摘要
复杂网络上弱耦合动力学及信号传输是复杂网络研究的主要问题,并已取得了长足进展,如耦合系统的弱同步化,信号传输过程中的相互作用及放大等。本文从混沌系统中的分立映象、流行病传播及随机共振三个方面入手,在复杂网络上的弱耦合动力学及信号传输方面进行了一些探讨,具体结果如下：
在耦合系统的弱同步化方面,我们侧重于混沌系统的分立映象——Logistic映象,主要研究了两个方面的内容——第一：网络的拓扑结构对于“反序”的影响；第二：无标度网络上耦合分立系统的双共振现象。
2000年,Logistic映象中“反序”的概念——在迭代时,其函数值连续出现两次“向上相”的比例——被科学家们提出。该定义可以用“方向相”来刻画。在本文中,我们通过研究复杂网络中耦合的Logistic映象,观察了网络结构是如何影响反序的。我们的数值模拟结果揭示了出现反序的临界值会随着耦合强度的增大而增大,然而随着网络结构非均匀性的增大,该临界值又会出现减小的现象。进一步,我们发现与正常序不同,在适当的耦合强度下,反序会使系统呈现出周期窗口。在出现周期窗口的范围内,网络的非均匀性会减少网络出现集团个数的最大值,也就是说,网络结构越不均匀,网络中集团个数的最大值越小。
Liu等人在2003年的一篇Physical Review E上指出混沌系统中,弱的耦合强度并不能使耦合的连续流系统达到同步化,有时候反而使得他们的行为变得更加复杂。在本文中,我们发现在耦合的混沌分立系统中也有类似的现象——弱耦合强度会使分立相集团个数增大。在不同的耦合强度下我们发现了双共振现象——第一个共振现象来源于耦合导致的周期现象,第二个共振现象来源于反序的消失。我们通过无序相边解释了第二个共振现象的机制。进一步我们继续研究了非全同振子的情形,并发现出现分立相集团个数最大值的临界耦合强度会随着分岔系数的增大而快速增大,而随着非全同振子的分岔系数分布范围的变大而缓慢增大。
在信号传输过程中的相互作用方面,本文主要研究了流行病在社区网络中的传播特性。在2004年的一篇Nature上,作者Cummings等人报道了流行病传播呈时空周期震荡的数据。为了理解其中的机制,并基于前人提出的社区网络的构成方法,我们分别用SIS模型和SIRS模型研究了流行病在社区网络中的传播,并发现：随着感染机率的增大,总感染人数可能会稳定在一个定值,可能出现震荡波,也可能出现周期环。同时,SIS模型的流行病传播可以用一个可解析的映象来解释。
在信号传输过程中的信号放大方面,本文基于双势阱系统,提出了一个通过自调节目标节点与其邻居的边权来达到放大信号目的的方法。该方法主要是通过提高与目标节点处于不同势阱中的邻节点的边权重,同时降低处于同一个势阱的邻节点的边权重来实现放大信号的效果的。数值模拟证明该方法十分有效,且在仅知道局域信息的系统中应用很广。在本文的最后我们用理论解析解释了该方法的机制。
我们的这些研究结果表明复杂网络本身与信号动力学互相影响,正如复杂网络的拓扑结构直接影响着网络中的信号动力学行为；相反地,动力学过程也影响着网络本身的拓扑结构及基本性质。研究这些复杂系统上的信号动力学行为是非线性科学研究的重要课题。
本论文将结合自己的工作着重介绍四个方面的内容：第一、网络拓扑结构对于“反序”的影响；第二、无标度网络上耦合分立系统的双共振现象；第三、流行病在社区网络中的传播；第四、边权的自适应调节放大振子信号。
The weak coupling dynamics and the signals transmission on the complex networks are significant issues of the complex-system studies, which are well studied in many fields, such as the weak synchronization of the coupled systems、the signals interaction and amplification in their transmission process, etc.. In this thesis, we study the weak coupling dynamics and the signals transmission based on three aspects:the discrete maps of chaotic systems、the epidemic spreading and the stochastic resonance. We obtain the following results:
Firstly, the abnormal phase order of coupled logistic maps, i.e., the ratio of two sequential "up phases" in the total iterations, can be characterized by the direction phase [Phys. Rev. Lett.,84 (2000) 2610]. We here consider the case of coupled logistic maps on complex networks and study how the network topology influences the abnormal phase order. Our numerical simulations reveal that the critical point for the appearance of abnormal phase order increases with the coupling strength but decreases with the degree of heterogeneity of complex networks. Moreover, we find that unlike in the case of normal phase order, it is possible for the system to show a periodic window in the case of abnormal phase order, but only within an appropriate range of coupling strengths, and finally, that the heterogeneity can reduce the maximum number of the phase clusters in a given periodic window.
