复杂动态网络中的同步

Synchronization of Complex Dynamic Networks

n 问题描述 Problem Description

同步现象广泛存在于我们的生活中，如夏日夜晚的青蛙齐鸣、一群萤火虫的同时闪光和不闪光以及剧场中观众鼓掌的频率逐渐相同等等。有些同步是有益的，如调和振子的生成、保密通讯、语言涌现及其发展(谈话的同步)、组织管理的协调及高效运行(代理同步)等，我们需要这种同步；有些同步是有害的, 如传输控制协议窗口的增加、因特网或通讯网络中的信息拥塞、周期路由信息的同步等，我们要尽量避免这种同步。

复杂动态网络同步化性能的研究已成为一个极富挑战性的课题。对同步现象的建模和控制是研究的热点，近十年来学者们研究了具有不同拓扑结构的复杂动态网络中各种不同类型的同步状态，比如完全同步(complete synchronization)，相位同步(phase synchronization)等，以及在参数改变（例如具有时滞或时变的网络）和结构扰动（比如在网络中加入少量的点和边或对网络中的点和边进行加权）的情况下网络同步化性能的变化。

n 典型例子 Typical Examples

钟摆同步： 1665年，物理学家惠更斯发现，挂在同一横梁上的两个钟的摆在一段时间以后会出现同步摆动现象。

萤火虫同步：1680年，荷兰旅行家肯普弗在泰国观察到了一个奇特的现象：停在同一棵树上的萤火虫有时候同时闪光又同时不闪光，很有规律而且在时间上很准确。

掌声同步： 当一场精彩的演出结束的时候，掌声在最初的时刻是凌乱的，节奏是不同的，但是在几秒钟后，大家鼓掌的节奏趋于一致。2000年，《Nature》上的一篇文章从非线性动力学的观点阐述了此现象的产生机理（Néda Z, Ravasz E, Vicsek T et al. The sound of many hands clapping, Nature, 2000, 403: 849-850.）

激光同步： 在特定条件下亿万个发光原子具有相同的相位和频率，它们产生的激光束的一致性会令我们非常吃惊，宇宙中竟然有如此美妙的事情！

纳米振子同步：纳米振子之间有旋转力矩的相互影响，最新研究表明正是这种影响使得振子的相位锁定一致，2005年9月《Nature》上的一篇文章描述并分析了这种同步现象(Mohanty P, Nano-oscillators get it together, Nature, 2005, 437: 325-326)。

路由器同步： Internet上的每一个路由器都是周期性地发布路由消息，各个路由器自己决定什么时候发布信息，但是研究人员发现不同的路由器最终会以同步的方式发送路由消息，从而引发网络交通堵塞。

共振：2000年6月10日伦敦千年桥落成，当成千上万人同时通过大桥时，共振使大桥开始振动，所引起的偏差甚至达到了20厘米，桥上的人们万分恐慌，大桥于是不得不临时关闭。2005年11月《Nature》上的一篇文章对这一共振现象做了理论分析（Strogatz S H, Abrams D M, McRobie A et al, Crowd Synchrony on the Millennium Bridge, Nature, 2005, 438: 42-43）。

n 参考文献 References

一般性介绍：

[1] Wang X F, Chen G. Synchronization in complex dynamical networks, Journal of Systems Science and Complexity, 2003, 16: 1-14

[2] Strogatz S H. Sync: The Emerging Science of Spontaneous Order. New York: Hyperion, 2003.

[3] Boccaletti S, Kurths J, Osipov G, Valladares D L, Zhou C S, The synchronization of chaotic systems, Physics Reports, 2002, 366:1-101.

[4] Wang X F. Complex networks: topology, dynamics and synchronization, International Journal of Bifurcation and Chaos, 2002, 12(5): 885-916

[5] Engel A K, Fries P, Singer W, Dynamic predictions: oscillations and synchrony in top-down processing, Nature, 2001, 2: 704-716.

[6] Glass L, Synchronization and rhythmic processes in physiology, Nature, 2001, 410:277-284.

[7] Strogatz S H, Stewart I, Coupled oscillators and biological synchronization, Scientific American, 1993, 269(6): 68-74.

[8] Kuramoto Y. Chemical Oscillations, Waves and Turbulence. Springer-Verlag, 1984.

[9] Winfree A T. Biological rhythms and the behavior of populations of coupled oscillators. J. Theo. Biol., 1967, 16: 15-42.

完全同步：

[1] Wu C W, Synchronizability of networks of chaotic systems coupled via a graph with a prescribed degree sequence, Physics Letters A, 2005, 346: 281-287.

[2] Donetti L, Hurtado P I, Muñoz M A, Entangled networks, synchronization, and optimal network topology, Phys. Rev. Lett, 2005, 95: 188701.

[3] Kim Y, Mesbahi M, On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian, 2005 American Control Conference, June 2005, Portland, OR, USA.

