复杂网络上传播动力学研究
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摘要
复杂网络上传播动力学的研究是复杂性科学中一个全新的研究方向,具有广阔的应用背景,往往需要借助于多学科的研究手段,具备多学科的交叉和融合的特征。目前,复杂网络上传播动力学的研究是复杂网络理论和实证研究的重要而具有挑战性的课题。本文重点在于研究复杂网络上的疾病和舆论传播动力学行为的一般规律。
本文综述了复杂网络的基本概念、典型的网络机制模型,尤其是阐述了目前在国际上流行的主要几种基于复杂网络的本传播模型及其传播行为特点。结合本文的研究内容,重点介绍了基于小世界和无标度网络上的传播特性。
对于疾病传播动力学的研究,首先我们建立了一个同时考虑人群的死亡、出生因素、人群扩散特性以及季节变化对传播概率的影响的疾病传播模型,在此模型中一个格点只能容纳一人。在此模型中,我们研究了扩散人群中疾病传播的一般规律:人群时空分布特性。研究表明,在此模型中存在两个重要的临界传播概率振幅。一个是维持疾病传播的传播概率振幅临界值ε_(ec),另一个是感染人群空间分布能呈现螺旋波图形的传播概率波动振幅临界值ε_c。当传播概率振幅达到ε_c时,同类人群空间分布呈现集聚现象,并出现稳定、清晰的螺旋状图案。同时,感染人群、健康人群,免疫人群随时间的演化规律也会出现同步自振荡现象。随着传播概率振幅的增加,各类人群的同步自振荡现象更明显,上述螺旋形时空分布和同步自振荡特性越来越明显、越来越稳定。
在上面模型的框架上,我们引入人口噪声因素,构建了另一个传染病模型,并假定一个格点可以容纳多人。重点研究了噪声对疾病传播规律的影响。发现当流行病传播达到稳态时,感染人群密度随时间演化呈现周期性的自振荡现象,并且噪声对疾病的振荡行为有强化作用。随着噪声增大,自振荡振幅加大。利用相图和对集体振荡同步序参量分析,进一步证实了上述结论的可靠性。研究还表明,不同状态(类)人群密度空间分布结构随噪声的增加,人群分布无序化随之增大,导致感染人群与健康人群共存的区域增大,相应的成团化趋势增强,从而感染者与健康者的实际接触概率增加,导致感染疾病和因病死亡人数增加。
我们还在一维链和二维离散欧氏空间中,构建了人群扩散的借助媒介传播的疾病传播模型。在此模型中,利用有限尺寸效应和临界驰豫特性两种途经研究了疾病传播规律的普适类问题。研究发现,感染人群密度对总人群密度的变化极其敏感:随着总人群密度的增加,疾病的传播由局部范围,会经历一个相变,达到大规模爆发;在相变过程中,感染人群密度分布满足严格幂律关系。通过对图形拟合得到相变过程的相关临界指数,分析表明,我们模型中的相关临界指数与一般的保守系统反应扩散过程中的相关临界指数吻合,表明我们的模型与一般的保守系统反应扩散过程的临界行为同属于一个普适类。事实上,我们的模型也可以看做保守系统,总人数不变。我们的工作证实,对于此类具有反应扩散流动行为的保守系统,其临界行为是相同的。
本文研究的第二大部分是基于复杂网络的舆论动力学研究。舆论动力学属于自然科学和社会科学的交叉领域。我们首先基于BA(解释)网络构建了拓扑结构随外场和惯性影响变化的舆论演化模型,网络在演化中,逐渐偏离BA网络,是一个自适应网络。在我们的模型中,引入恒定外场φ_0和形如e~(γk_h)的惯性作用。其特点是舆论和网络的演化受外场φ_0和节点的惯性e~(γk_h)的乘积,即演化序参量φ_0 e~(γk_i)制约。模型显示了舆论的演化与网络的拓扑结构变化有密切关系,受到外场与本身惯性因素的相互影响。在这个模型中,网络结构和舆论演化相互适应,是一个动态的演化过程。
当不考虑惯性作用时,研究表明,在外场作用下,舆论的演化影响了网络结构,网络中的度分布不再满足幂律分布,演化后的度分布服从泊松分布,即网络结构不再是初始的无标度网络。从不同种舆论分布随时间演化的规律分析,舆论的演化表现出很强的趋同效应。原来初始状态的50个舆论值,在长时间的演化后,大部分舆论灭亡,只有少数的舆论存留,且发展壮大。这一种情况与社会上的舆论、意见、信仰的演化大体上是吻合的。通过对舆论分布随外场作用变化规律的研究表明,总的来看,整个网络中舆论分布呈现明显的趋同效应,但随着外场φ_0越大,系统中趋同效应产生越慢。还有就是网络尺寸的大小也影响了舆论产生趋同效应的快慢,网络尺寸越大,系统中趋同效应现象产生越慢,舆论达到统一所需时间越长。
当考虑惯性作用时,研究发现惯性对舆论分布的影响不仅与惯性调节因子有关,而且还取决于外场φ_0。系统中网络节点度分布在节点惯性和外场的影响下,逐渐偏离无标度网络的幂率分布,而呈现头部下垂的幂率分布,即“重头”型的幂率分布。舆论总数在加入惯性的作用后,呈现丰富多彩的变化,调节惯性可以控制舆论总数的变化,甚至于还可以控制变化的快慢。在γ=0.3的时候,无论时间怎么变化,网络的舆论的总数和舆论所对应的社团的数目都是不变的。在γ=-10的时候,舆论数目开始时迅速减少,后来减少的趋势减缓,直至剩下一个,同时趋同现象随着时间的演化会不断的强化,以至于实现舆论的完全一律化。这充分表明,在节点惯性和外场的影响下,系统中的网络结构和舆论演化进行相互作用,相互适应,因此,这些因素改变了网络的结构,也影响系统中的舆论分布。
Transmission dynamics of complex network research is a very new and broader issue, and it is a new direction in the complexity of scientific research section. At present, it has been widely used in many areas. In this paper we study the transmission dynamics of disease and public opinion on the complex networks.
We summarize the basic concept and typical mechanism model of complex networks, and we especially introduce some main transmission models and the spreading behavior of transmission model on complex networks, and we pay more attention to the newest investigation of spreading behavior in small world and scale free networks.
At first we establish an epidemical model of mobile individual, which takes into account births and deaths of people as well as seasonal variation of the transmission probability. And we study the propagation behaviors and the people's distribution trait of epidemic spreading in mobile individuals. Simulation results show that there exists a critical value of infected rate fluctuating amplitude, above which the epidemic can spread in whole population. Moreover, with the value of infected rate fluctuating amplitude increasing, the spatial distribution of infected population exhibit the spontaneous formation of irregular spiral waves and cluster phenomena, at the same time, the density of different population will oscillate automatically with time. What's more, the traits of dynamic grow clearly and stably when the time and the value of infected rate fluctuating amplitude increasing.