Secondly, it is known that for chaotic flows, a weak coupling does not always make the coupled systems approach synchronization but sometimes make them become more complicated [PRE 67,045203(R)]. We here report that a similar situation also occurs in the coupled chaotic maps, where a weak coupling will make the number of direction-phase clusters increase. We find a double resonance effect on the coupling strength where the first resonance comes from the coupling induced periodic behaviors and the second one owing to the disappearance of disorder phase. The mechanism of the second resonance is revealed through the out-of-phase links. Moreover, we show that the critical coupling strength of the maximum of the number of direction-phase clusters will increase rapidly with the bifurcation parameter but slowly with the range of the distribution of non-identical oscillators.
Thirdly, It is reported in Nature (London) 427,344(2004) that there are periodic waves in the spatiotemporal data of epidemic. For understanding its mechanism, we study the epidemic spreading on community networks by both the SIS model and the SIRS model. We find that with the increase of infection rate, the number of total infected nodes may be stabilized at a fixed point, oscillatory waves, and periodic cycles. Moreover, the epidemic spreading in the SIS model can be explained by an analytic map.
Finally, we proposed a method to amplify the signal in a double well system by adaptively adjusting the weight of the connections between the objective and its neighbors. This method is to increase the weights of the connections which link the objective with its neighbors in the different well while to decrease the weights of the connections which link the objective with its neighbors in the same well. This method is proved to be quite effective and could be utilized to a wide range of nodes while only local information of the system is needed. Our theoretical analysis provides the explanation of the mechanism of this method.
All the results show that topological structure and the dynamical process can interact each other, i.e., the topological structure of complex networks may influence the dynamics on it; and the coupled signals dynamics may, in contrast, also influence the topology of complex networks. Hence, the dynamics of the coupled signals on the complex networks is a significant issue in the nonlinear physics.
This thesis mainly consists of four parts. Firstly, we introduce the influence of network topology on the abnormal phase order. Secondly, we discuss the resonance effect of direction-phase clusters in a scale-free network. Thirdly, we introduce the periodic wave of epidemic spreading in community networks. At last, we discuss the signal amplification by adaption on weighted network.
引文
[1]Watts D.-J. and Strogatz S. H., Nature,393 (1998) 440.
[2]A. L. Barabasi and R. Albert, Science 286,509 (1999).
[3]复杂系统与复杂网络,何大韧,刘宗华,汪秉宏,高等教育出版社(2008)。
[4]Albert R. and Barab'asi A.-L., Rev. Mod. Phys.,74 (2002) 47.
[5]M. J. Feigenbaum, J. Stat. Phys.19,25 (1978).
[6]B. L. Hao, Elementary Symbolic Dynamics and Chaos in Dissipative Systems (World Scientific, Singapore,1989).
[7]E. Ott, Chaos in Dynamical System (Cambridge University Press, New York, 1993); H. A. Lauwerier, in Chaos, edited by A.V. Holden (Manchester University Press, Manchester,1986).
[8]S. K. Ma, Modern Theory of Critical Phenomena (Benjamin, London, 1976).
[9]M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.65,851 (1993).
[10]M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev. Lett.76,1804 (1996).
[11]Wang W., Liu Z. and Hu B., Phys. Rev. Lett.,84 (2000) 2610;
[12]Ho M.-C., Hung Y.-C. and Jiang I-M., Phys. Lett. A,324 (2004) 450.
[13]Chen J. Y., Wong K. W., Zheng H. Y. and Shuai J. W., Phys. Rev. E,64 (2001)016212.
[14]Shrimali M. D. and Ramaswamy R., Phys. Lett. A,295 (2002) 273.
[15]Jalan S., Amritkar R. E. and Hu C.-K., Phys. Rev. E,72 (2005) 016211; 016212.
[1.6]Amritkar R. E. and Jalan S., Physica A,346 (2005) 13.
[17]Zhuang G. C., Wang J., Shi Y. and Wang W., Phys. Rev. E,66 (2002) 046201.
[18]Batista A. M., Pinto S. E., Viana R. L. and Lopes S. R., Phys. Rev. E,65 (2002) 056209.
[19]Gomez-Gardenes J., Moreno Y. and Arenas A., Phys. Rev. Lett.,98 (2007) 034101.