[4] Stilwell D J, Bollt E M, Roberson D G, Sufficient conditions for fast switching synchronization in time varying network topologies, nlin/0502055.

[5] Matias M A, Synchronization in complex networks: a comment on two recent PRL papers, arXiv:cond-mat/0507471.

[6] Bernardo M, Garofalo F, Sorrentino F, Synchronization of degree correlated physical networks, arXiv:cond-mat/0506236.

[7] Bernardo M, Garofalo F, Sorrentino F, Synchronizability of degree correlated networks, arXiv:cond-mat/0504335.

[8] Belykh I, Hasler M, Lauret M, and Nijmeijer H, Synchronization and graph topology, Int. Journal of Bifurcation and Chaos, 2005, 15(11) (inpress) (invited tutorial).

[9] Belykh I V, Lange E, Hasler M. Synchronization of bursting neurons: what matters in the network topology, Phys. Rev. Lett, 2005, 94: 188101.

[10] Atay F M, Biyikoglu T, Graph operations and synchronization of complex networks, Phys. Rev. E, 2005, 72: 016217.

[11] Kocarev L, Amato P. Synchronization in power-law networks, Chaos, 2005, 15: 024101

[12] Li X, Sync in complex dynamical networks: stability, evolution, control, and application, Internal Journal of Computational Cognition, 2005, 3(4): 16-26.

[13] Fan J, Li X, Wang X F. On synchronous preference of complex dynamical networks, Physica A, 2005, 355:657-666.

[14] Fan J, Wang X F. On synchronization in scale-free dynamical networks, Physica A, 2005, 349: 443-451

[15] Hong H, Kim B J, Choi M Y et al. Factors that predict better synchronizability on complex networks, Phys. Rev. E, 2004, 69: 067105

[16] Belykh I. V, Belykh V N, Hasler M. Blinking model and synchronization in small-world networks with a time-varying coupling, Physica D, 2004, 195: 188-206

[17] Belykh I V, Belykh V N, Hasler M. Connection graph stability method for synchronized coupled chaotic systems, Physica D, 2004, 195: 159-187

[18] Barahona M, Pecora L M. Synchronization in small-world systems, Phys. Rev. Lett., 2004, 89(5): 054101

[19] Li C, Chen G. Synchronization in general complex dynamical networks with coupling delays, Physica A, 2004, 343: 236-278

[20] Wu C W. Perturbation of coupling matrices and its effect on the synchronizability in arrays of coupled chaotic systems, Physics Letters A, 2003, 319: 495-503

[21] Nishikawa T, Motter A E, Lai Y-C et al. Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? Phys. Rev. Lett., 2003, 91: 014101

[22] Li X, Wang X F, Chen G. Synchronization in complex dynamical networks and its applications, Conference on Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems, Rome, Italy, 2003

[23] Li X, Chen G. Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint, IEEE Trans. Circuits & Systems-I, 2003, 50(11): 1381-1390

[24] Li X, Jin Y Y, Chen G. Complexity and synchronization of the world trade web, Phyisca A, 2003, 328: 287-296.

[25] Wei G W, Zhan M, Lai C-H. Tailoring wavelets for chaos control, Phys. Rev. Lett, 2002, 89(28): 284103

[26] Wang X F, Chen G. Synchronization in complex dynamical networks, Journal of Systems Science and Complexity, 2003, 16: 1-14

[27] Wang X F, Chen G. Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circuits & Systems–I, 2002, 49(1): 54-62

[28] Wang X F, Chen G. Synchronization in small-world dynamical networks, International Journal of Bifurcation & Chaos, 2002, 12(1): 187-192

[29] Fink K S, Johnson G, Carroll T L, Mar D, Pecora L M, Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays, Phys. Rev. E., 2000, 61(5): 5080-5090.

[30] Pecora L M, Carroll T L. Master stability functions for synchronized coupled systems, Phys. Rev. Lett, 1998, 80: 2109

[31] Pecora L M. Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems, Phys. Rev. E, 1998, 58(10): 347-360

[32] Brown R, Rulkov N F, Synchronization of chaotic systems: transverse stability of trajectories in invariant manifolds, Chaos, 1997, 7(3): 395-413.

[33] Wu C W, Chua L O. Application of graph theory to the synchronization in an array of coupled nonlinear oscillators, IEEE Trans. Circuits & Systems–I, 1995, 42: 494–497.

[34] Wu C W, Chua L O. Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuits & Systems–I, 1995, 42: 430-447

[35] Heagy J F, Carroll T L, Pecora L M, Desynchronization by periodic orbits, Phys. Rev. E., 1995, 52(2): 1253-1256

[36] Heagy J F, Pecora L M, Carroll T L, Short wavelength bifurcations and size instabilities in coupled oscillator systems, Phys. Rev. Lett., 1995, 74(21): 4185-4188.