Then we introduce noisy population to above model, and establish epidemic spreading model in diffusion population considering the effects of noisy population. A node can contain more than one people in system. It is investigated that how to influence the epidemic spreading in diffusion population by the noisy. It is found that infected population density changing with time evolution occurre a clearly cyclical synchrotron self-oscillation phenomenon under steady state in epidemic spreading, and the synchrotron self-oscillation behavior is strengthen along with the noisy increasing, which is confirmed through analyzing phase pattern and collective synchronization order parameter. At the same time, the different population density spatial distribution are more disorder with noisy increased, which induce to the infected population and susceptible population contacting more frequently, and it lead to the number of infected population and the death for infected are increased with the noisy increased.
We investigate the critical behavior of an epidemical model in a diffusive population mediated by a static vector environment on 2D network using Finite-size and short-time dynamic scaling relations. We find that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. Finite-size and short-time dynamic scaling relations are used to determine the critical population density and the critical exponents characterizing the behavior near the critical point. The results are compatible with the universality class of directed percolation coupled to a conserved diffusive field with equal diffusion constants.
At the same time, public opinion evolution model is investigated on scale free network in which the topology structure is always changing in this paper. In the model we consider the influence of a constant external fieldφ_0 and node's inertia e~(γk_h) on publicopinion evolution, here we define the public opinion evolution parameter asφ_0e~(γk_h). It is indicated that public opinion evolution and network topological structure interact with each other, and they are all influenced by the external fieldφ_0, and the node's inertiafactorγ.
Without considering the influence of node's inertia, it is found that the time evolution of the public opinion not only is controlled by the topological structure , but also induces the change of the topological structure: The network structure is no longer the initial scale-free network, but Poisson distribution. With the time evolution in system, there is obvious convergenve effect of the public opinions. The dozens of opinions in initial state evolve with the time, Most of them perish and only a few of them remain and development after a long time. The trend is coincident with the evolution of the public opinion, views and beliefs in the society.
Considering the influence of node's inertia, it is found that degree distribution gradually gets away typical power law distribution with the public opinion time evolution in our model. The number of public opinion will decrease by the time evolution without considering node's inertia. In other words, the evolution of network itself must obliterate some public opinion value. The result will take on colorful changes with time evolution.when considering node's inertia. As the parameterγ= 0.3, no matter how the time evolves, both the opinion's number and distribution are changeless. But when the parameterγ=-10, the number of opinion decrease sharply at the beginning, then this tendency slow down, until there is only one opinion value remaining. We draw the conclusion that adjust of node's inertia factorγcan control the opinion value's number, which even controls the rate of change.
本文综述了复杂网络的基本概念、典型的网络机制模型,尤其是阐述了目前在国际上流行的主要几种基于复杂网络的本传播模型及其传播行为特点。结合本文的研究内容,重点介绍了基于小世界和无标度网络上的传播特性。
对于疾病传播动力学的研究,首先我们建立了一个同时考虑人群的死亡、出生因素、人群扩散特性以及季节变化对传播概率的影响的疾病传播模型,在此模型中一个格点只能容纳一人。在此模型中,我们研究了扩散人群中疾病传播的一般规律:人群时空分布特性。研究表明,在此模型中存在两个重要的临界传播概率振幅。一个是维持疾病传播的传播概率振幅临界值ε_(ec),另一个是感染人群空间分布能呈现螺旋波图形的传播概率波动振幅临界值ε_c。当传播概率振幅达到ε_c时,同类人群空间分布呈现集聚现象,并出现稳定、清晰的螺旋状图案。同时,感染人群、健康人群,免疫人群随时间的演化规律也会出现同步自振荡现象。随着传播概率振幅的增加,各类人群的同步自振荡现象更明显,上述螺旋形时空分布和同步自振荡特性越来越明显、越来越稳定。
在上面模型的框架上,我们引入人口噪声因素,构建了另一个传染病模型,并假定一个格点可以容纳多人。重点研究了噪声对疾病传播规律的影响。发现当流行病传播达到稳态时,感染人群密度随时间演化呈现周期性的自振荡现象,并且噪声对疾病的振荡行为有强化作用。随着噪声增大,自振荡振幅加大。利用相图和对集体振荡同步序参量分析,进一步证实了上述结论的可靠性。研究还表明,不同状态(类)人群密度空间分布结构随噪声的增加,人群分布无序化随之增大,导致感染人群与健康人群共存的区域增大,相应的成团化趋势增强,从而感染者与健康者的实际接触概率增加,导致感染疾病和因病死亡人数增加。
我们还在一维链和二维离散欧氏空间中,构建了人群扩散的借助媒介传播的疾病传播模型。在此模型中,利用有限尺寸效应和临界驰豫特性两种途经研究了疾病传播规律的普适类问题。研究发现,感染人群密度对总人群密度的变化极其敏感:随着总人群密度的增加,疾病的传播由局部范围,会经历一个相变,达到大规模爆发;在相变过程中,感染人群密度分布满足严格幂律关系。通过对图形拟合得到相变过程的相关临界指数,分析表明,我们模型中的相关临界指数与一般的保守系统反应扩散过程中的相关临界指数吻合,表明我们的模型与一般的保守系统反应扩散过程的临界行为同属于一个普适类。事实上,我们的模型也可以看做保守系统,总人数不变。我们的工作证实,对于此类具有反应扩散流动行为的保守系统,其临界行为是相同的。
本文研究的第二大部分是基于复杂网络的舆论动力学研究。舆论动力学属于自然科学和社会科学的交叉领域。我们首先基于BA(解释)网络构建了拓扑结构随外场和惯性影响变化的舆论演化模型,网络在演化中,逐渐偏离BA网络,是一个自适应网络。在我们的模型中,引入恒定外场φ_0和形如e~(γk_h)的惯性作用。其特点是舆论和网络的演化受外场φ_0和节点的惯性e~(γk_h)的乘积,即演化序参量φ_0 e~(γk_i)制约。模型显示了舆论的演化与网络的拓扑结构变化有密切关系,受到外场与本身惯性因素的相互影响。在这个模型中,网络结构和舆论演化相互适应,是一个动态的演化过程。
当不考虑惯性作用时,研究表明,在外场作用下,舆论的演化影响了网络结构,网络中的度分布不再满足幂律分布,演化后的度分布服从泊松分布,即网络结构不再是初始的无标度网络。从不同种舆论分布随时间演化的规律分析,舆论的演化表现出很强的趋同效应。原来初始状态的50个舆论值,在长时间的演化后,大部分舆论灭亡,只有少数的舆论存留,且发展壮大。这一种情况与社会上的舆论、意见、信仰的演化大体上是吻合的。通过对舆论分布随外场作用变化规律的研究表明,总的来看,整个网络中舆论分布呈现明显的趋同效应,但随着外场φ_0越大,系统中趋同效应产生越慢。还有就是网络尺寸的大小也影响了舆论产生趋同效应的快慢,网络尺寸越大,系统中趋同效应现象产生越慢,舆论达到统一所需时间越长。
当考虑惯性作用时,研究发现惯性对舆论分布的影响不仅与惯性调节因子有关,而且还取决于外场φ_0。系统中网络节点度分布在节点惯性和外场的影响下,逐渐偏离无标度网络的幂率分布,而呈现头部下垂的幂率分布,即“重头”型的幂率分布。舆论总数在加入惯性的作用后,呈现丰富多彩的变化,调节惯性可以控制舆论总数的变化,甚至于还可以控制变化的快慢。在γ=0.3的时候,无论时间怎么变化,网络的舆论的总数和舆论所对应的社团的数目都是不变的。在γ=-10的时候,舆论数目开始时迅速减少,后来减少的趋势减缓,直至剩下一个,同时趋同现象随着时间的演化会不断的强化,以至于实现舆论的完全一律化。这充分表明,在节点惯性和外场的影响下,系统中的网络结构和舆论演化进行相互作用,相互适应,因此,这些因素改变了网络的结构,也影响系统中的舆论分布。
Transmission dynamics of complex network research is a very new and broader issue, and it is a new direction in the complexity of scientific research section. At present, it has been widely used in many areas. In this paper we study the transmission dynamics of disease and public opinion on the complex networks.