[20]McGraw P. N. and Menzinger M., Phys. Rev. E,72 (2005) 015101.
[21]Barahona M. and Pecora L. M., Phys. Rev. Lett.,89 (2002) 054101.
[22]Hasegawa H., Phys. Rev. E,72 (2005) 056139.
[23]Motter A. E., New J. Phys.,9 (2007) 182.
[24]Wang X., Lai Y.-C. and Lai C. H., Phys. Rev. E,75 (2007) 056205.
[25]Chavez M., Hwang D.-U., Amann A., Hentschel H. G. E. and Boccaletti S., Phys. Rev. Lett.,94 (2005) 218701.
[26]Belykh I., Lange E. and Hasler M., Phys. Rev. Lett.,94 (2005) 188101.
[27]Gade P. M. and Hu C.-K., Phys. Rev. E,73 (2006) 036212.
[28]Willeboordse F. H., Phys. Rev. E,47 (1993) 1419.
[29]Willeboordse F. H., Phys. Rev. E,65 (2002) 026202.
[30]Boccaletti S., Latora V., Moreno Y., Chavez M. and Hwang D.-U., Phys. Rep.,424 (2006) 175.
[31]Zhou Y., Zhou J. and Liu Z., Europhysics Letters Vol 84, (2008) 60001.
[32]Liu Z., Lai Y.-C. and Matias M. A., Phys. Rev. E,67 (2003) 045203(R).
[33]Zhou Y., Zhou J., Wang X., Guan S., Liu Z. and Lai C.H., (to be accepted by Europhysics Letters)
[34]F. Liljeros, C. R. Edling, L. A. Amaral, H. E. Stanley and Y. Aberg, Nature, 411 (2001)907.
[35]Ferguson N M, Keeling M J, Edmunds W J, Gant R, Grenfell B T, Anderson R M, Leach S. Nature. (2003);425:681-685.
[36]Cummings, D. A. T., Irizarry, R. A., Huang, N. E., Endy, T. P., Nisalak, A. Ungchusak, K. & Burke, D. S. Nature 427, (2004) 344-347.
[37]Stone, L., Olinky, R. and Huppert, A. Seasonal dynamics of recurrent epidemics. Nature 446, (2007).533-536.
[38]Pastor-Satorras R. and Vespignani A., Phys. Rev. Lett.86,3200 (2001).
[39]Eguiluz V.M. and Klemm, Phys. Rev. Lett.89,108701 (2002).
[40]Anderson R M, May R M. Infections Diseases of Humans. Oxford:Oxford University Press,1992.
[41]Zonghua Liu and Bambi Hu, Europhys.Lett.,72(2),315-321(2005).
[42]M.Boguna, R. Pastor-Satorras and A Vespignani, Phys.Rev.Lett.90, 028701(2003).
[43]Shao Z.G. and Sang J.P., Physica A 351,662(2005).
[44]M. Barthelemy, R. Pastor-Satorras and A Vespignani, Phys.Rev.Lett.92, 178701(2004).
[45]Anderson R. M. and May R. M., Infectious Diseases in Humans (Oxford University Press, Oxford) 1992.
[46]Pastor-Satorras R. and Vespignani A., Phys. Rev. Lett.,86 (2001) 3200; Phys. Rev. E,63 (2001) 066117; 65 (2002) 035108.
[47]R. Pastor-Satorras and A.Vespignani, Phys. Rev. E.63,066117(2001).
[48]Tessone C. J., Toral R., Amengual P., Wio H. S. and Miguel M. S., Eur. Phys. J. B,39 (2004) 535.
[49]Galam S., Physica A,320 (2003) 571; J. Stat. Phys,61 (1990) 943; J. Math. Psychol.,30 (1986) 426.
[50]Newman M. E. J., Phys. Rev. Lett.,89 (2002) 208701; Rev. E,67 (2003) 026126; Newman M. E. J. and Park J., Phys. Rev. E,68 (2003) 036122; Newman M. E. J. and Girvan M., Phys. Rev. E,69 (2004) 026113.
[51]Jin E. M., Girvan M. and Newman M. E. J., Phys. Rev. E,64 (2001) 046132.
[52]Zhou Y., Liu Z. and Zhou J., Chin. Phys. Lett. Vol 24, No.2 (2007) 581.
[53]D. Bray, J. Theor. Biol.143,215 (1990).
[54]U. Alon, Science 301,1866 (2003).
