of adaptive coupling.For z = 3 the width is significantly enhanced comparedto z = 2. Note that, the values of Ω and R are neg-ligibly small before the forward critical coupling. Thereason is as follows: adaptive function R(z−1), acting oneach node of the network, decreases the effective cou-pling (Λeff = λ × R(z−1) where 0 < R < 1 and z > 1)substantially. Therefore, higher strength is required toestablish locked phase. Numerical values of backward Rand Ω closely match with the ones obtained from thesemi-analytical expression of R and Ω derived from theEqn. (12). We therefore establish the fact, that a tun-ing parameter associated with adaptive term in powercan enhance the width of hysteresis irrespective of thefrustration value or network structure. For better real-ization we also simulate our model Eq. 1 in Fig. (4) withWatts-Strogatz small world network size N = 5 × 103,

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 0 1 2 3 4 5

FIG. 4. Realization of the expansion of explosive width

with z for three different values of phase frustration α.

Both the figures are constructed using Watts-Strogatz smallworld network of size N = 500, 〈k〉 = 4, and β = 0.5. Doubleheaded arrow indicates the explosive width of the hysterisistransition.

〈k〉 = 4, and β = 0.5 for three different values of phasefrustration α = 0, 0.3, and 0.6. In every panel of Fig. (4)we have simulate the model with increasing z. We per-form the numerical analysis for diverse set of frequencydistributions, and we show that the enhancement prop-erties appear for all the cases which only depend on thez. The changes in forward and backward points dependon the choices of frequency distributions. Next, we val-idate our proposition in frustrated networks for a cer-tain choice of frequency distribution directly correlatedwith its own degree in which a global order parameterachieves perfect synchronization35 (R = 1). This is im-portant, as z mainly contributes in the enhancement ofhysteresis width irrespective of α. Although α frustratesthe system by not allowing the order parameter to reachperfect synchronization. We seek a selection of frequencydistribution which can avoid the impact of frustration αby reaching perfect synchronization (R = 1) as well asit will create a broader explosive width in presence ofhigher values of z.

III. AMPLIFICATION OF HYSTERESIS WIDTHS IN

DEGREE-FREQUENCY ENVIRONMENT

We have already shown that our adaptive strategyworks efficiently when the frequencies are drawn fromLorentzian distribution. A degree-frequency environ-ment naturally induces ES15,33, however a frustrated net-work (α > 0.334) cannot reveal ES even in the presenceof degree-frequency correlated dynamics. Here we seekto understand the impact of such coupling configurationin frustrated dynamics.We consider a SF network of size N = 2000 with

γ = 2.8 and average degree 〈k〉 = 14, where naturalfrequency of each oscillator scales with its own degreeas ωi = aki where a is a proportionality constant. Weset the phase-frustration parameter α = 1, and we takea = sinα. In this case order parameter R does not ex-hibit the first-order transition for non-adaptive environ-

5

0 1 2 3

0.2

0.4

0.6

0.8

1

0 1 2 3 0 1 2 3

FIG. 5. Amplification of explosive width in case of

degree-frequency correlation. Order parameter R as afunction of coupling strength λ for a SF network of networksize N = 2000, 〈k〉 = 14, and γ = 2.8 for different valuesof z. Blue line indicates the forward transition and red lineindicates backward transition. Magenta dot on the cyan colorline indicates our targeted point at (1, 1)

ment (z = 1) (Fig. 5, shown in left column with blueline). Interestingly one additional characteristic appearsin the behavior of the order parameter, i.e., R approach-ing 1 at λ = 1. This is due to the emergence of perfectsynchronization (R = 1)35,36 for choosing a specific kindof frequency distribution.As we know, in perfect synchronization state, |θj−θi| =

0 and R = 1. Using Taylor’s series approximation, theEqn. (1) can be written as

dθi

dt= ωi − λ cos(α)

∑

Lijθj , (13)

where ωi = ωi−λ sin(α)ki and L is the Laplacian matrix.

In vector form, one can write θ = ω − λ cos(α)Lθ. As-sume that, in global synchronization, all the oscillatorsfollow a common frequency Ω and if we consider a Ω ro-tating frame, the phases will be freezed in this frame bysetting themselves into steady states (dθ

dt= 0). This fur-

ther implies θ∗ = L†ω

λ cos(α) , where L† is the pseudo-inverse

operator of the Laplacian matrix L. Now if we chooseωi = sin(α)ki which sets ω = 0 at λ = 1, and that even-tually gives θ

∗ = 0. This signifies that the frequencyensures perfect synchronization of oscillators (R = 1) atλ = 1 (Magenta dot in each panel of the Fig. 5 on thecyan line). Although the choice of the higher value ofz (= 2) in Rz−1 yields the explosive width (Fig. 5 mid-dle panel). The hysteresis width is significantly increasedfor higher values of z (= 3) (shown in the right panel).