We summarize the basic concept and typical mechanism model of complex networks, and we especially introduce some main transmission models and the spreading behavior of transmission model on complex networks, and we pay more attention to the newest investigation of spreading behavior in small world and scale free networks.
At first we establish an epidemical model of mobile individual, which takes into account births and deaths of people as well as seasonal variation of the transmission probability. And we study the propagation behaviors and the people's distribution trait of epidemic spreading in mobile individuals. Simulation results show that there exists a critical value of infected rate fluctuating amplitude, above which the epidemic can spread in whole population. Moreover, with the value of infected rate fluctuating amplitude increasing, the spatial distribution of infected population exhibit the spontaneous formation of irregular spiral waves and cluster phenomena, at the same time, the density of different population will oscillate automatically with time. What's more, the traits of dynamic grow clearly and stably when the time and the value of infected rate fluctuating amplitude increasing.
Then we introduce noisy population to above model, and establish epidemic spreading model in diffusion population considering the effects of noisy population. A node can contain more than one people in system. It is investigated that how to influence the epidemic spreading in diffusion population by the noisy. It is found that infected population density changing with time evolution occurre a clearly cyclical synchrotron self-oscillation phenomenon under steady state in epidemic spreading, and the synchrotron self-oscillation behavior is strengthen along with the noisy increasing, which is confirmed through analyzing phase pattern and collective synchronization order parameter. At the same time, the different population density spatial distribution are more disorder with noisy increased, which induce to the infected population and susceptible population contacting more frequently, and it lead to the number of infected population and the death for infected are increased with the noisy increased.
We investigate the critical behavior of an epidemical model in a diffusive population mediated by a static vector environment on 2D network using Finite-size and short-time dynamic scaling relations. We find that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. Finite-size and short-time dynamic scaling relations are used to determine the critical population density and the critical exponents characterizing the behavior near the critical point. The results are compatible with the universality class of directed percolation coupled to a conserved diffusive field with equal diffusion constants.
At the same time, public opinion evolution model is investigated on scale free network in which the topology structure is always changing in this paper. In the model we consider the influence of a constant external fieldφ_0 and node's inertia e~(γk_h) on publicopinion evolution, here we define the public opinion evolution parameter asφ_0e~(γk_h). It is indicated that public opinion evolution and network topological structure interact with each other, and they are all influenced by the external fieldφ_0, and the node's inertiafactorγ.
Without considering the influence of node's inertia, it is found that the time evolution of the public opinion not only is controlled by the topological structure , but also induces the change of the topological structure: The network structure is no longer the initial scale-free network, but Poisson distribution. With the time evolution in system, there is obvious convergenve effect of the public opinions. The dozens of opinions in initial state evolve with the time, Most of them perish and only a few of them remain and development after a long time. The trend is coincident with the evolution of the public opinion, views and beliefs in the society.
Considering the influence of node's inertia, it is found that degree distribution gradually gets away typical power law distribution with the public opinion time evolution in our model. The number of public opinion will decrease by the time evolution without considering node's inertia. In other words, the evolution of network itself must obliterate some public opinion value. The result will take on colorful changes with time evolution.when considering node's inertia. As the parameterγ= 0.3, no matter how the time evolves, both the opinion's number and distribution are changeless. But when the parameterγ=-10, the number of opinion decrease sharply at the beginning, then this tendency slow down, until there is only one opinion value remaining. We draw the conclusion that adjust of node's inertia factorγcan control the opinion value's number, which even controls the rate of change.
引文
[1]D.J.Watts and S.H.Strogaz,Collective dynamics of 'small-wofld'networks,Nature(london),1998,393:440
[2]A-L Barabasi and R.Albert,Emergence of scaling in random networks,Science,1999,286:509
[3]R.Albert and A-L Barabasi,Statistical mechanics of complex networks[J],Rev.Mod.Phys.,2002,74:47-97
[4]M.E.J.Newman,The structure and function of complex networks,SIAM Review 2003,45:167-224
[5]X.-F.Wang,Complex networks:topology,dynamics and synchronization,Int.J.Bifurcation & Chaos,2002,12:885
[6]T.S.Evans,Complex Networks.arXiv:cond-mat/0405123.