[55]B. Ando, S. Baglio, A. R. Bulsara, and V. Sacco, IEEE Instrum. & Meas. Mag.8,64 (2005).
[56]V. In, A. R. Bulsara, A. Palacios, P. Longhini, and A. Kho, Phys. Rev. E 72,045104(R)(2005).
[5.7]A. R. Bulsara, J. F. Lindner, V. In, A. Kho, S. Baglio, V. Sacco, B. Ando, P. Longhini, A. Palacios, and W.-J. Rappel, Phys. Lett. A 353,4 (2006).
[58]R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, Tellus 34,10 (1982).
[59]L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70,223 (1998).
[60]M. Morillo, J. Go'mez-Ordon-ez, and J. M. Casado, Phys. Rev. E 52,316 (1995).
[61]J. F. Lindner, B.K. Meadows, W. L. Ditto, M. E. Inchiosa, and A. R. Bulsara, Phys. Rev. Lett.75,3 (1995).
[62]C. J. Tessone, C. R. Mirasso, R. Toral, and J. D. Gunton, Phys. Rev. Lett. 97,194101 (2006).
[63]J. A. Acebron, S. Lozano, and A. Arenas, Phys. Rev. Lett.99,128701 (2007).
[64]Z. Liu, and T. Munakata, Phys. Rev. E 78,046111 (2008).
[65]A. A. Zaikin, J. Kurths, and L. Schimansky-Geier, Phys. Rev. Lett.85,227 (2000).
[66]A. S. Pikovsky, A. Zaikin, and M. A. de la Casa, Phys. Rev. Lett.88, 050601 (2002).
[67]B. Seo, R. Krishnan, and T. Munakata, Phys. Rev. E 75,056106 (2007); T. Munakata, T. Hada, and M. Ueda, PhysicaA 375,492 (2007).
[68]Zhou Y., Zhou J. and Liu Z., Signal Amplification by Adaption on Weighted Network, (in preparation)
[1]S. Boccaletti, V. Latora, Y. Moreno, M. Chavezf and U. Hwang, Physics Reports.424,175,2006.
[2]B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics, Springer, New York,1998.
[3]S. Wasserman and K. Fause, Social Networks Analysis, Cambridge University Press, Cambridge,1994.
[4]V. Latora and M. Marchiori, Physica A 314,109,2002.
[5]R. Albert, I. Albert and G. L. Nakarado, Phys. Rev. E 69,025103(R),2004.
[6]V. Latora and M. Marchiori, Phys. Rev. E 71,015103(R),2005.
[7]S. H.Yook, H. Jeong, A.-L. Barabasi and Y. Tu, Phys. Rev. Lett.86,5835, 2001.
[8]J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertesz and A. Kanto, Phys.Rev.E 68,056110,2003.
[9]Dijkstra E.W., Numerische. Math,1(1959):269-271.
[lO]Floyd R.W., Comm.ACM.,5(1962):345.
[11]D.J.Watts, S.H.Strogatz, Nature 393(1998)440.
[12]M.Marchjori, V.Latora, Physica A 285(2000)539.
[13]V.Latora, M.Marchiori, Phys.Rev.Lett.87(2001)198701.
[14]V.Latora, M.Marchiori, Eur.Phys.J.B 32(2003)249.
[15]S. Milgram, Psychol. Today 1,60 (1967).
[16]M.E.J.Newman, Phys.Rev.E 64(2001)016132.
[17]R.Albert,H.Jeong,A.L.Barabasi, Nature,401(1999),130-131.
[18]M.E.J.Newman. SIAM Review,45(2003):167-256.
[19]R. Pastor-Satorras, A. Vazquez, A.Vespignani, Phys.Rev.Lett.87,258701 (2001).
[20]P. Erdos, A. Renyi, Publ. Math. Debrecen 6,290(1959).
[21]R. Albert and A.L. Barabasi, Rev. Mod. Phys.74,47(2002).
[22]A.BalTat, M.weigt, Eur.Phys.J.B 13(2000)547.
[23]A.BalTat, M.weigt, Eur.Phys.J.B 13(2000)547.
[24]M.E.J.Newman, D.J.Watts, Phys.Lett. A 263(1999)341.
[25]M.Barth6lemy, L.A.N.Amaral, Phys.Rev.Lett.82(1999)3180.
[26]Barabasi, A.-L.,(1999).Albert, R., Jeong, H., PhysicaA 272,173-187(1999).
[27]Bollobds,B.and Riordan,O. University of Memphis (2002).
