复杂动力网络的同步分析与计算
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摘要
复杂网络是由具有一定特征和功能的、相互关联及相互影响的基本单元所构成的复杂集合体。近十年来,国内外掀起了研究复杂网络的热潮。许多来自物理、生物、数学和计算机领域的研究者都开始致力于复杂网络的研究。由于现实社会中大规模网络的存在,促使人们去研究这些网络的拓扑结构及其动力学行为。本论文主要运用动力系统理论与数值计算方法与来研究复杂网络的同步问题,探讨网络的拓扑结构与同步之间的关系。大致说来,包括时滞复杂网络的同步研究,加权网络的同步与分岔分析以及网络间的同步问题。这些问题的研究为设计具有良好性能的网络提供了一定的理论依据和指导意义。具体来说,我们的工作如下:
1.第二章、第三章具体介绍了时滞复杂网络的同步行为,给出了网络达到同步的充分条件。首先我们拓展了网络的耦合时滞,引入了“时滞向量”和“时滞矩阵”;其次讨论带有时滞和非线性内部耦合函数的网络同步问题,利用线性矩阵不等式,我们得出了网络实现同步的理论结果;这里我们提供了一定量的数值例子来说明理论的有效性。
2.第四章介绍了加权网络的同步态跃迁。在研究复杂网络的同步问题时,许多学者都假设网络耦合矩阵满足耗散条件(耦合矩阵的每行和都为零),网络的同步态仅由节点方程所确定。本章令耦合矩阵的每行和为一个常数,这样网络可以达到一个新的同步态,而这个同步态并不是由原节点方程所确定。当我们把行和看作分岔参数时,发现了一个有趣的现象,网络的同步态发生跃迁,出现了分岔行为。这一章是我们研究的重要内容之一。
3.第五章研究了两个耦合网络间的同步问题。在现实世界中,我们容易把节点性质类似的节点当成一个网络来看待,而把具有不同性质的节点当作多个网络来分析,比如考虑各种传染病(Mad Cows、AIDS、SARS)是如何在人群和动物之间传播的,此时需要把人群和动物当作两个网络来看待,因此考虑网络间的动力学问题是有现实意义的。本章中我们提出了“网络外部同步”(即发生在网络间的同步),而把以前研究的网络同步称作是“内部同步”即发生在一个网络内的同步。这一章内容是我们研究的主要内容之一。
4.第六章作为网络研究的应用部分。我们根据2004和2005这两年的中国火炬计划统计资料和年度报告上的统计数据构造出了国家高新技术园区网络。具体分析了三种类型高新园区网络的拓扑结构,包括平均路径长度、聚类系数和度分布。这里我们发现无权无向(有向)网络具有小世界特性,而加权无向网络的边权分布呈幂率分布;其次运用CFinder软件来研究高新园区网络的社团结构演变。由数值计算的结果可以看出,构建的国家高新园区网络能够说明国家高新园区的实际发展趋势。第七章总结全文并指出进一步研究的问题。
Generally speaking, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. The recent decade has witnessed the birth of a new movement of interest and research on the study of complex networks throughout the world. The researchers in physics, biology, mathematics and computer science dedicate to the study of complex networks. In this dissertation, we apply the theory of dynamical systems and method of numerical computation to studying the synchronization of complex networks, explaining the relations between topological structures and synchronization. The main work is to study synchronization in delayed complex networks, the synchronization and bifurcation of weighted network and synchronization between two coupled networks. In details, the main contents of the dissertation are organized as follows:
1. Synchronization analysis in delayed complex network is studied in Chapters 2 and 3, where some sufficient conditions on synchronization are derived. Firstly we expand the coupling delay by introducing the "delay vector" and "delay matrix"; secondly synchronization of complex networks with time delay and nonlinear inner-coupling function is also discussed, we use the linear matrix equality to obtain the theory of synchronization; the numerical examples taken here show the efficiency of the derived theory.
2. Chapter 4 deals with the transition of synchronized states in weighted networks. In many works, the authors often confine the coupling matrix satisfying the dissipation condition (the sum of every row equals zero), and the synchronization state is determined only by the node function. In this Chapter, we needn't confine the sum of each row to be zero, but to equal a nonzero constant. Such a network can synchronize to a new state, not to that of limit set determined by the original node equation. It is interesting to find that the synchronized state appears bifurcation if we regard the constant as a bifurcation parameter.
3. Synchronization between two coupled networks is discussed in Chapter 5. If network nodes are of similar properties, we can regard it as one network; otherwise, as more networks. For example, how the infectious diseases (Mad Cows, SARS, AIDS) spread between the human beings and animals, here regarding the human and animals as two networks. In this Chapter, we propose "network outer synchronization". If synchronization happens in a network, we may regard it as "inner synchronization"; while we may call it as "outer synchronization" if synchronization exists between two or more networks.