[7]J.E.Cohen,F.Briand,and C.M.Newman,Community food webs:Date and theory(Berlin:Springer-Verlag) 1990
[8]吴金闪,狄增如.从统计物理看复杂网络研究.Prog.PhyS,2004:24,18-46
[9]M.Faloutsos,P.Faloutsos and C.Faloutsos.Computer Communications Review,1999,29:251
[10]S.Boccaletti,V.Latora,Y.Moreno,M.Chavez,D.U.Hwang,Complex networks:Structure and dynamics,Physics Reports 2006,424:175-308
[11]P.Erdos and A.Renyi,Random Graph,Publ.Math.,1959,6:290-310
[12]R.Albert,H.Jeong,and A.-L.Barabasi,Emergence of Scaling in Random Networks,Nature(London),1999,401:130-132
[13]C.Moore,and M.E.J.Newman,Exact solution of site and bond percolation on small-world networks,Phys.Rev.E,2000,62:7059-7064
[14]A.L.Lloyd,and R.M.May,Science,292,1316(2001)
[15]R.Paster-Satorras,and A.Vespignani,Epidemic Spreading in Scale-Free Networks,Phys.Rev.Lett.,2001,86:3200-3203
[16]S.N.Dorogovtsev,A.V.Goltsev,and J.F.F.Mendes,Ising model on networks with an arbitrary distribution of connections,Phys.Rev.E,2002,66:016104-016108
[17]S.N.Dorogovtsev,A.V.Goltsev,and J.F.F.Mendesl,Ising model in small-world networks,Phys.Rev.E,2002,65:066110-066115
[18]A.Scala,L.A.N.Amaral,and M.Barthe1emy,Small-World Networks and the Conformation Space of a Short Lattice Polymer Chain,Europhys.Lett.,2001,55:594-600
[19]R.Monasson,Diffusion,localization and dispersion relations on "small-world"lattices,Eur.Phys.J.B,1999,12:555-510
[20]S.Jespersen,I.M.Sokolov,and A.Blumen,Relaxation properties of small-world networks,Phys.Rev.E,2000,62:4405-4408
[21]F.Jasch and A.Blumen,Target problem on small-world networks,Phys.Rev.E,2001,63:041108-041112
[22]J.Lahtinen,J.Kertesz,and K.Kaski,Scaling of random spreading in small world networks,Phys.Rev.E,2001,64:057105-057107
[23]S.A.Pandit and R.E.Amritkar,Random spread on the family of small-world networks,Phys.Rev.E,2001,63:041104-041110
[24]汪小帆,李翔,陈关荣著,复杂网络理论及其应用,清华大学出版社,2006
[25]J.Lahtinen,J.Kertesz,and K.Kaski,Random spreading phenomena in annealed small world networks,Physica A,2002,311:571-580
[26]J.D.Noh and H.Rieger,Random Walks on Complex Networks,Phys.Rev.Lett.,2004,92:118701-118704
[27]N T J Bailey The Mathematical Theroy of Infection Disease,1975,2nd ed.Hafner New York
[28]马知恩,周义仓,王稳地,靳祯.传染病动力学的数学建模与研究.北京科学出版社,2003
[29]陈关荣,许晓鸣.复杂网络理论与应用.上海系统科学出版社.2008年6月
[30]T.Zhou,J.G.Liu,W.J.Bai,G.R.Chen,B.H.Wang,Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity,Phys.Rev.E 2006,74:056109
[31]J.Joo and J.L.Lebowitz,Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation,Phys.Rev.E 2004,69,066105
[32]M.Y.li,James S.Muldowney,Global stability for the SEIR model in epidemiology,Math.Biosci.,1995,125:155-164
[33]M.Y.li,J.R.Greaf,L.C.Wang,J.Karsai,Global dynamics of a SEIR model with a varying total population size,Math.Biosci.,1999,160:191-213
[34]Meng Fan,M.Y.Li,Ke Wang,Global stability of an SEIS epidemic model with recruitment and a verying total population size,Math.Biosci.,2001,170:199-208
[35]K.L.Cooke,P.Van den Driessche and X.Zou,Interaction of maturation delay and nonlinear birth in population and epidemic models,J.Math.Biol.,1999,39:332-352
[36]H.W.Hethcote,P.Van den Driessche,Two SIS epidemiological models with delays,J.Math.Biol.,2000,40:2-26
[37]P.Nelson,A.Perelson,Mathematical analysis of delay differential equation models of HIV-1 intection,Math.Biosci.,2002,179:73-94
[38]原三领,含时滞流行病模型的研究.西安交通大学博士论文,2001
[39]H.Hethcote.The mathematics of infectious diseases,SIAM Review,2000,42:599-653
[40]Z.Feng,H.R.Thieme.Recurrent outbreaks of childhood diseases revisted:the impact of isolation,Math.Biosci.,1995,128:93-130
[41]Fred Brauer,P.Van den Driessche,Models for transmission of disease with immigration of infectives,Math.Biosci.,2002,180:141-160
[42]Li Jianquan,Zhang Juan,Ma zhien,Global analysis of some epidemic models with general contact rate and constant immigration,Applied Math.Mech.,2004,4:396-404
[43]H.R.Thieme,Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations.Math.Biosci.,1992,111:99-130
[44]J Hyman,J Li,An intuitive formulation for the reprodictive number for the spread of diseases in heterogeneous populations,Math.Biosci.2000,167:65-86
[45]R.M.Anderson,R.M.May,The invasion,persistence,and spread of infectious diseases within animal and plant communites,Phil.Trans.R.Soc,London,1986,B314:533-570
[46]Y.N.Xiao,L.S.Chen,Modeling and analysis of a predator-prey model with disease in the prey,Math.Biosci,2001,171:59-82
[47]Y.N.Xiao,L.S.Chen,A ratio-dependent predator-prey model with disease in the prey,Appl.Math.Comput.,2002,131:397-414
[48]L.T.Han,Z.E.Ma,S.Tan,An SIRS epidemic model of two competitive species,Mathl.Commput.Modelling,2003,37:87-108
[49]Sznajd W.K.,Sznajd J.,Opinion evolution in closed community,Int.J.Mod.Phys.C,2000,11(6):1157-1165
[50]Amblard F.,Deffuant guillaume the role of network topology on extremism propagation with the relative agreement opinion dynamics,Physica A,2004,343:725-738
[51]Galam S.,Contrarian deterministic effects on opinion dynamics:"The hung elections scenario",Physica A,2004,333:453-460
[52]Guan J.Y.,Wu Z.X.,Wang Y.H.,Effects of inhomogeneous influence of individuals on an order-disorder transition in opinion dynamics,Phys.Rev.E.,2007,76:042102
[53]秦希德,饶德江,舆论学教程,武汉大学出版社,1989,89-102
[54]Sznajd W.K.,Controlling simple dynamics by a disagreement function,Phys.Rev.E.,2002,66:046131
[55]Elgazzar A.S.,Applications of small-world networks to some socio-economic systems,Physica A,2002,324:402-407
