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cate/physa

Physica A 360 (2006) 629–636

www.elsevier.com/lo

A

0

d

Uniform synchronous criticality of diversely

a

random complex networks

Xiang Li�

Department of Automation, Shanghai Jiao Tong University, Shanghai 200030, PR Chin

Received 28 December 2004; received in revised form 31 May 2005

Available online 14 July 2005

bstract

imit-cycle

coupling

pologies.

nclusion.

bed by ahas everars. Tillorks [3]risingly,f small-systems,

networknetworks

We investigate collective synchronous behaviors in random complex networks of l

oscillators with the non-identical asymmetric coupling scheme, and find a uniform

criticality of collective synchronization which is independent of complexity of network to

Numerical simulations on categories of random complex networks have verified this co

r 2005 Elsevier B.V. All rights reserved.

Keywords: Collective synchronization; Asymmetry coupling; Random graph; Scale-free

1. Introduction

Traditionally, a network with complex topology is mathematically descrirandom graph from the ER model proposed by Erdos and Renyi [1], whichdominated scientists’ thinking about complex networks for almost 40 yerecently, the discoveries of small-world networks [2] and scale-free netwpioneered a revolution in the theoretic study of complex networks, and, surpmany real-life large-scale complex networking systems are in the categories oworld and(or) scale-free networks, ranging from biological to engineeringand even to social and economic systems [4–8].

It has been argued that network topologies significantly affect the emergentdynamics [9], and the small-world phenomena and scale-free features of complex

�Fax: +86 21 62932344.

378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved.

oi:10.1016/j.physa.2005.06.041

E-mail address: xli@sjtu.edu.cn.

have led to a fascinating set of common problems concerning how the complexity of[10].natural

terature,ing [11].study byics anded limit-etely de-

nding of[15–17].a largergloballyf criticalnnected

play anetworksrence ofout thatodes aregeneraluplings.hronousschemesuestion,ength to

evolvingre non-couplingthe sameorks byation.

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X. Li / Physica A 360 (2006) 629–636630

network topology facilitates and constrains synchronous behaviors of networksCollective synchronization in a large population of oscillators having

different frequencies is one of the focal points in the synchronization liwhich is a typical phenomenon in the fields of biology, physics, and engineerAfter the first mathematical description by Wiener [12] and the later fruitfulWinfree [13], Kuramoto refined this connection between nonlinear dynamstatical physics, and formalized the solution to a network of globally couplcycle oscillators [14], answering the situation why the oscillators are complsynchronized until the coupling strength overcomes a criticality Csyn.

The recent discoveries of topological complexity accelerate the understathis cooperative phenomenon in the frame of random complex networksEspecially, Hong et al. reported their synchronization observations ofcriticality of coupling strength on small-world networks than that ofcoupled networks [15]. And, the latest investigation stated the absence ocoupling strength in frequency synchronization of a swarm of oscillators coas a scale-free network having a power-law exponent 2ogp3 [16].

All these fruits indicate that the topology of complex networks doesimportant role in collective synchronization, and different categories of nmay hold significantly different critical coupling strengths for the occursynchrony in mutually coupled oscillators. However, it should be pointedmost of these issues hold an assumption that every pair of connected ncoupled together with the identical coupling strength. In practice, it is morethat pairs of nodes are connected with non-identical and asymmetric coTherefore, it is natural to ask the question that whether the collective syncbehaviors of complex networks with non-identical and asymmetric couplingstill depend on the network topologies? In this paper, we try to explore this qand the answer is that there does exist a uniform criticality of coupling strsynchronize diversely random networks of limit-cycle oscillators.

The whole paper is organized as follows. Section 2 describes the phasemodel of a complex network of limit-cycle oscillators, whose couplings aidentical and asymmetric as specified in this work. A uniform criticalstrength for collective synchrony among oscillators is analytically arrived atsection, whose validity is verified for categories of random complex netwnumerical simulations in Section 3. Section 4 finally concludes this investig

2. Model description and main result

s yi; i ¼

(1)

We consider a network of N coupled limit-cycle oscillators whose phase1; 2; . . . ;N; evolve as

dyi

dt¼ oi þ

XN

j¼1

Cijaij sinðyj � yiÞ ,

where Cij is the coupling strength between node (oscillator) i and node (oscillator) j,quenciesdistribu-

(2)

he given

(3)

e classic

(4)

he given

(5)

(3), andequencyerefore,

(6)

ð7Þ

(8)

(9)

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X. Li / Physica A 360 (2006) 629–636 631

and aij is 1(or 0) if node i is connected (or disconnected) with node j. Freoi; i ¼ 1; 2; . . . ;N; are randomly distributed following the given frequencytion gðoÞ, which assumes that gðoÞ ¼ gð�oÞ.