[1]Wang W., Liu Z. and Hu B., Phys. Rev. Lett.,84 (2000) 2610; Hu B. and Liu Z., Phys. Rev. E,62 (2000) 2114.
[2]Perc M. and Marhl M., Chaos, Solitons Fractals,18 (2003) 759; Gosak M. and Perc M., Phys. Rev. E,76 (2007) 037201.
[3]Grebogi C., Ott E., York J. A., Phys. Rev. Lett.,48 (1982) 1507; Grebogi C., Ott E., York J. A., Physica D,7 (1982) 181.
[4]黄润生,混沌及其应用,武汉：武汉大学出版社 (1999)。
[5]Abraham R., Gardini L. and Mira C., Chaos in Discrete Dynamical Systems: A Visual Introduction in 2 Dimensions (The Electronic Library of Science, Santa Clara) 1996.
[6]Pikovsky A., Rosenblum M. and Kurths J., Synchronization:A Universal
Concept in Nonlinear Sciences (Cambridge University Press, Cambridge) 2001.
[7]Liu Z., Europhys. Lett.,68 (2004) 19; Liu Z., Hu B. and Iasemidis L. D., Europhys. Lett.,71 (2005) 200.
[8]Ho M.-C., Hung Y.-C. and Jiang I-M., Phys. Lett. A,324 (2004) 450.
[9]Chen J. Y., Wong K. W., Zheng H. Y. and Shuai J. W., Phys. Rev. E,64 (2001)016212.
[10]Shrimali M. D. and Ramaswamy R., Phys. Lett. A,295 (2002) 273.
[1.1]Jalan S., Amritkar R. E. and Hu C.-K., Phys. Rev. E,72 (2005) 016211; 016212.
[12]Amritkar R. E. and Jalan S., Physica A,346 (2005) 13.
[13]Zhuang G. C., Wang J., Shi Y. and Wang W., Phys. Rev. E,66 (2002) 046201.
[14]Batista A. M., Pinto S. E., Viana R. L. and Lopes S. R., Phys. Rev. E,65 (2002)056209.
[15]Albert R. and Barab'asi A.-L., Rev. Mod. Phys.,74 (2002) 47.
[16]Boccaletti S., Latora V., Moreno Y., Chavez M. and Hwang D.-U., Phys. Rep.,424 (2006) 175.
[17]Gomez-Gardenes J., Moreno Y. and Arenas A., Phys. Rev. Lett.,98 (2007) 034101.
[18]McGraw P. N. and Menzinger M., Phys. Rev. E,72 (2005) 015101.
[19]Barahona M. and Pecora L. M., Phys. Rev. Lett.,89 (2002) 054101.
[20]Hasegawa H., Phys. Rev. E,72 (2005) 056139.
[21]Motter A. E., New J. Phys.,9 (2007) 182.
[22]Wang X., Lai Y.-C. and Lai C. H., Phys. Rev. E,75 (2007) 056205.
[23]Chavez M., Hwang D.-U., Amann A., Hentschel H. G. E. and Boccaletti S., Phys. Rev. Lett.,94 (2005) 218701.
[24]Belykh I., Lange E. and Hasler M., Phys. Rev. Lett.,94 (2005) 188101.
[25]Gade P. M. and Hu C.-K., Phys. Rev. E,73 (2006) 036212.
[26]Willeboordse F. H., Phys. Rev. E,47 (1993) 1419.
[27]Willeboordse F. H., Phys. Rev. E,65 (2002) 026202.
[28]Watts D.-J. and Strogatz S. H., Nature,393 (1998) 440.
[29]Barab'asi A.-L. and Albert R., Science,286 (1999) 509.
[30]Liu Z., Lai Y.-C., Ye N. and Dasgupta P., Phys. Lett. A,303 (2002) 337.
[31]Sinha S., Phys. Rev. E,66 (2002) 016209.
[32]Rangarajan G., Chen Y. and Ding M., Phys. Lett. A,310 (2003) 415.
[33]Zhang X., Fu M., Xiao J. and Hu G., Phys. Rev. E,74 (2006) 015202.
[34]LiuZ. and Hu B., Int. J. Bifurcat. Chaos,11 (2001)3137; Liu Z., Lai Y. and Hoppensteadt F. C., Phys. Rev. E,63 (2001) 055201.
[35]Liang X., Lv H. and Liu Z., Chin. Phys. Lett.,25 (2008) 409.
[36]Zhou Y., Zhou J. and Liu Z., Europhysics Letters Vol 84, (2008) 60001.