[56] Jiang L. L., Hua D. Y., Zhu J. F. et al, Opinion dynamics on directed small-world networks, Eur. Phys. J. B, 2008,65(2): 251-255
[57] Pluchino A., Latora V., Rapisarda A., Compromise and synchronization in opinion dynamics, Eur. Phys. J. B, 2006,50: 169-176
[58] Ben N. E., Krapivsky P. L., Vazquez F., et al, Unity and discord in opinion dynamics, Phys A, 2003,330: 99-106
[59] Bagnoli F., Carletti T., Fanelli D. et al, Dynamical affinity in opinion dynamics modeling, Phys. Rev. E, 2007,76:066105
[60] Guzman V. L., Hernandez P. R., Small-world topology and memory effects on decision time in opinion dynamics, Physica A, 2006,372(2): 326-332
[61] Fortunato S., Damage spreading and opinion dynamics on scale-free networks, Physica A, 2005, 348: 683-690
[62] Galam S., The dynamics of minority opinions in democratic debate, Physica A, 2004,336: 56-62
[63] Deffuant G., Neau D., Amblard F., Weisbuch G., Mixing beliefs among interacting agents, Adv. Complex Syst., 2000,3: 87-98
[64] Hegselmann R.,Krause U., Opinion dynamics and bounded confidence models, analysis, and simulations. Journal of Artifical Societies and Social Simulation, 2002
[65] P.Clifford and A.Sudbury, Biometrika 60,581(1973)
[66] R.Holley and T.Liggett, Annals of Probability, 3, 643(1975)
[67] J.Lorenz, ar Xiv:0707.1762vl(2007)
[68] D.Stauffer, A.Sousa and C.Schulze, JASSS 7,3(2004)
[69] G. Weisbuch, G.Deffuant,F.Amblard, and J.P.Nadal, in Lecture Notes in Economics and Mathematical Systems, edited by G.Fandel and W. Trockel(Springer Verlag, Berlin-Heidelberg, Germany), Vol 521, pp.225-242
[70] Sznajd W. K., Dynamical model of Ising spins , Phys. Rev. E 2004, 70: 037104
[71] Schulze C., Advertising, consensus, and aging in multilayer Sznajd model, Int. J. Mod.Phys.C,2004,15:569-573
[72]Santo F.,The Sznajd consensus model with continuous opinions,International Journal of Modern Physics C,2005,16(1):17-24
[73]Wooszyn M.,Stauffer D.,Kuakowski K.,Phase transitions in Nowak-Sznajd opinion dynamics,Physica A,2007,338:453-458
[74]郭龙,外场中复杂网络的拓扑结构对舆论动力学的影响,华中师范大学硕士毕业论文,2007,13-16
[75]Holme P.,Newman M.E.J.,Nonequilibrium phase transition in the coevolution of networks and opinions,Phys.Rev.E,2006,74:056108
[76]R.M.Anderson and R.M.May,Infectious Disease of Humans,Oxford University Press(1992)
[77]P.S.Romualdo,V.Alessandro.Epidemic Spreading in Scale-Free Networks[J],Physical Review Letters,2001,86:3200-3203
[78]M.Kuperman and G.Abramson,Small world effect in an epidemiological model.Phys.Rev.Let.2001,86:2909-1913
[79]R.M.May and A.L.Lioyd,Infection dynamics on scale-free networks,Phys.Rev.E.2001,64:066112
[80]R.Pastor-Satorras and A.Vespignani,Epidemic dynamics in finite size Scale-free networks,Phys.Rev.E,2002,65:035108
[81]R.Pastor-Satorras and A.Vespignani,Immunization of complex networks,Phys.Rev.E,2002,65:036104
[82]R.Pastor-Satorras and A.Vespignani,Epidemic dynamics and endemic states in complex networks,Phys.Rev.E,2001,63:066117
[83]R.Pastor-Satorras and A.Vespignani,Epidemics and immunization in scale-free networks.In Bornholdt S,Schuster H G(eds.) Handbook of graph and networks.Berlin:Wiley-VCH,2003
[84]M.E.J.Newman,The spread of epidemic disease on networks,Phys.Rev.E,2002, 66: 016128
[85] C. Moore and M. E. J. Newman, Epidemics and percolation in small-world networks, Phys. Rev. E, 2000,61: 5678-5682
[86] F.C.SantoS,J.f.Rodrigues and J.M.pacheco, Epidemic spreading and cooperation dynamics on homogeneous small-world networks, Phys.Rev.E, 2005, 72: 056128
[87] Y. Moreno and A. Vazquez, Disease spreading in structured scale-free networks, Euro. Phys.J.B,2003, 31:265
[88] P. Crepey, F. P. Alvarez, and M. Barthelemy, Epidemic variability in complex networks, Phys. Rev. E, 2006,73:046131
[89] M. Barthelemy, A.Barrat, R. Pastor-Satorras, and A. Vespignani, Velocity and Hierarchical Spread of Epidemic Outbreaks in Scale-Free Networks, Phys. Rev. Lett., 2004,90: 178701
[90] M.Boguna, R.Pastor-Satorras and A.Vespignani, Absence of Epidemic Threshold in Scale-Free Networks with Degree Correlations, Phys. Rev. Lett., 2003, 90: 028701
[91] E. Ben-Naim and P. L. Krapivsky, Size of outbreaks near the epidemic threshold, Phys. Rev. E, 2004, 69:050901
[92] T. Petermann and P. De Los Rios, Role of clustering and gridlike ordering in epidemic spreading, Phys. Rev. E, 2004,69: 066116
[93] H.-K. Janssen, M. Müller, and O. Stenull, Generalized epidemic process and tricritical dynamic percolation, Phys. Rev. E, 2004, 70: 026114
[94] M. Ballard, V. M. Kenkre, and M. N. Kuperman, Periodically varying externally imposed environmental effects on population dynamics, Phys. Rev. E, 2004, 70:031912
[95] I. B. Schwartz, L. Billings, and E. M. Bollt, Dynamical epidemic suppression using stochastic prediction and control, Phys. Rev. E, 2004, 70: 046220
[96] R. M. Ziff, Simple algorithm to test for linking to Wilson loops in percolation, Phys. Rev. E, 2005,72: 017104
[97] C. Castellano and R. Pastor-Satorras, Non-Mean-Field Behavior of the Contact Process on Scale-Free Networks, Phys. Rev. Lett., 2006,96: 038701
[98] S. N. Dorogovtsev, J. F. Mendes, Evolution of networks, Adv. Phys. 51, 2002:1079-1187
[99] A. L.Lloyd, R. M. May, Science, 2001, 292: 1316-1317
[100] F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley and Y. Aberg, The Web of Human Sexual Contacts, Nature(London) 2001,411: 907-908
[101] S. H. Strogatz, Exploring complex networks, Nature (London) 2001,410: 268-270.