Therefore, we have shown that our adaptive strategy cancreate ES and enhance the ES width in these types of net-works irrespective of the choice of the frequencies. Theanalytical prediction is confirmed in Fig. 5 assuring per-fect synchronization at λ = 1 which also nullifies thecontrarian effect of frustration parameter α. The for-ward line misses the perfect synchronization point (blueline) where as backward line coincides with R = 1 atλ = 1 for each cases of z.IV. CONCLUSIONS

We have investigated the phenomenon of explosive syn-chronization in complex networks of phase oscillatorsboth numerically and analytically, based on an adap-tive coupling strategy. Numerical simulations for differ-ent networks both in frustrated and non frustrated envi-ronment show that, as a parameter (z) in the couplingfunction is tuned appropriately, networks undergo abrupttransition to (explosive) synchronization and width of theassociated hysteresis loop is greatly enhanced. It is ob-served that the proposed strategy is quite general and isapplicable for any type of networks and frequency distri-butions for the amplification of hysteresis width. We haveconfirmed that the strategy can successfully induce theemergence and enhancement of explosive synchronizationin diverse frustrated environments. Based on the Ott-Antonsen ansatz, we have derived analytical expressionsfor the global order parameter and global frequency. Theresults obtained from the semi-analytical approach basedon Ott-Antonsen ansatz match with the backward tran-sition points obtained from the numerical simulation ofthe entire complex networks.

V. ACKNOWLEDGEMENTS

PK acknowledges support from DST, India under theDST-INSPIRE scheme (Code: IF140880). PJ acknowl-edges support from National Key R&D Program of China(2018YFB0904500), NSF of Shanghai, Eastern Scholarand by NSFC 269 (11701096). CH is supported byINSPIRE-Faculty grant (Code: IFA17-PH193).

VI. APPENDIX: COUPLED EQUATION OF R AND Ω

The contribution of locked oscillators to the order pa-rameter is

Rl =(cosα− i sinα)

〈k〉

∫ ∞

kmin

∫

kP (k)g(ω)

√

1−(

ω − Ω

λkRz

)2

+ iω − Ω

λkRz

dkdωH

(

1−∣

∣

∣

∣

ω − Ω

λkRz

∣

∣

∣

∣

)

(14)

6

Now the contribution of drift oscillators to the order parameter is given by

Rd =(sinα+ i cosα)

〈k〉

∫ ∞

kmin

∫

kP (k)g(ω)ω − Ω

λkRz

1−

√

1−(

λkRz

ω − Ω

)2

dkdωH

(∣

∣

∣

∣

ω − Ω

λkRz

∣

∣

∣

∣

− 1

)

(15)

Hence we get R = Rl+Rd, where Rl and Rd are given by Eqn.(14) and Eqn.(15) respectively.Now comparing the real and imaginary parts we get

R〈k〉 = cosα

∫ ∞

kmin

∫

kP (k)g(ω)

√

1−(

ω − Ω

λkRz

)2

H

(

1−∣

∣

∣

∣

ω − Ω

λkRz

∣

∣

∣

∣

)

dkdω +sinα

λRz(〈ω〉 − Ω)

− sinα

∫ ∞

kmin

∫

kP (k)g(ω)ω − Ω

λkRz

√

1−(

λkRz

ω − Ω

)2

H

(∣

∣

∣

∣

ω − Ω

λkRz

∣

∣

∣

∣

− 1

)

dkdω (16)

and

〈ω〉 − Ω = λRz tanα

∫ ∞

kmin

∫

kP (k)g(ω)

√

1−(

ω − Ω

λkRz

)2

H

(

1−∣

∣

∣

∣

ω − Ω

λkRz

∣

∣

∣

∣

)

dkdω

+

∫ ∞

kmin

∫

P (k)g(ω)(ω − Ω)

√

1−(

λkRz

ω − Ω

)2

H

(∣

∣

∣

∣

ω − Ω

λkRz

∣

∣

∣

∣

− 1

)

dkdω (17)

By solving the above two equations we can get the set of values for order parameter R corresponding to couplingstrength λ.

1T. Aoki, and T. Aoyagi, Phys. Rev. Lett. 102, 034101 (2009).2S-Y Ha, S. E. Noh, and J. Park, SIAM 15, 162-194 (2016).3D. V. Kasatkin, and V. I. Nekorkin, EPJST 227, 1051–1061(2018).