[1]Liu Z., Lai Y.-C. and Matias M. A., Phys. Rev. E,67 (2003) 045203(R).
[2]Ott E., Chaos in dynamical systems (Cambridge University Press, New York) 1993.
[3]Alligood K. T., Sauer T. D. and York J. A., Chaos:An introduction to dynamical systems (Springer-Verlag, New York) 1996.
[4]Pikovsky A., Rosenblum M. and Kurths J., Synchronization:A universal concept in nonlinear sciences (Cambridge University Press, Cambridge) 2001.
[5]Pecora L. M. and Carroll T. L., Phys. Rev. Lett.,64 (1990) 821.
[6]Rosenblum M. G., Pikovsky A. S. and Kurths J., Phys. Rev. Lett.,76 (1996) 1804; Phys. Rev. Lett.,78 (1997) 4193.
[7]Wang W., Liu Z. and Hu B., Phys. Rev. Lett.,84 (2000) 2610; Phys. Rev. E, 62(2000)2114.
[8]Rulkov N. F., Sushchik M. M., Tsimring L. S. and Abarbanel H. D. I., Phys. Rev. E,51 (1995)980.
[9]Abarbanel H. D. I., Rulkov N. F., Sushchik M. M. and Tsimring L. S., Phys. Rev. E,53 (1996)4528.
[10]Kocarer L. and Parlitz U., Phys. Rev. Lett.,76 (1996) 1816.
[11]Hunt B. R., Ott E. and Yorke J. A., Phys. Rev. E,55 (1997) 4029.
[12]Pyragas K., Phys. Rev. E,56 (1997) 5183; Phys. Rev. E,54 (1996) R4508.
[13]Brown R., Phys. Rev. Lett.,81 (1998) 4835.
[14]Zheng Z. and Hu G., Phys. Rev. E,62 (2000) 7882.
[15]Liu Z. and Chen S., Phys. Rev. E,56 (1997) 7297; Phys. Rev. E,55 (1997) 6651.
[16]Liu Z., Zhou J. and Munakata T., Europhys. Lett.,87 (2009) 50002.
[17]Ho M.-C., Hung Y.-C. and Jiang I-M., Phys. Lett. A,324 (2004) 450.
[18]Chen J. Y., Wong K. W., Zheng H. Y. and Shuai J. W., Phys. Rev. E,64 (2001) 016212.
[19]Shrimali M. D. and Ramaswamy R., Phys. Lett. A,295 (2002) 273.
[20]Liang X., Lv H. and Liu Z., Chin. Phys. Lett.,25 (2008) 409.
[21]Jalan S., Amritkar R. E. and Hu C.-K., Phys. Rev. E,72 (2005) 016211; ibid,72 (2005)016212;
[22]Amritkar R. E. and Jalan S., Physica A,346 (2005) 13.
[23]Zhuang G. C., Wang J., Shi Y. and Wang W., Phys. Rev. E,66 (2002) 046201.
[24]Batista A. M., Pinto S. E., Viana R. L. and Lopes S. R., Phys. Rev. E,65 (2002) 056209.
[25]Zhou Y., Zhou J. and Liu Z., Euro. Phys. Lett.,84 (2008) 60001.
[26]Osipov G., Hu B., Zhou C. S., Ivanchenko M. V. and Kurths J., Phys. Rev. Lett.,91 (2003)024101.
[27]Barabasi A.-L. and Albert R., Science,286 (1999) 509.
[28]Liu Z., Lai Y.-C., Ye N. and Dasgupta P., Phys. Lett. A,303 (2002) 337.
[29]Zhou Y., Zhou J., Wang X., Guan S., Liu Z. and Lai C.H., (to be accepted by Europhysics Letters)
[1]Lilienfeld A M.流行病学基础。俞焕文,译。上海：上海科技出版社,1981。
[2]华西医科大学流行病教研室,流行病学研究方法,成都：华西医科大学内部教材,1985。
[3]Bailey N T J. The Mathematical Theory of Infections Diseases and Its Application. New York:Hafner Press,1975.
[4]Anderson R M, May R M. Infections Diseases of Humans. Oxford:Oxford University Press,1992.
[5]Ferguson N M, Keeling M J, Edmunds W J, Gant R, Grenfell B T, Anderson R M, Leach S. Nature. (2003);425:681-685.
[6]Stone, L., Olinky, R. and Huppert, A. Seasonal dynamics of recurrent epidemics. Nature 446, (2007).553-536.