[102] A. L. Barabasi and R .Albert, Emergence of scaling in random networks, Science, 1999,286: 509
[103] A.Grabowski and R.A.Kosinski, Epidemic spreading in a hierarchical social network, Phys. Rev. E. 2004, 70: 031908
[104] M. E. J. Newman, Threshold Effects for Two Pathogens Spreading on a Network, Phys.Rev.Lett., 2005,95:108701
[105] R. M. Anderson and R. M. May, Infectious Diseases of Humans (Oxford: Oxford University Press, 1992)
[106] Newman M E J, Watts D J, Scaling and percolation in the small-world network model, Phys Rev E, 1999, 60: 7332-7342
[107] Newman M E J, Jensen I, Ziff R M. Percolation and epidemics in a two-dimensional small world, Phys Rev E, 2002,65: 021904
[108] Kuperman M, Abramson G, Small world effect in an epidemiological model, Phys Rev. Lett, 2001, 86: 2909-2912
[109] Agiza H N, Elgazzar A S, Youssef S A, Phase transitions in some epidemic models defined on small-world networks, arXiv: cond-mat/0301004
[110] Barabasi A L, Albert R, Jeong H., Mean-field theory for scale-free random networks, Physica A, 1999, 272: 173-187
[111] Prashant M.Gade, Dynamic transitions in small world networks: Approach to equilibrium limit, Phys Rev.E 2005,72: 052903
[112] Gang Yan, Zhong-Qian Fu, Jie Ren and Wen-Xu Wang, Collective synchronization induced by epidemic dynamics on complex networks with communities,Phys.Rev.E 2007,75:016108
[113]Z.H.Liu and B.Hu,Epidemie spreadingin community networks,Euro.Phys.Lett.,2005,72(2):315-321
[114]J.Liu,J.Wu,and Z.R.Yang,The spread of infeetious disease on complex networks with household-strueture,Physiea A,2004,341:273-280
[115]Ming Tang,Li Liu,and Zonghua Liu,Influence of dynamical condensation on epidemic spreading in scale-free networks,Phys Rev.E 2009,79:016108
[116]YAN Gang,ZHOU Tao,WANG Jie,FU Zhong-Qian and WANG Bing-Hong,Epidemic Spread in Weighted Scale-Free Networks,CHIN.PHYS.LETT.2005,22(2):510
[117]Ballegooijen W M and Boerlijst M C Proc.Natl.Acad.Sci.U.S.A.2004,101:18246
[118]Liu Q X,Jin Z,and Liu M X Phys.Rev.E.2006,74:031110
[119]Zhou T,Fu Z Q and Wang B H Prog.Nat.Sci.2006,16:452
[120]Sabrina B L Araujo and de Aguiar M A M Phys.Rev.E 2007,75:061908
[121]Sabrina B LAraujo and de Aguiar M A M Phys.Rev.E 2008,77:022903
[122]J.J.Ramasco,M.A.Mu(?)oz,and C.A.da Silva Santos,Numerical study of the Langevin theory for fixed-energy sandpiles,Phys.Rev.E 2004,69:045105
[123]I.Dornic,H.Chate,and M.A.Mu(?)oz,Integration of Langevin Equations with Multiplicative Noise and the Viability of Field Theories for Absorbing Phase Transitions,Phys.Rev.Lett.2005,94:100601
[124]Candia J,Irreversible opinion spreading on scale-free networks,Phys.Rev.E 2007,75:026110
[125]Bartolozzi M.,Leinweber D.B.,Thomas A.W.,Stochastic opinion formation in scale-free networks,Phys.Rev.E,2005,72:046113
[126]罗萨里奥.N.曼特尼亚,H.尤金.斯坦尼,经济物理学导论--金融中的相关性和复杂性,2006,中国人民大学出版社P55
[2]A-L Barabasi and R.Albert,Emergence of scaling in random networks,Science,1999,286:509
[3]R.Albert and A-L Barabasi,Statistical mechanics of complex networks[J],Rev.Mod.Phys.,2002,74:47-97
[4]M.E.J.Newman,The structure and function of complex networks,SIAM Review 2003,45:167-224
[5]X.-F.Wang,Complex networks:topology,dynamics and synchronization,Int.J.Bifurcation & Chaos,2002,12:885
[6]T.S.Evans,Complex Networks.arXiv:cond-mat/0405123.
[7]J.E.Cohen,F.Briand,and C.M.Newman,Community food webs:Date and theory(Berlin:Springer-Verlag) 1990
[8]吴金闪,狄增如.从统计物理看复杂网络研究.Prog.PhyS,2004:24,18-46
[9]M.Faloutsos,P.Faloutsos and C.Faloutsos.Computer Communications Review,1999,29:251
[10]S.Boccaletti,V.Latora,Y.Moreno,M.Chavez,D.U.Hwang,Complex networks:Structure and dynamics,Physics Reports 2006,424:175-308
[11]P.Erdos and A.Renyi,Random Graph,Publ.Math.,1959,6:290-310
[12]R.Albert,H.Jeong,and A.-L.Barabasi,Emergence of Scaling in Random Networks,Nature(London),1999,401:130-132
[13]C.Moore,and M.E.J.Newman,Exact solution of site and bond percolation on small-world networks,Phys.Rev.E,2000,62:7059-7064
[14]A.L.Lloyd,and R.M.May,Science,292,1316(2001)
[15]R.Paster-Satorras,and A.Vespignani,Epidemic Spreading in Scale-Free Networks,Phys.Rev.Lett.,2001,86:3200-3203
[16]S.N.Dorogovtsev,A.V.Goltsev,and J.F.F.Mendes,Ising model on networks with an arbitrary distribution of connections,Phys.Rev.E,2002,66:016104-016108
[17]S.N.Dorogovtsev,A.V.Goltsev,and J.F.F.Mendesl,Ising model in small-world networks,Phys.Rev.E,2002,65:066110-066115
[18]A.Scala,L.A.N.Amaral,and M.Barthe1emy,Small-World Networks and the Conformation Space of a Short Lattice Polymer Chain,Europhys.Lett.,2001,55:594-600
[19]R.Monasson,Diffusion,localization and dispersion relations on "small-world"lattices,Eur.Phys.J.B,1999,12:555-510
[20]S.Jespersen,I.M.Sokolov,and A.Blumen,Relaxation properties of small-world networks,Phys.Rev.E,2000,62:4405-4408
[21]F.Jasch and A.Blumen,Target problem on small-world networks,Phys.Rev.E,2001,63:041108-041112
[22]J.Lahtinen,J.Kertesz,and K.Kaski,Scaling of random spreading in small world networks,Phys.Rev.E,2001,64:057105-057107
[23]S.A.Pandit and R.E.Amritkar,Random spread on the family of small-world networks,Phys.Rev.E,2001,63:041104-041110
[24]汪小帆,李翔,陈关荣著,复杂网络理论及其应用,清华大学出版社,2006
[25]J.Lahtinen,J.Kertesz,and K.Kaski,Random spreading phenomena in annealed small world networks,Physica A,2002,311:571-580
[26]J.D.Noh and H.Rieger,Random Walks on Complex Networks,Phys.Rev.Lett.,2004,92:118701-118704
[27]N T J Bailey The Mathematical Theroy of Infection Disease,1975,2nd ed.Hafner New York
[28]马知恩,周义仓,王稳地,靳祯.传染病动力学的数学建模与研究.北京科学出版社,2003
[29]陈关荣,许晓鸣.复杂网络理论与应用.上海系统科学出版社.2008年6月
[30]T.Zhou,J.G.Liu,W.J.Bai,G.R.Chen,B.H.Wang,Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity,Phys.Rev.E 2006,74:056109