Define the non-identical asymmetric coupling scheme

Cij ¼ Cj ¼C

ki

; i; j ¼ 1; . . . ;N ,

where C is a positive constant, and ki is the degree of node i, which fits tdegree distribution PðkÞ of a network. Therefore, we have

dyi

dt¼ oi þ

C

ki

XN

j¼1

aij sinðyj � yiÞ .

If for every node i, its degree ki ¼ N; i ¼ 1; 2; . . . ;N, model (3) equals thKuramoto model for globally coupled networks in Refs. [9,11,14].

Define the order parameter ðr;CÞ as [16]

reiC ¼

RdoRdkRdy gðoÞPðkÞkrðk;o; t; yÞeiyR

dkPðkÞk,

where rðk;o; t; yÞ is the density of oscillators with phase y at time t with tfrequency o and degree k, which satisfies the normalization asZ 2p

0

rðk;o; t; yÞdy ¼ 1 .

Assume v to be the continuum limit of the right-hand side (r.h.s.) of Eq.each randomly selected edge is connected to the oscillator having degree k, fro, and phase y with probability kPðkÞgðoÞrðk;o; t; yÞ=

Rdk PðkÞk. Th

determined by the continuity equation

@r@t¼ �

@ðrvÞ

@y,

the density rðk;o; t; yÞ evolves as

@rðk;o; t; yÞ@t

¼ �@

@yrðk;o; t; yÞ½

� oþC

kk

Rdo0

Rdk0Rdy0gðo0ÞPðk0Þk0rðk0;o0; t; y0ÞR

dk0 Pðk0Þk0sinðy0 � yÞ

� ��.

Substituting Eq. (4) into Eq. (7) yields

@rðk;o; t; yÞ@t

¼ �@

@yfrðk;o; t; yÞ½oþ Cr sinðC� yÞ�g ,

whose solution independent of time is

@

@yfrðk;o; yÞ½oþ Cr sinðC� yÞ�g ¼ 0 ,

where rðk;o; yÞ could be assumed to be8(10)

rmalizederefore,

ð11Þ

ð12Þ

ð13Þ

(14)

coupled

mmetricollectivedegree

is point

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X. Li / Physica A 360 (2006) 629–636632

rðk;o; yÞ ¼d y� arcsin o

Cr

� �� �if joj

Crp1 ;

Dðk;oÞjo�Cr sin yj otherwise :

<:

Here, we assume C ¼ 0 without loss of generality, and Dðk;oÞ is the nofactor. Recall gð�oÞ ¼ gðoÞ, and substitute Eq. (10) into Eq. (4). We, tharrive at

r ¼

RdkRCr

�Crdo gðoÞkPðkÞei arcsinðo=CrÞR

dk kPðkÞ,

¼

Z Cr

�Cr

do gðoÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

oCr

� �2r,

¼ Cr

Z p2

�p2

cos2 ygðCr sin yÞdy .

From this self-consistency equation we finally come to

Csyn ¼2

pgð0Þ,

which is surprisingly the same as the critical coupling strength of globallynetworks investigated by Kuramoto [11,14].

The main point of criticality (14) is that, with the non-identical and asycoupling (2), different networks may exhibit the same transition of csynchronous behaviors regardless of the significant difference of theirdistributions, i.e., the complexity of network topology. We will verify ththrough extensive simulations.

3. Numerical simulations

(15)

intrinsicly. In allhe samewith the

(16)

Define the average order parameter as

rav ¼

PNi¼1 kie

iyiPNi¼1 ki

" #* +,

where h� � �i and ½� � �� denote the averages over different realizations ofnetworks and over different realizations of intrinsic frequencies, respectivethe simulations, rav is averaged over ten groups of networks satisfying tdegree distribution PðkÞ, and each network has five sets of frequenciesdistribution gðoÞ. We further specify

gðoÞ ¼0:5 if � 1ooo1 ;

0 otherwise :

Therefore, Csyn � 1:273 by Eqs. (14) and (16).

We first numerically simulate the case of scale-free networks of the BA model [18]�3 1 and 2

m0 ¼ 3,at evenaverage

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X. Li / Physica A 360 (2006) 629–636 633

whose degree distribution is in the power-law form of PðkÞ / k . Figuresshow the simulating outcomes on scale-free networks of BA model with m ¼

and N ¼ 256; 512; 1024; 2048, respectively. It can be clearly observed thhaving the finite-size effect for N ¼ 256; 512; 1024; 2048, respectively, the

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

r av

N=256N=512N=1024N=2048

Csyn=2/g(0)π

Fig. 1. The average order parameter rav vs the coupling strength C for scale-free networks of the BA

model with N ¼ 256; 512; 1024; 2048, respectively. All networks are started from m ¼ m0 ¼ 3.