[37] Heagy J F, Carroll T L, Pecora L M, Synchronous chaos in coupled oscillator systems, Phys. Rev. E., 1994, 50(3): 1874-1885.

[38] Pecora L M, Carroll T L, Synchronization in chaotic systems, Phys. Rev. Lett., 1990, 64(8): 821-825.

[39] Awerbuch B, Complexity of network synchronization, Journal of the Association for Computing Machinery, 1985, 32(4): 804-823.

[40] Fujisaka H, Yamada T, Stability theory of synchronized motion in coupled-oscillator systems, Progress of Theoretical Physics, 1983, 69(1): 32-47.

相位同步：

[41] Li X. Uniform synchronous criticality of diversely random complex networks, Physica A, 2006, 359: 827-834.

[42] Trees B R, Saranathan V, Stroud D, Synchronization in disordered Josephson junction arrays: small-world connections and the Kuramoto model, Phys. Rev. E., 2005, 71: 016215.

[43] Jadbabaie A, Motee N, Barahona M, On the stability of the Kuramoto model of coupled nonlinear oscillators, math.OC/0504419

[44] Moreno Y, Vàzquez-Prada M, Pacheco A F. Fitness for synchronization of network motifs. Physica A, 2004, 343: 279-287

[45] Li C, Chen G. Phase synchronization in small-world networks of chaotic oscillators, Physica A, 2004, 341: 73-79.

[46] Ichinomiya T. Frequency synchronization in random oscillator network, Phys. Rev. E, 2004, 70: 026116

[47] Moreno Y, Pacheco A F. Synchronization of Kuramoto oscillators in scale-free networks, Europhysics Letters, 2004, 68: 603-609.

[48] Zhao L, Lai Y –L, Wang R, Gao J –Y, Limits to chaotic phase synchronization, Europhysics Letters, 2004, 66(3):324-330.

[49] Ivanchenko M V, Osipov G V, Shalf-eev V D, Kurths J, Phase synchronization of chaotic intermittent oscillations, Phys. Rev. Lett., 2004, 92(13): 134101.

[50] Jalan S, Amritkar R E, Self-organized and driven phase synchronization in coupled maps, Phys. Rev. Lett, 2003, 90(1): 014101

[51] Amritkar R E, Jalan S, Self-organized and driven phase synchronization in coupled map networks, Physica A, 2003, 321: 220-225.

[52] Hong H, Choi M Y, Kim B J. Synchronization on small-world networks, Phys. Rev. E, 2002, 65: 026139

[53] Strogatz S H. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 2000, 143: 1–20

[54] Néda Z, Ravasz E, Vicsek T, Brechet Y, Barabási A L, Physics of the rhythmic applause, Phys. Rev. E, 2000, 61(6):6987-6992.

[55] Rosenblum M, Pikovsky A S, Kurths J, Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 1996, 76(11): 1804-1807.

加权网络同步：

[56] Wu C W, Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling, IEEE Trans. Circuits & Systems-II: Express Briefs, 2005, 52(5): 282-286.

[57] Chavez M, Hwang D-U, Amann A et al. Synchronization is enhanced in weighted complex networks. Phys. Rev. Lett, 2005, 94: 218701.

[58] Hwang D.U. et al., Synchronization in complex networks with age ordering, Phys. Rev. Lett., 94, 2005,138701.

[59] Motter A E, Zhou C, Kurths, J. Network synchronization, diffusion, and the paradox of heterogeneity. Phys. Rev. E, 2005, 71: 016116.

[60] Motter A E, Zhou C, Kurths J. Enhancing complex-network synchronization. Europhysics Letters, 2005, 69: 334-340.

[61] Wu C W, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 2005, 18: 1057-1064.

[62] Motter A E, Zhou C, Jurths J, Weighted networks are more synchronizable: how and why, AIP Conference Proceedings 2005, 776:201.

时变动态复杂网络同步：

[63] Lv J, Chen G. A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Transactions on Automatic Control, 2005,50(6):841-846.

[64] Lv J, Yu X, Chen G, Characterizing the synchronizability of small-world dynamical networks, IEEE Transactions on Automatic Control, 2005,51(4): 787-796.

[65] Li C, Chen G, Synchronization in general complex dynamical networks with coupling delays, Physica A, 2004, 343: 263-278.

[66] Lv J, Yu X, Chen G. Chaos synchronization of general complex dynamical networks. Physica A, 2004, 334: 281-302.

时滞动态复杂网络同步：

[67] Li C, Sun W, Xu D, Synchronization of complex dynamical network with nonlinear inner-coupling functions and time delays, Progress of Theoretical Physics, 2005, 114(4).

[68] Zhou J, Chen T, Synchronization in general complex delayed dynamical networks, to appear in IEEETCAS (in press).

[69] Rosenblum M G, Pikovsky A S, Controlling synchroniz