复杂网络稳定性研究
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摘要
本文从复杂网络的拓扑统计特性和动力学机制出发,将非平衡统计理论、非线性系统理论、随机微分方程方法以及Fokker-Plank方程的定态解应用到复杂网络的研究中。主要研究了在非平衡涨落驱动下复杂网络的动力学稳定性,以及在蓄意攻击情况下病毒在复杂网络上的传播机理,这两个方面的研究无论在理论上还是在实践中都有着十分重要的意义。通过对复杂复杂网络的稳定性的研究,一方面可以使我们更清晰的认识和理解实际网络所表现出的各种动力学现象;另一方面,我们可以将复杂网络稳定性研究的成果应用到实际中去,可设计出更好性质的网络或使网络处于更有利的状态。对在蓄意攻击情况下病毒在复杂网络上的传播机理的研究,一方面可为研究和开发新的免疫机制与策略提供理论基础,另一方面也有利于提高网络信息传输能力。另外,复杂网络在许多领域都得到了较为广泛而深入的应用,作为本课题的一个主要内容,我们综述了复杂网络理论在电子电路与微电子学方面的应用。
本文的主要内容和创新之处可作如下概述:
1.非平衡涨落与小世界网络稳定性的研究
针对小世界特性是复杂网络的普遍特性,我们首先研究了故障对小世界网络稳定性的影响。在把实际网络元素的差错、失效等等“故障”抽象为非平衡随机涨落的前提下,利用非平衡统计理论、随机微分方程方法以及Fokker-Plank方程的定态解,来研究小世界网络稳定性。研究从小世界网络模型的非线性相互作用系数λ和NW长度标度ζ两个具体参数展开。
(1)通过理论研究,我们首次发现了,小世界网络的特性参量NW长度标度ζ是网络突变发生的敏感因子。当非平衡涨落发生在特性参量NW长度标度ζ上时,NW长度标度的平均值(?)在涨落σ~2的驱动下迅速增长。在随机化连接概率p不变时,网络的度值k将迅速减小。当k减小到某一值时(比如k＜1),复杂网络系统的稳定性将发生突变。由NW长度标度ζ的定义可知,复杂网络的节点这时应仍然存在,但是节点之间的连接边基本被删除。这种情况,对应于整个网络系统出现了崩溃。这一研究结论充分说明,作为小世界网络特性参量的NW长度标度ζ传播稳定性的主要决定因素。电力网络由于个别节点跳闸而发生大停电事故,是这一发现的最好例证。
(2)相比之下,当非线性相互作用系数λ发生非平衡涨落时,NW长度标度ζ随涨落σ~2呈迅速下降趋势。在随机化连接概率p不变的情况下,小世界网络系统的度值k会缓慢增加。这表示网络系统连通性会增强,网络会向更稳定的方向发展。也就是说,系统的非线性相互作用系数发生的涨落,不是引起复杂网络系统稳定性发生宏观突变的主要因素。
2.蓄意攻击与复杂网络稳定性的研究
或许蓄意攻击会有较多的方式,但病毒的攻击方式在网络上是如此的泛滥,必然引起我们的高度重视。因此,本课题中我们主要研究了病毒的传播机理及其稳定性。
(1)我们提出了一种描述病毒传播的新模型SIS-BD模型。利用这一模型以及非平衡统计理论,随机过程的理论和方法,从病毒传播的前期、后期及传播的全过程三个方面进行了研究。得到了病毒传播的三个时期病毒感染密度函数的分布规律。比较实际数据可知,这个分布过程与实际数据符合得较好。
(2)研究过程中我们发现,病毒的有效传播速率并不是一个常数,而是一个可以用类Logistic微分方程描述的函数。这与传统传染病研究中,病毒的传播速率通常是一个常数明显不同。通过求解类Logistic微分方程和用实际数据拟合出相关的常数,并结合前两部分的研究结果,得到了病毒传播全过程的感染密度分布函数。这个分布函数曲线与实际病毒传播数据曲线十分吻合。
(3)进一步分析有效传播速率我们知道,它与网络的结构参量密切相关。由此,我们用理论分析的方法讨论了有效传播速率与复杂网络结构参量的关系。经过分析,推论了网络节点的平均度值与有效传播速率变化率成正比;网络的聚类系数与有效传播速率成正比。从而得到病毒传播的有效速率与网络结构参量之间的更为普适的函数关系式。
3.非平衡统计理论与方法
利用非平衡统计理论与方法研究复杂网络,从研究方法上来说本身与是一种创新。
在本文的最后,我们介绍了复杂网络理论在电子电路与微电子学方面结合实际的可能应用。重点对目前引起了广泛研究兴趣,并成为最新研究热点的量子相干网络、纳米线网络等作了简要介绍,这是复杂网络理论应用在量子信息技术上的一些探索。
This thesis is concerned with the study of the dynamics stability of complex network driven by non-equilibrium fluctuation and virus's spreading mechanism on complex network under the intentional attack, which use the non-equilibrium statistical theory, the stochastic differential equation method as well as the Fokker-Plank equation steady-state solution. These studies are very important both in theory and in practical applications. By studying stability of complex network, on the one hand, we can understand and explain the dynamic properties presented in real-world networks; and on the other hand, we can apply these theoretical results of complex network stability to some practical applications, for instance, we may design of real network to achieve a better performance by using this results. By studying virus's spreading mechanism on complex network under the intentional attack, on the one hand, we can provide the rationale to study and develop the new immunity mechanism and the strategy; on the other hand, there are advantageous in raising the network transmission effection. In addition, complex network theory more and more widespread applies in many domains. As this topic main contents, we summarized the application of complex network theory in the electronic circuit and microelectronics.
The main contents and originalities in this thesis can be summarized as follows:
1. Non-equilibrium fluctuation and small-world network stability
Using the non-equilibrium statistical theory, the stochastic differential equation method as well as the Fokker-Plank equation steady-state solution, we have studied the small-world network stability by abstracting the real network factors of element mistake, node breakdown and so on as the non-equilibrium stochastic fluctuation. We discuss separately two kinds of situations to unify structure parameters of small-world network model, one is the fluctuation to occur in the non-linearity affects the coefficient and another is in the NW length scale.