[31]J.Joo and J.L.Lebowitz,Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation,Phys.Rev.E 2004,69,066105
[32]M.Y.li,James S.Muldowney,Global stability for the SEIR model in epidemiology,Math.Biosci.,1995,125:155-164
[33]M.Y.li,J.R.Greaf,L.C.Wang,J.Karsai,Global dynamics of a SEIR model with a varying total population size,Math.Biosci.,1999,160:191-213
[34]Meng Fan,M.Y.Li,Ke Wang,Global stability of an SEIS epidemic model with recruitment and a verying total population size,Math.Biosci.,2001,170:199-208
[35]K.L.Cooke,P.Van den Driessche and X.Zou,Interaction of maturation delay and nonlinear birth in population and epidemic models,J.Math.Biol.,1999,39:332-352
[36]H.W.Hethcote,P.Van den Driessche,Two SIS epidemiological models with delays,J.Math.Biol.,2000,40:2-26
[37]P.Nelson,A.Perelson,Mathematical analysis of delay differential equation models of HIV-1 intection,Math.Biosci.,2002,179:73-94
[38]原三领,含时滞流行病模型的研究.西安交通大学博士论文,2001
[39]H.Hethcote.The mathematics of infectious diseases,SIAM Review,2000,42:599-653
[40]Z.Feng,H.R.Thieme.Recurrent outbreaks of childhood diseases revisted:the impact of isolation,Math.Biosci.,1995,128:93-130
[41]Fred Brauer,P.Van den Driessche,Models for transmission of disease with immigration of infectives,Math.Biosci.,2002,180:141-160
[42]Li Jianquan,Zhang Juan,Ma zhien,Global analysis of some epidemic models with general contact rate and constant immigration,Applied Math.Mech.,2004,4:396-404
[43]H.R.Thieme,Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations.Math.Biosci.,1992,111:99-130
[44]J Hyman,J Li,An intuitive formulation for the reprodictive number for the spread of diseases in heterogeneous populations,Math.Biosci.2000,167:65-86
[45]R.M.Anderson,R.M.May,The invasion,persistence,and spread of infectious diseases within animal and plant communites,Phil.Trans.R.Soc,London,1986,B314:533-570
[46]Y.N.Xiao,L.S.Chen,Modeling and analysis of a predator-prey model with disease in the prey,Math.Biosci,2001,171:59-82
[47]Y.N.Xiao,L.S.Chen,A ratio-dependent predator-prey model with disease in the prey,Appl.Math.Comput.,2002,131:397-414
[48]L.T.Han,Z.E.Ma,S.Tan,An SIRS epidemic model of two competitive species,Mathl.Commput.Modelling,2003,37:87-108
[49]Sznajd W.K.,Sznajd J.,Opinion evolution in closed community,Int.J.Mod.Phys.C,2000,11(6):1157-1165
[50]Amblard F.,Deffuant guillaume the role of network topology on extremism propagation with the relative agreement opinion dynamics,Physica A,2004,343:725-738
[51]Galam S.,Contrarian deterministic effects on opinion dynamics:"The hung elections scenario",Physica A,2004,333:453-460
[52]Guan J.Y.,Wu Z.X.,Wang Y.H.,Effects of inhomogeneous influence of individuals on an order-disorder transition in opinion dynamics,Phys.Rev.E.,2007,76:042102
[53]秦希德,饶德江,舆论学教程,武汉大学出版社,1989,89-102
[54]Sznajd W.K.,Controlling simple dynamics by a disagreement function,Phys.Rev.E.,2002,66:046131
[55]Elgazzar A.S.,Applications of small-world networks to some socio-economic systems,Physica A,2002,324:402-407
[56] Jiang L. L., Hua D. Y., Zhu J. F. et al, Opinion dynamics on directed small-world networks, Eur. Phys. J. B, 2008,65(2): 251-255
[57] Pluchino A., Latora V., Rapisarda A., Compromise and synchronization in opinion dynamics, Eur. Phys. J. B, 2006,50: 169-176
[58] Ben N. E., Krapivsky P. L., Vazquez F., et al, Unity and discord in opinion dynamics, Phys A, 2003,330: 99-106
[59] Bagnoli F., Carletti T., Fanelli D. et al, Dynamical affinity in opinion dynamics modeling, Phys. Rev. E, 2007,76:066105
[60] Guzman V. L., Hernandez P. R., Small-world topology and memory effects on decision time in opinion dynamics, Physica A, 2006,372(2): 326-332
[61] Fortunato S., Damage spreading and opinion dynamics on scale-free networks, Physica A, 2005, 348: 683-690
[62] Galam S., The dynamics of minority opinions in democratic debate, Physica A, 2004,336: 56-62
[63] Deffuant G., Neau D., Amblard F., Weisbuch G., Mixing beliefs among interacting agents, Adv. Complex Syst., 2000,3: 87-98
[64] Hegselmann R.,Krause U., Opinion dynamics and bounded confidence models, analysis, and simulations. Journal of Artifical Societies and Social Simulation, 2002
[65] P.Clifford and A.Sudbury, Biometrika 60,581(1973)
[66] R.Holley and T.Liggett, Annals of Probability, 3, 643(1975)
[67] J.Lorenz, ar Xiv:0707.1762vl(2007)
[68] D.Stauffer, A.Sousa and C.Schulze, JASSS 7,3(2004)
[69] G. Weisbuch, G.Deffuant,F.Amblard, and J.P.Nadal, in Lecture Notes in Economics and Mathematical Systems, edited by G.Fandel and W. Trockel(Springer Verlag, Berlin-Heidelberg, Germany), Vol 521, pp.225-242
[70] Sznajd W. K., Dynamical model of Ising spins , Phys. Rev. E 2004, 70: 037104
[71] Schulze C., Advertising, consensus, and aging in multilayer Sznajd model, Int. J. Mod.Phys.C,2004,15:569-573
[72]Santo F.,The Sznajd consensus model with continuous opinions,International Journal of Modern Physics C,2005,16(1):17-24
[73]Wooszyn M.,Stauffer D.,Kuakowski K.,Phase transitions in Nowak-Sznajd opinion dynamics,Physica A,2007,338:453-458
[74]郭龙,外场中复杂网络的拓扑结构对舆论动力学的影响,华中师范大学硕士毕业论文,2007,13-16
[75]Holme P.,Newman M.E.J.,Nonequilibrium phase transition in the coevolution of networks and opinions,Phys.Rev.E,2006,74:056108
[76]R.M.Anderson and R.M.May,Infectious Disease of Humans,Oxford University Press(1992)
[77]P.S.Romualdo,V.Alessandro.Epidemic Spreading in Scale-Free Networks[J],Physical Review Letters,2001,86:3200-3203
[78]M.Kuperman and G.Abramson,Small world effect in an epidemiological model.Phys.Rev.Let.2001,86:2909-1913
[79]R.M.May and A.L.Lioyd,Infection dynamics on scale-free networks,Phys.Rev.E.2001,64:066112
