I show that the conclusions of [Hwang, Chavez, Amann, & Boccaletti, PRL 94,
138701 (2005); Chavez, Hwang, Amann, Hentschel, & Boccaletti, PRL 94, 218701
(2005)] are closely related to those of previous publications.Comment: 2 page

Matias, Manuel A.

English

arXiv

2 pages.-- PACS nr.: 89.75Hc.-- ArXiv pre-print: http://arxiv.org/abs/cond-mat/0507471.-- Both versions available: v1 (submitted on 20 Jul 2005), v2 (last revised 11 Aug 2005).I show that the conclusions of [Hwang, Chavez, Amann, & Boccaletti, PRL 94, 138701 (2005); Chavez, Hwang, Amann, Hentschel, & Boccaletti, PRL 94, 218701 (2005)] are closely related to those of previous publications.

arxiv: cond-mat/050747

Matías, Manuel A.

Digital.CSIC

arXiv:cond-mat/0507471 v2   11 Aug 2005Synchronization in Complex Networks: a Comment on two recent PRL papersManuel A. Mat´ıas1, ∗1Instituto Mediterra´neo de Estudios Avanzados, IMEDEA (CSIC-UIB), E-07122 Palma de Mallorca, Spain(Dated: June 20, 2006)I show that the conclusions of [Hwang, Chavez, Amann, & Boccaletti, PRL 94, 138701 (2005);Chavez, Hwang, Amann, Hentschel, & Boccaletti, PRL 94, 218701 (2005)] are closely related tothose of previous publications.PACS numbers: 89.75HcRecent interest on dynamical networks regards the syn-chronizability properties of networks with coupled identi-cal oscillators [1]. According to the standard formalism inthe study of synchronization in a class of coupled iden-tical oscillators [1], the eigenratio R = λN/λ2, definedfrom the eigenvalues of the laplacian matrix is a measureof the synchronizability of the network, and has to besmaller than a certain dynamical ratio corresponding tothe oscillators. In [2] it was shown that networks thatare heterogeneous in the degree distribution (e.g. scale-free networks (SFNs)) are more difficult to synchronize.In Ref. [3] it was shown that synchronization can beenhanced in SFNs by constructing a weighted network.More precisely, the idea is weight all the oscillators inthe network such that the total strength of input con-nections is the same for all the oscillators [3]. Thus, inRef. [3] the strength of connections is weighted suchthat wi→j = 1/kβj (and analogously for wj→i). It wasproven [3] that synchronizability is maximum for β = 1,and for this case it follows immediately that the totalstrength of input connections∑j wj→i is the same forany node i of the network, while for output connec-tions∑i wj→i the distribution of strengths is identicalto the degree distribution (cf. comment reference [30]in [4]). On the other hand, because of the asymmetrywi→j = 1/kβj < wj→i = 1/kβi when kj > ki and β > 0,then it is clear that the strength of the output connec-tions is positively correlated with the degree of the node,implying that when synchronizability is improved thedominant coupling direction is from high-degree nodesto low-degree ones.In particular, in the 1024-node random SFN [5] consid-ered in [3] it is shown, Fig. 2, that a ten-fold, i.e. 1000%improvement in synchronizability is attained, comparedto the unweighted case considered in Ref. [2]. Moreover,as the number of oscillators N grows, this synchroniza-tion enhancement improves even further (check the ratiofor weighted networks, Eq. [4] of Ref. [3], with the un-weighted case, Eq. 2 in Ref. [2]). This improvementin synchronizability is shown to be significant when thetotal strength of input connections is equal for all theoscillators in the network.In more recent work, Ref. [6], the same idea has beensuggested: a weighting scheme that lowers the eigenratioR by lowering the connection strength of the most highlyconnected nodes, while satisfying also the condition un-covered in Ref. [3] that the total strength of input con-nections is the same for all the oscillators in the network.Relaxing constraints on the individual connections, theywere able to improve the synchronizability uncovered byMotter et al. by a factor of upto 1.2 [8], i.e., 20% (cf.Fig.2 in Ref. [6]). On the other hand, the dependence onthe system sizeN of the obtained improvement is difficultto ascertain.In closely connected work, Ref. [7], a more refinedversion of the weighted and directed coupling of [6] hasbeen introduced. The main conclusion of Ref. [7] is thatpropensity for synchronization is enhanced in networks ofasymmetrically coupled units and that in growing SFNssuch enhancement is particularly evident when the dom-inant coupling direction is from older to younger nodes.But for the kind of growing SFNs considered in Ref. [7]older nodes have a larger degree (more connections)[9].So, this means that the concepts of age and numberof connections (or degree) are actually equivalent, and,thus, the improvements in the propensity for synchro-nization and synchronizability in Refs. [7] and [3, 4] are,thus, intimately related.∗ Electronic address: manuel@imedea.uib.es[1] M. Barahona and L.M. Pecora, Synchronization in Small-World Systems, Phys. Rev. Lett. 89, 054101 (2002),nlin.CD/0112023.[2] T. Nishikawa, A. E. Motter, Y.-C. Lai, and F. C. Hoppen-steadt, Heterogeneity in oscillator networks: are smallerworlds easier to synchronize? Phys. Rev. Lett. 91, 014101(2003), cond-mat/0306625.[3] A. E. Motter, C. S. Zhou, and J. Kurths, Enhancingcomplex-network synchronization, Europhys. Lett. 69,334 (2005); cond-mat/0406207.[4] A. E. Motter, C. S. Zhou, and J. Kurths, Network syn-chronization, diffusion, and the paradox of heterogeneity,Phys. Rev. E 71, 016116 (2005), cond-mat/0502309.[5] A.-L. Baraba´si and R. Albert, Emergence of scal-ing in random networks, Science 286, 509 (1999),cond-mat/9910332.2[6] M. Chavez, D.-U. Hwang, A. Amann, H. G. E. Hentschel,and S. Boccaletti, Synchronization is enhanced in weightedcomplex networks, Phys. Rev. Lett. 94, 218701 (2005).[7] D.-U. Hwang, M. Chavez, A. Amann, and S. Boccaletti,Synchronization in complex networks with age ordering,Phys. Rev. Lett. 94, 138701 (2005), cond-mat/0412728.[8] Notice, Eq. 2 in Ref. [6], that the normalization impliedby the denominator makes this scheme very close indeedto the case β = 1 of Ref. [3][9] This was shown analytically in the very first paper aboutscale-free networks [5]: ki(t) = m(t/ti)1/2

Synchronization in Complex Networks: a Comment on two recent PRL papers

Abstract

Similar works

Full text

Available Versions

Digital.CSIC