(1)Through fundamental research, our initial find out: if the non-equilibrium fluctuation occurs in the NW length scaleζ, the NW length scale's mean value rapid growth driven by fluctuationσ~2. It implies that randomization connection probability p is invariable; the complex network degree's value k will be rapidly reduced. When k reduces to some value (for instance: k< 1), the stability of complex network system structure will have a fierce change. May know by the NW length scaleζdefinition, the complex network node should still exist, but the connection between the nodes is deleted completely. This kind of situation might correspond in the entire complex network system collapse. This conclusion full explains that the NW length scaleζis a primary factor to decide the dissemination stable as the small-world network characteristic parameter. When it appears fluctuation and approach a certain threshold, the complex network stability and the macroscopic dynamics behavior will have the fierce change, until will appear the entire net collapse. It is revealed that the NW length scaleζis one of extremely sensitive factors in the complex network dynamics process.
(2)Comparatively, if the non-linearity mutually coefficient A, has the non-equibirum fluctuation, the NW length scaleζalong with the fluctuationσ~2 has the tendency to drop rapidly. if randomisation connection probability p is invariable, complex network degree k can increase slowly. These phenomenons show the network system connectivity strongly and complex network can develop to a stabler direction. It is to say, if the fluctuation occurs in the system non-linear mutually coefficient, it can not lead to the complex network system stability to sudden change.
2. Intentional attacks and complex network stability
Perhaps intentional attacks have many ways, but virus attacks on network are like this being in flood, we have to take it seriously.
(1) We proposed one kind of new model SIS-BD model which represent virus dissemination. Using this model and non-equibirum statistical theory as well as stochastic process theory and method, we study the virus spreading from three aspects: earlier period, the later period and the dissemination entire process. We get the virus infection density function distributed rule of virus spreading in three periods. To compare with the actual data, we know that this distribution function accord with the actual data well.
(2) In the research process we discovered virus's effective dissemination speed is not a constant, it is a kind of function which can describe a similar Logistic differential equation. It is obvious different in the traditional infectious disease studies in which virus's dissemination speed is usually a constant. By solution the similar Logistic differential equation and useing the actual data fiting the correlation constant, and union two part of findings, we obtain the infection density distribution function of virus spreading on entire process.
(3) By further analyzes the effective dissemination speed we know that it close correlate with network structure parameter. Therefor, we used the theoretical analysis the method to discuss the effective dissemination speed and the complex network architecture parameter relation. Finally, we get the virus dissemination effective speed to relate with the network architecture parameter more general function.