4M. Brede and A. C. Kalloniatis Phys. Rev. E 99, 032303 (2019).5M. Chen, Y. Shang, C. Zhou, Y. Wu, and J. Kurths, Chaos 19,013105 (2009).

6S. Chakravartula, P. Indic, B. Sundaram, and T. Killingback,Plos One 12, e0178975 (2017).

7R. K. Niyogi, and L. Q. English, Phys. Rev. E 80, 066213 (2009).8P.S. Skardal, D. Taylor, and J. G. Restrepo, Physica D 267 27-35(2014).

9M. Li, S. Guan, and C-H Lai, New Journal of Physics 12 103032(2010).

10P. Khanra, P. Kundu, C. Hens, and P. Pal, Phys. Rev. E 98,052315 (2018).

11A. Kachhvah and S. Jalan, Euro Phys. Lett. 119, 60005 (2017).12M. M. Danziger, I. Bonamassa, S. Boccaletti, and S. Havlin, Nat.

Phys 15 178–185 (2019).13M. M. Danziger, O. I. Moskalenko, S. A. Kurkin, X. Zhang, S.Havlin, and S. Boccaletti, Chaos 26 065307 (2016).

14R. S. Pinto and A. Saa, Phys. Rev. E 92, 062801 (2015).15J. Gomez-Gardenes, S. Gomez, A. Arenas, and Y. Moreno,Phys.Rev. Lett. 106, 128701 (2011). P. Ji, T. Peron, P. J. Menck,F. A. Rodrigues, and J. Kurths, Phys.Rev. Lett. 110, 218701(2013)

16S. Boccaletti, J.A. Almendral, S. Guan, I. Leyva, Z. Liu, I.Sendina-Nadal, Z. Wan, and Y. Zou, Phys. Rep. 660, 1-94(2016);

17I. Leyva, A. Navas, I. Sendina-Nadal, J. A. Almendral, J. M.Buldu, M. Zanin, D. Papo, and S. Boccaletti, Scientific Reports

3 1281 (2013). Y. Zou, T. Pereira, M. Small, Z. Liu, and J.Kurths, Phys. Rev. Lett. 112 114102 (2014).

18G. Filatrella, N. F. Pedersen, and K. Wiesenfeld, Phys. Rev. E

75, 017201 (2007).19X. Zhang, S. Boccaletti, S. Guan, and Z. Liu, Phys. Rev. Lett.

114 038701 (2015).20Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence,(Springer, New York, 1984).

21J. A. Acebron, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R.Spigler, Rev. Mod. Phys. 77, 137-185 (2005).

22F. Dorfler and F. Bullo, SIAM Journal on Control and Opti-

mization 50(3), 1616-1642 (2012).23F. A. Rodrigues, T. K. D. M. Peron, P. Ji, and J. Kurths, Phys.

Rep. 610, 1-98 (2016).24H. Sakaguchi and Y. Kuramoto, Prog. Theor. Phys. 76, 576(1986).

25O. E. Omel’chenko and M. Wolfrum, Phys. Rev. Lett. 109,164101 (2012).

26M. A. Lohe, Automatica 54, 114-123 (2015).27P. S. Skardal, D. Taylor, J. Sun, and A. Arenas, Phys. Rev. E

91, 010802(R) (2015).28P. S. Skardal, D. Taylor, J. Sun, and A. Arenas, Physica D:

Nonlinear Phenomena 323, 40-48 (2015).29E. Ott, and T. M. Antonsen, Chaos 18, 037113 (2008).30T. K. D. Peron and F. A. Rodrigues, Phys. Rev. E 86, 016102(2012).

31T. Ichinomiya, Phys. Rev. E 70, 026116 (2004).32S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev.

Mod. Phys. 80, 1275 (2008).33B. C. Coutinho, A. V. Goltsev, S. N. Dorogovtsev, and J. F. F.Mendes, Phys. Rev. E 87, 032106 (2013); S. Yoon, M. SorbaroSindaci, A. V. Goltsev, and J. F. F. Mendes, Phys. Rev. E 91,032814 (2015).

34P. Kundu, P. Khanra, C. Hens, and P. Pal, Phys. Rev. E 96,052216 (2017).

7

35P. Kundu, C. Hens, B. Barzel, and P. Pal, Europhys. Lett. 120,40002 (2018).

36M. Brede and A. C. Kalloniatis, Phys. Rev. E 93, 062315 (2016).

37A. Ben-Israel, T. N. E. Greville, Generalized Inverses Theory

and Applications, (Canadian Mathematical Society, Springer,2nd edition, 2002).