4. Application part of network research is shown in Chapter 6. Based on the data in statistics and annual reports on China torch program in 2004-2005, we apply the network knowledge to construct the China Hi-Tec Park Networks (CHTPN). From the viewpoints of complex networks, firstly we investigate and initially analyze its topological properties, including the average path length, clustering coefficient and degree distribution. Secondly using the software CFinder, we analyze the evolvement of network community structures. The constructed networks basically show the trend of real CHTPNs through the numerical results. Chapter 7 summarizes the conclusions and gives further studies in the future.
1.第二章、第三章具体介绍了时滞复杂网络的同步行为,给出了网络达到同步的充分条件。首先我们拓展了网络的耦合时滞,引入了“时滞向量”和“时滞矩阵”;其次讨论带有时滞和非线性内部耦合函数的网络同步问题,利用线性矩阵不等式,我们得出了网络实现同步的理论结果;这里我们提供了一定量的数值例子来说明理论的有效性。
2.第四章介绍了加权网络的同步态跃迁。在研究复杂网络的同步问题时,许多学者都假设网络耦合矩阵满足耗散条件(耦合矩阵的每行和都为零),网络的同步态仅由节点方程所确定。本章令耦合矩阵的每行和为一个常数,这样网络可以达到一个新的同步态,而这个同步态并不是由原节点方程所确定。当我们把行和看作分岔参数时,发现了一个有趣的现象,网络的同步态发生跃迁,出现了分岔行为。这一章是我们研究的重要内容之一。
3.第五章研究了两个耦合网络间的同步问题。在现实世界中,我们容易把节点性质类似的节点当成一个网络来看待,而把具有不同性质的节点当作多个网络来分析,比如考虑各种传染病(Mad Cows、AIDS、SARS)是如何在人群和动物之间传播的,此时需要把人群和动物当作两个网络来看待,因此考虑网络间的动力学问题是有现实意义的。本章中我们提出了“网络外部同步”(即发生在网络间的同步),而把以前研究的网络同步称作是“内部同步”即发生在一个网络内的同步。这一章内容是我们研究的主要内容之一。
4.第六章作为网络研究的应用部分。我们根据2004和2005这两年的中国火炬计划统计资料和年度报告上的统计数据构造出了国家高新技术园区网络。具体分析了三种类型高新园区网络的拓扑结构,包括平均路径长度、聚类系数和度分布。这里我们发现无权无向(有向)网络具有小世界特性,而加权无向网络的边权分布呈幂率分布;其次运用CFinder软件来研究高新园区网络的社团结构演变。由数值计算的结果可以看出,构建的国家高新园区网络能够说明国家高新园区的实际发展趋势。第七章总结全文并指出进一步研究的问题。
Generally speaking, a complex network is a large set of interconnected nodes, in which a node is a fundamental unit with specific contents. The recent decade has witnessed the birth of a new movement of interest and research on the study of complex networks throughout the world. The researchers in physics, biology, mathematics and computer science dedicate to the study of complex networks. In this dissertation, we apply the theory of dynamical systems and method of numerical computation to studying the synchronization of complex networks, explaining the relations between topological structures and synchronization. The main work is to study synchronization in delayed complex networks, the synchronization and bifurcation of weighted network and synchronization between two coupled networks. In details, the main contents of the dissertation are organized as follows:
1. Synchronization analysis in delayed complex network is studied in Chapters 2 and 3, where some sufficient conditions on synchronization are derived. Firstly we expand the coupling delay by introducing the "delay vector" and "delay matrix"; secondly synchronization of complex networks with time delay and nonlinear inner-coupling function is also discussed, we use the linear matrix equality to obtain the theory of synchronization; the numerical examples taken here show the efficiency of the derived theory.
2. Chapter 4 deals with the transition of synchronized states in weighted networks. In many works, the authors often confine the coupling matrix satisfying the dissipation condition (the sum of every row equals zero), and the synchronization state is determined only by the node function. In this Chapter, we needn't confine the sum of each row to be zero, but to equal a nonzero constant. Such a network can synchronize to a new state, not to that of limit set determined by the original node equation. It is interesting to find that the synchronized state appears bifurcation if we regard the constant as a bifurcation parameter.
3. Synchronization between two coupled networks is discussed in Chapter 5. If network nodes are of similar properties, we can regard it as one network; otherwise, as more networks. For example, how the infectious diseases (Mad Cows, SARS, AIDS) spread between the human beings and animals, here regarding the human and animals as two networks. In this Chapter, we propose "network outer synchronization". If synchronization happens in a network, we may regard it as "inner synchronization"; while we may call it as "outer synchronization" if synchronization exists between two or more networks.
4. Application part of network research is shown in Chapter 6. Based on the data in statistics and annual reports on China torch program in 2004-2005, we apply the network knowledge to construct the China Hi-Tec Park Networks (CHTPN). From the viewpoints of complex networks, firstly we investigate and initially analyze its topological properties, including the average path length, clustering coefficient and degree distribution. Secondly using the software CFinder, we analyze the evolvement of network community structures. The constructed networks basically show the trend of real CHTPNs through the numerical results. Chapter 7 summarizes the conclusions and gives further studies in the future.
引文
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