3. Non-equilibrium statistical theory and method
It is one kind of innovation technique that uses the non-equilibrium statistical theory and method to research complex network.
Finally, we discussed the application of complex network theory in the electronic circuit and microelectronics. First, we introduce the main application and achievement of complex network on electronic circuit. Second, we make to a brief report about the complex network application in the quantum coherent network and nanometer wire nets, which become the newest research hot spot and attract to how widespread research interest at present.
本文的主要内容和创新之处可作如下概述:
1.非平衡涨落与小世界网络稳定性的研究
针对小世界特性是复杂网络的普遍特性,我们首先研究了故障对小世界网络稳定性的影响。在把实际网络元素的差错、失效等等“故障”抽象为非平衡随机涨落的前提下,利用非平衡统计理论、随机微分方程方法以及Fokker-Plank方程的定态解,来研究小世界网络稳定性。研究从小世界网络模型的非线性相互作用系数λ和NW长度标度ζ两个具体参数展开。
(1)通过理论研究,我们首次发现了,小世界网络的特性参量NW长度标度ζ是网络突变发生的敏感因子。当非平衡涨落发生在特性参量NW长度标度ζ上时,NW长度标度的平均值(?)在涨落σ~2的驱动下迅速增长。在随机化连接概率p不变时,网络的度值k将迅速减小。当k减小到某一值时(比如k＜1),复杂网络系统的稳定性将发生突变。由NW长度标度ζ的定义可知,复杂网络的节点这时应仍然存在,但是节点之间的连接边基本被删除。这种情况,对应于整个网络系统出现了崩溃。这一研究结论充分说明,作为小世界网络特性参量的NW长度标度ζ传播稳定性的主要决定因素。电力网络由于个别节点跳闸而发生大停电事故,是这一发现的最好例证。
(2)相比之下,当非线性相互作用系数λ发生非平衡涨落时,NW长度标度ζ随涨落σ~2呈迅速下降趋势。在随机化连接概率p不变的情况下,小世界网络系统的度值k会缓慢增加。这表示网络系统连通性会增强,网络会向更稳定的方向发展。也就是说,系统的非线性相互作用系数发生的涨落,不是引起复杂网络系统稳定性发生宏观突变的主要因素。
2.蓄意攻击与复杂网络稳定性的研究
或许蓄意攻击会有较多的方式,但病毒的攻击方式在网络上是如此的泛滥,必然引起我们的高度重视。因此,本课题中我们主要研究了病毒的传播机理及其稳定性。
(1)我们提出了一种描述病毒传播的新模型SIS-BD模型。利用这一模型以及非平衡统计理论,随机过程的理论和方法,从病毒传播的前期、后期及传播的全过程三个方面进行了研究。得到了病毒传播的三个时期病毒感染密度函数的分布规律。比较实际数据可知,这个分布过程与实际数据符合得较好。
(2)研究过程中我们发现,病毒的有效传播速率并不是一个常数,而是一个可以用类Logistic微分方程描述的函数。这与传统传染病研究中,病毒的传播速率通常是一个常数明显不同。通过求解类Logistic微分方程和用实际数据拟合出相关的常数,并结合前两部分的研究结果,得到了病毒传播全过程的感染密度分布函数。这个分布函数曲线与实际病毒传播数据曲线十分吻合。
(3)进一步分析有效传播速率我们知道,它与网络的结构参量密切相关。由此,我们用理论分析的方法讨论了有效传播速率与复杂网络结构参量的关系。经过分析,推论了网络节点的平均度值与有效传播速率变化率成正比;网络的聚类系数与有效传播速率成正比。从而得到病毒传播的有效速率与网络结构参量之间的更为普适的函数关系式。
3.非平衡统计理论与方法
利用非平衡统计理论与方法研究复杂网络,从研究方法上来说本身与是一种创新。
在本文的最后,我们介绍了复杂网络理论在电子电路与微电子学方面结合实际的可能应用。重点对目前引起了广泛研究兴趣,并成为最新研究热点的量子相干网络、纳米线网络等作了简要介绍,这是复杂网络理论应用在量子信息技术上的一些探索。
This thesis is concerned with the study of the dynamics stability of complex network driven by non-equilibrium fluctuation and virus's spreading mechanism on complex network under the intentional attack, which use the non-equilibrium statistical theory, the stochastic differential equation method as well as the Fokker-Plank equation steady-state solution. These studies are very important both in theory and in practical applications. By studying stability of complex network, on the one hand, we can understand and explain the dynamic properties presented in real-world networks; and on the other hand, we can apply these theoretical results of complex network stability to some practical applications, for instance, we may design of real network to achieve a better performance by using this results. By studying virus's spreading mechanism on complex network under the intentional attack, on the one hand, we can provide the rationale to study and develop the new immunity mechanism and the strategy; on the other hand, there are advantageous in raising the network transmission effection. In addition, complex network theory more and more widespread applies in many domains. As this topic main contents, we summarized the application of complex network theory in the electronic circuit and microelectronics.