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

C

r avN

0.25

N=256N=512N=1024N=2048

Csyn=2/g(0)π

Fig. 2. The average ravN0:25 vs the coupling strength C for scale-free networks of the BA model with

N ¼ 256; 512; 1024; 2048, respectively. All networks are started from m ¼ m0 ¼ 3.

order parameter rav shows a rapid increase after the critical coupling strengthcillatorse on the

degreethat theetwork,-coupled

PðkÞ, werks havethe next

s are inous ERributiongeneratekÞ / k�g

] owns adegree

ulations

lity (14).in thesee orderIn otherot showcoupledno suchcomplex

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X. Li / Physica A 360 (2006) 629–636634

Csyn ¼ 2=gð0Þp, indicating that when CsynpC, the scale-free network of osbegin to cluster with synchrony. In Fig. 2, ravN0:25 shows a similar dependenccoupling strength C, which rapidly increased when CsynpC.

From this simulation example of the same scale-invariant power-lawdistribution PðkÞ / k�3 with different network scale N, we clearly concludenonzero constant criticality Csyn is independent of the finite size of a nshowing a difference from the investigated case of scale-free identicaloscillators in Ref. [16].

To investigate the independence of Csyn on different degree distributionsfix the scale N ¼ 2048, and the average degree hki ¼ 6. Therefore, all netwothe same number of nodes and edges with different connectivity patterns insimulation studies.

Two main categories of degree distributions of random complex networkthe forms of PpowerðkÞ / k�g and PexpðkÞ / e�k. Therefore, we select the fammodel [1] to generate networks having an exponential degree distPexpðkÞ / e�k, and nominate the BA model and the GKK model [20] tonetworks having the scale-invariant power-law degree distribution Ppowerð

with g ¼ 3; 6, respectively. The proposed local-world evolving network [19transition between the exponential degree distribution and the power-lawdistribution, therefore it is adopted as the final prototype for the coming simwith the parameters M ¼ 10, m ¼ m0 ¼ 3.

The simulating results of Figs. 3 and 4 are not out of expectation of criticaThere is a common critical coupling strength of the value Csyn ¼ 2=gð0Þpfour categories of networked limit-cycle oscillators, and both the averagparameter rav and the average ravN

0:25 increased sharply when CsynpC.words, the significant difference among complex network topologies does neffect on the collective synchrony of the non-identically asymmetricallylimit-cycle oscillators (3). However, as concluded from Refs. [15–17], there isa same critical coupling strength of collective synchrony in different randomnetworks of identically symmetrically coupled limit-cycle oscillators.

4. Conclusion

viors ofrengths,exity ofetworks,roposedriticalityte is theplicatednetwork

To summarize, we have investigated the collective synchronous behacomplex networks of oscillators having non-identical asymmetric coupling stwhich show a uniform synchrony criticality regardless of the complnetworking topologies. One of the direct extensions is that for scale-free nwhen synchronization is not preferred in some practical situations [21], the pasymmetric coupling (2) could be applied to obtain a nonzero synchrony cwhich is independent of the network topologies. More important to noinvestigation in this paper indicates that, with the embedment of comcoupling schemes, the interactions previously observed between the

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0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

C

r av

ER modelLocal World modelBA model GKK model

Csyn=2/g(0)π

Fig. 3. The average order parameter rav vs the coupling strength C for networks generated by the ER

model (dashed line with circle markers), the Local-World model (dotted line with square markers), the BA

model (solid line with diamond markers), and the GKK model having power-law exponent g ¼ 6 (dash-

dot line with right triangle markers). All networks have the same scale N ¼ 2048 and the same average

degree hki ¼ 6.

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

C

r avN

0.25

ER modelLocal World modelBA modelGKK model

Csyn=2/g(0)π

Fig. 4. The average ravN0:25 vs the coupling strength C for networks generated by the ER model (dashed

line with circle markers), the Local-World model (dotted line with square markers), the BA model (solid

line with diamond markers), and the GKK model having power-law exponent g ¼ 6 (dash-dot line with

right triangle markers). All networks have the same scale N ¼ 2048 and the same average degree hki ¼ 6.

X. Li / Physica A 360 (2006) 629–636 635

collective behaviors and network complex topologies are still a small tip of an huge

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X. Li / Physica A 360 (2006) 629–636636

iceberg, most of which are left for further explorations.

Acknowledgements

his workChina

dges theROCS,

The author thanks Dr. Takashi Ichinomiya for his valuable discussion. Twas partially supported by the National Natural Science Foundation of PRunder Grants No. 90412004 and 70431002, and the author also acknowlesupport from the Alexander von Humboldt Foundation and the SRF forSEM.

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