[7]Cummings, D. A. T., Irizarry, R. A., Huang, N. E., Endy, T. P., Nisalak, A., Ungchusak, K. & Burke, D. S. Nature 427, (2004) 344-347.
[8]Pastor-Satorras R. and Vespignani A., Phys. Rev. Lett.86,3200 (2001).
[9]Eguiluz V.M. and Klemm, Phys. Rev. Lett.89,108701 (2002).
[10]Zonghua Liu and Bambi Hu, Europhys.Lett.,72(2),315-321(2005).
[11]M.Boguna, R. Pastor-Satorras and A Vespignani, Phys.Rev.Lett.90, 028701(2003).
[12]Shao Z.G. and Sang J.P., Physica A 351,662(2005).
[13]M. Barthelemy, R. Pastor-Satorras and A Vespignani, Phys.Rev.Lett.92, 178701(2004).
[14]Ramon Huerta and Lev S. Tsimring, Phys. Rev. E.66,056115(2002).
[15]Thilo Gross, Carlos J. Dommar D'Lima, and Bernd Blasius, Phys. Rev. Lett. 96,208701(2006).
[16]M. Barthelemy, A. Barrat, R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett.92,178701(2002).
[17]T. Zhou, J.-G. Liu, W.-J. Bai, G.R. Chen, B.-H. Wang, Phys. Rev. E.74, 056109(2006).
[18]R. Pastor-Satorras and A. Vespignani, Phys.Rev.Lett.86,3200(2000); R. Pastor-Satorras and A. Vespignani,Phys.Rev.E.65,035108(2002).
[19]J. Joo and J. L. Lebowitz, Phys.Rev.E.69,066105(2004).
[20]M. Barthelemy and A. Barrat, J.Theor.Biol.235 275(2005).
[21]Z. Liu, Y. C. Lai, and N. Ye, Phys.Rev E.67,031911(2003).
[22]M.E.J. Newman, Phys.Rev.E.66,016128(2002).
[23]Y. Moreno, J. B. Gomez, and A. F. Pacheco, Phys.Rev.E.68,035103(R) (2003).
[24]H.Zhang, Z.Liu, and W.Ma, Chin.Phys.Lett.23,1050(2006).
[25]A. Grabowski and R. A. Kosinski, Phys.Rev.E.70,031908 (2004).
[26]V. Colizza, A. Barrat, M. Barthelemy and Veapignani, Inter. J. of Bif. Chaos,17 (2007)2491.
[27]M. Kuperman, G. Abramson, Phys. Rev. Lett.86,2909 (2001).
[28]G. Yan, Z. Fu, J. Ren and W. Wang, Phys.Rev.E.75,016108 (2007).
[29]J. Zhou, Z. Liu, and B. Li, Phys. Lett. A 368,458 (2007).
[30]Y. Moreno, M. Nekovee, A.F. Pacheco, Phys. Rev. E.69,066130 (2004).
[31]A. Sudbury, J. Appl. Prob.22,443. (1985).
[32]D.H. Zanette, Phys. Rev. E.64,050901(R) (2001); D.H. Zanette, Phys. Rev. E.65,041908 (2002).
[33]R. Pastor-Satorras and A.Vespignani, Phys. Rev. E.63,066117(2001).
[34]Tessone C. J., Toral R., Amengual P., Wio H. S. and Miguel M. S., Eur. Phys. J. B,39,535 (2004).
[35]Galam S., Physica A,320,571 (2003); J. Stat. Phys,61,943 (1990); J. Math. Psychol.,30,426(1986).
[36]Newman M. E. J., Phys. Rev. Lett.,89 (2002) 208701; Phys. Rev. E, 67,026126 (2003); Newman M. E. J. and Park J., Phys. Rev. E,68,036122 (2003); Newman M. E. J. and Girvan M., Phys. Rev. E,69,026113 (2004).
[37]Jin E. M., Girvan M. and Newman M. E. J., Phys. Rev. E,64,046132 (2001).
[38]M.E. J. Newman, M. Girvan, Phys. Rev. E.69,026113(2004).
[39]Pastor-Satorras R. and Vespignani A., Phys. Rev. E,65,035108(2002).
[40]Eguiluz V. M. and Klemm K., Phys. Rev. Lett.,89,108701 (2002).