The main contents and originalities in this thesis can be summarized as follows:
1. Non-equilibrium fluctuation and small-world network stability
Using the non-equilibrium statistical theory, the stochastic differential equation method as well as the Fokker-Plank equation steady-state solution, we have studied the small-world network stability by abstracting the real network factors of element mistake, node breakdown and so on as the non-equilibrium stochastic fluctuation. We discuss separately two kinds of situations to unify structure parameters of small-world network model, one is the fluctuation to occur in the non-linearity affects the coefficient and another is in the NW length scale.
(1)Through fundamental research, our initial find out: if the non-equilibrium fluctuation occurs in the NW length scaleζ, the NW length scale's mean value rapid growth driven by fluctuationσ~2. It implies that randomization connection probability p is invariable; the complex network degree's value k will be rapidly reduced. When k reduces to some value (for instance: k< 1), the stability of complex network system structure will have a fierce change. May know by the NW length scaleζdefinition, the complex network node should still exist, but the connection between the nodes is deleted completely. This kind of situation might correspond in the entire complex network system collapse. This conclusion full explains that the NW length scaleζis a primary factor to decide the dissemination stable as the small-world network characteristic parameter. When it appears fluctuation and approach a certain threshold, the complex network stability and the macroscopic dynamics behavior will have the fierce change, until will appear the entire net collapse. It is revealed that the NW length scaleζis one of extremely sensitive factors in the complex network dynamics process.
(2)Comparatively, if the non-linearity mutually coefficient A, has the non-equibirum fluctuation, the NW length scaleζalong with the fluctuationσ~2 has the tendency to drop rapidly. if randomisation connection probability p is invariable, complex network degree k can increase slowly. These phenomenons show the network system connectivity strongly and complex network can develop to a stabler direction. It is to say, if the fluctuation occurs in the system non-linear mutually coefficient, it can not lead to the complex network system stability to sudden change.
2. Intentional attacks and complex network stability
Perhaps intentional attacks have many ways, but virus attacks on network are like this being in flood, we have to take it seriously.
(1) We proposed one kind of new model SIS-BD model which represent virus dissemination. Using this model and non-equibirum statistical theory as well as stochastic process theory and method, we study the virus spreading from three aspects: earlier period, the later period and the dissemination entire process. We get the virus infection density function distributed rule of virus spreading in three periods. To compare with the actual data, we know that this distribution function accord with the actual data well.
(2) In the research process we discovered virus's effective dissemination speed is not a constant, it is a kind of function which can describe a similar Logistic differential equation. It is obvious different in the traditional infectious disease studies in which virus's dissemination speed is usually a constant. By solution the similar Logistic differential equation and useing the actual data fiting the correlation constant, and union two part of findings, we obtain the infection density distribution function of virus spreading on entire process.
(3) By further analyzes the effective dissemination speed we know that it close correlate with network structure parameter. Therefor, we used the theoretical analysis the method to discuss the effective dissemination speed and the complex network architecture parameter relation. Finally, we get the virus dissemination effective speed to relate with the network architecture parameter more general function.
3. Non-equilibrium statistical theory and method
It is one kind of innovation technique that uses the non-equilibrium statistical theory and method to research complex network.
Finally, we discussed the application of complex network theory in the electronic circuit and microelectronics. First, we introduce the main application and achievement of complex network on electronic circuit. Second, we make to a brief report about the complex network application in the quantum coherent network and nanometer wire nets, which become the newest research hot spot and attract to how widespread research interest at present.
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