[80]R.Pastor-Satorras and A.Vespignani,Epidemic dynamics in finite size Scale-free networks,Phys.Rev.E,2002,65:035108
[81]R.Pastor-Satorras and A.Vespignani,Immunization of complex networks,Phys.Rev.E,2002,65:036104
[82]R.Pastor-Satorras and A.Vespignani,Epidemic dynamics and endemic states in complex networks,Phys.Rev.E,2001,63:066117
[83]R.Pastor-Satorras and A.Vespignani,Epidemics and immunization in scale-free networks.In Bornholdt S,Schuster H G(eds.) Handbook of graph and networks.Berlin:Wiley-VCH,2003
[84]M.E.J.Newman,The spread of epidemic disease on networks,Phys.Rev.E,2002, 66: 016128
[85] C. Moore and M. E. J. Newman, Epidemics and percolation in small-world networks, Phys. Rev. E, 2000,61: 5678-5682
[86] F.C.SantoS,J.f.Rodrigues and J.M.pacheco, Epidemic spreading and cooperation dynamics on homogeneous small-world networks, Phys.Rev.E, 2005, 72: 056128
[87] Y. Moreno and A. Vazquez, Disease spreading in structured scale-free networks, Euro. Phys.J.B,2003, 31:265
[88] P. Crepey, F. P. Alvarez, and M. Barthelemy, Epidemic variability in complex networks, Phys. Rev. E, 2006,73:046131
[89] M. Barthelemy, A.Barrat, R. Pastor-Satorras, and A. Vespignani, Velocity and Hierarchical Spread of Epidemic Outbreaks in Scale-Free Networks, Phys. Rev. Lett., 2004,90: 178701
[90] M.Boguna, R.Pastor-Satorras and A.Vespignani, Absence of Epidemic Threshold in Scale-Free Networks with Degree Correlations, Phys. Rev. Lett., 2003, 90: 028701
[91] E. Ben-Naim and P. L. Krapivsky, Size of outbreaks near the epidemic threshold, Phys. Rev. E, 2004, 69:050901
[92] T. Petermann and P. De Los Rios, Role of clustering and gridlike ordering in epidemic spreading, Phys. Rev. E, 2004,69: 066116
[93] H.-K. Janssen, M. Müller, and O. Stenull, Generalized epidemic process and tricritical dynamic percolation, Phys. Rev. E, 2004, 70: 026114
[94] M. Ballard, V. M. Kenkre, and M. N. Kuperman, Periodically varying externally imposed environmental effects on population dynamics, Phys. Rev. E, 2004, 70:031912
[95] I. B. Schwartz, L. Billings, and E. M. Bollt, Dynamical epidemic suppression using stochastic prediction and control, Phys. Rev. E, 2004, 70: 046220
[96] R. M. Ziff, Simple algorithm to test for linking to Wilson loops in percolation, Phys. Rev. E, 2005,72: 017104
[97] C. Castellano and R. Pastor-Satorras, Non-Mean-Field Behavior of the Contact Process on Scale-Free Networks, Phys. Rev. Lett., 2006,96: 038701
[98] S. N. Dorogovtsev, J. F. Mendes, Evolution of networks, Adv. Phys. 51, 2002:1079-1187
[99] A. L.Lloyd, R. M. May, Science, 2001, 292: 1316-1317
[100] F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley and Y. Aberg, The Web of Human Sexual Contacts, Nature(London) 2001,411: 907-908
[101] S. H. Strogatz, Exploring complex networks, Nature (London) 2001,410: 268-270.
[102] A. L. Barabasi and R .Albert, Emergence of scaling in random networks, Science, 1999,286: 509
[103] A.Grabowski and R.A.Kosinski, Epidemic spreading in a hierarchical social network, Phys. Rev. E. 2004, 70: 031908
[104] M. E. J. Newman, Threshold Effects for Two Pathogens Spreading on a Network, Phys.Rev.Lett., 2005,95:108701
[105] R. M. Anderson and R. M. May, Infectious Diseases of Humans (Oxford: Oxford University Press, 1992)
[106] Newman M E J, Watts D J, Scaling and percolation in the small-world network model, Phys Rev E, 1999, 60: 7332-7342
[107] Newman M E J, Jensen I, Ziff R M. Percolation and epidemics in a two-dimensional small world, Phys Rev E, 2002,65: 021904
[108] Kuperman M, Abramson G, Small world effect in an epidemiological model, Phys Rev. Lett, 2001, 86: 2909-2912
[109] Agiza H N, Elgazzar A S, Youssef S A, Phase transitions in some epidemic models defined on small-world networks, arXiv: cond-mat/0301004
[110] Barabasi A L, Albert R, Jeong H., Mean-field theory for scale-free random networks, Physica A, 1999, 272: 173-187
[111] Prashant M.Gade, Dynamic transitions in small world networks: Approach to equilibrium limit, Phys Rev.E 2005,72: 052903
[112] Gang Yan, Zhong-Qian Fu, Jie Ren and Wen-Xu Wang, Collective synchronization induced by epidemic dynamics on complex networks with communities,Phys.Rev.E 2007,75:016108
[113]Z.H.Liu and B.Hu,Epidemie spreadingin community networks,Euro.Phys.Lett.,2005,72(2):315-321
[114]J.Liu,J.Wu,and Z.R.Yang,The spread of infeetious disease on complex networks with household-strueture,Physiea A,2004,341:273-280
[115]Ming Tang,Li Liu,and Zonghua Liu,Influence of dynamical condensation on epidemic spreading in scale-free networks,Phys Rev.E 2009,79:016108
[116]YAN Gang,ZHOU Tao,WANG Jie,FU Zhong-Qian and WANG Bing-Hong,Epidemic Spread in Weighted Scale-Free Networks,CHIN.PHYS.LETT.2005,22(2):510
[117]Ballegooijen W M and Boerlijst M C Proc.Natl.Acad.Sci.U.S.A.2004,101:18246
[118]Liu Q X,Jin Z,and Liu M X Phys.Rev.E.2006,74:031110
[119]Zhou T,Fu Z Q and Wang B H Prog.Nat.Sci.2006,16:452
[120]Sabrina B L Araujo and de Aguiar M A M Phys.Rev.E 2007,75:061908
[121]Sabrina B LAraujo and de Aguiar M A M Phys.Rev.E 2008,77:022903
[122]J.J.Ramasco,M.A.Mu(?)oz,and C.A.da Silva Santos,Numerical study of the Langevin theory for fixed-energy sandpiles,Phys.Rev.E 2004,69:045105
[123]I.Dornic,H.Chate,and M.A.Mu(?)oz,Integration of Langevin Equations with Multiplicative Noise and the Viability of Field Theories for Absorbing Phase Transitions,Phys.Rev.Lett.2005,94:100601
[124]Candia J,Irreversible opinion spreading on scale-free networks,Phys.Rev.E 2007,75:026110
[125]Bartolozzi M.,Leinweber D.B.,Thomas A.W.,Stochastic opinion formation in scale-free networks,Phys.Rev.E,2005,72:046113
[126]罗萨里奥.N.曼特尼亚,H.尤金.斯坦尼,经济物理学导论--金融中的相关性和复杂性,2006,中国人民大学出版社P55