[41]Zheng D.-F., Hui P. M., Trimper S. and Zheng B.2005 Physica A 352 659
[42]Naim E. B.2004 Phys. Rev. E 69 050901 (R)
[43]Yan G., Zhou T.,Wang J., Fu Z.-Q. and Wang B.-H.2005 Chin. Phys. Lett. 22 510
[44]Zhang H., Liu Z. and Ma W.2006 Chin. Phys. Lett.23 1050
[45]Vecchio A., Primavera L. and Carbone V.2006 Phys. Rev. E 73 031913
[46]Zhou Y., Liu Z. and Zhou J., Chin. Phys. Lett. Vol 24, No.2 (2007) 581.
[1]R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14, L453 (1981).
[2]B. McNamara and K. Wiesenfeld, Phys. Rev. A 39,4854(1989).
[3]A. Longtin, J. Stat. Phys.70,309 (1993).
[4]L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys.70, 223 (1998).
[5]D. F. Russell, L. A. Wilkens, and F. Moss, Nature (London) 402,291 (1999).
[6]B. Lindner, J. Garcia-Ojalvo, A. Neiman, and L. Schimansky-Geier, Phys. Rep.392,321 (2004).
[7]J. F. Lindner, B. K. Meadows, W. L. Ditto, M. E. Inchiosa, and A. R. Bulsara, Phys. Rev. Lett.75,3 (1995).
[8]Z. Liu, Y.-C. Lai, and A. Nachman, Phys. Lett. A 297,75(2002); Int. J. Bifurcation Chaos Appl. Sci. Eng.14,1655(2004).
[9]A. A. Zaikin, J. Kurths, and L. Schimansky-Geier, Phys. Rev. Lett.85,227 (2000).
[10]J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett.77,2863(1996).
[11]Hu Gang, T. Ditzinger, C. Z. Ning, and H. Haken, Phys. Rev. Lett.71,807 (1993).
[12]A. S. Pikovsky and J. Kurths, Phys. Rev. Lett.78,775 (1997).
[13]Z. Liu and Y.-C. Lai, Phys. Rev. Lett.86,4737 (2001); Y.-C. Lai and Z. Liu, Phys. Rev. E 64,066202 (2001); L. Zhu, Y.-C. Lai, Z. Liu, and A. Raghu, Phys. Rev. E 66,015204(R) (2002).
[14]A. S. Pikovsky, A. Zaikin, and M. A. de la Casa, Phys. Rev. Lett.88, 050601 (2002).
[15]J. A. Acebron, S. Lozano, and A. Arenas, Phys. Rev. Lett.99,128701 (2007).
[16]A. Longtin, A. Bulsara, D. Pierson, and F. Moss, Biol. Cybern.70,569 (1994).
[17]S. Camalet, T. Duke, F. Julicher, and J. Prost, Proc. Natl. Acad. Sci. U.S.A. 97,3183 (2000).
[18]A. Vilfan and T. Duke, Biophys. J.85,191 (2003).
[19]R. Stoop, A. Kern, M. C. Gopfert, D. A. Smirnov, T. V. Dikanev, and B. P. Bezrucko, Eur. Biophys. J.35,511 (2006).
[20]L. Moreau and E. Sontag, Phys. Rev. E 68,020901 (R) (2003).
[21]B. Seo, R. Krishnan, and T. Munakata, Phys. Rev. E 75,056106 (2007); T. Munakata, T. Hada, and M. Ueda, PhysicaA 375,492 (2007).
[22]Z. Liu, and T. Munakata, Phys. Rev. E 78,046111 (2008).
[23]A.-L. Barabasi and R. Albert, Science 286,509 (1999).
[24]Z. Liu, Y.-C. Lai, N. Ye, and P. Dasgupta, Phys. Lett. A 303,337 (2002); Z. Liu, Y.-C. Lai, N. Ye, and P. Dasgupta, Phys. Rev. E 66,036112 (2002).
[25]R. Albert and A.-L. Barabasi, Rev. Mod. Phys.74,47 (2002).
[26]S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Phys. Rep.424,175 (2006).
[27]O. Sporns, D. R. Chialvo, M. Kaiser, and C. C. Hilgetag, Trends Cogn. Sci. 9,418(2004).
[28]V. M. Eguiluz, D. R. Chialvo, G. A. Cecchi, M. Baliki, and A. V. Apkarian, Phys. Rev. Lett.94,018102 (2005).
[29]C. Zhou, L. Zemanova, G. Zamora, C. C. Hilgetag, and J. Kurths, Phys. Rev. Lett.97,238103 (2006).
[30]Zhou Y., Zhou J. and Liu Z., Signal Amplification by Adaption on Weighted Network, (in preparation)