基于V稳定性理论的复杂网络稳定性分析与牵制控制方法研究
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摘要
自上世纪九十年代开始,随着互联网技术的迅猛发展,网络与人们之间的联系日益紧密,并逐渐渗透到人类社会生活的各个角落。这些形形色色的网络,大到Internet网络、电力网络、通讯交通网络,小到神经元网络、新陈代谢网络、分子结构网络,甚至是食物链网络、亲友网络、合作网络、引文网络等都在不断影响着人类社会的生产和生活。因此,了解和利用自然界和人类社会中存在的各种关系网络,研究复杂网络模型的动力学传播机理,并实现对网络动态行为的可控,对于促进和提高社会的生产水平和人们的生活质量将起到重要的作用。
目前,稳定性问题作为复杂网络理论中的研究热点,得到了研究者越来越多的关注和重视。为了更好地研究复杂网络的传播机理及其动力学特性,并解决含有不确定性耦合的复杂网络模型的稳定性问题,基于复杂网络的牵制控制思想和社团结构特征,本文进一步拓展了李亚普诺夫V稳定性理论,在复杂非线性动态网络的同步稳定性方面进行了理论研究,提出了稳定性判据以及牵制控制中的节点选择策略,实现了目标网络的稳定。
本文的研究工作主要包括:
1.对复杂网络理论的研究内容和研究现状进行了综述,并具体总结了本文在研究复杂网络理论中涉及的网络同步稳定特性和网络控制方面的应用情况。
2.拓展了李亚普诺夫V稳定性理论,通过用节点的无源度来描述各节点的自动态对网络整体行为的影响,将复杂网络的同步稳定性问题转化成为判断网络特征矩阵不等式是否成立,从而解决了含不确定性耦合的复杂网络各节点动态轨迹与孤立节点轨迹同步稳定的问题,大大简化了此类复杂网络稳定性问题的分析难度。
3.根据含不确定性耦合的复杂网络同步判据,针对复杂网络模型设计相应的牵制控制策略,使得受控后的网络特征矩阵不等式成立,从而实现了复杂网络各节点动态轨迹与孤立节点轨迹的同步稳定,解决了牵制控制中受控节点的选择问题。实验仿真验证了牵制控制策略的正确性和有效性。
4.针对具有典型社团结构特征的复杂网络,在李亚普诺夫V稳定性理论的基础上,分析了聚合网络的稳定性与各子群落稳定性之间的关系。简要介绍了寻找复杂网络社团结构的几种主要算法,提出了一种对大尺度网络先进行社团结构划分,再施加特定牵制控制的新方法。实验仿真验证了当网络社团结构明显时,该方法与原先直接施加特定牵制控制的方法相比能够提高算法的有效性。
5.研究了复杂网络中的几种典型拓扑模型,通过考察各网络结构的统计特性,将新的牵制控制方法与原有控制策略进行对比,提出了一个策略衡量标准,以便在实际模型的应用中更好地选择牵制控制策略。实验仿真验证了在该衡量标准下实施牵制控制策略,能够减少网络达到稳定所需牵制控制器的个数,并可以避免进行无意义的群落划分,提高了牵制控制算法的有效性。
最后是全文的总结与展望。
Since the 90's of the last century, the networks have been related to human beings more and more closely as the result of the highly development of Internet technology. These various kinds of networks, such as Internet, power grids, communication networks, neuron networks, metabolic networks, food webs, social networks, collaboration networks and citation networks have greatly infiltrated every corner of our life and affected the production and life of human society constantly. Therefore, comprehending and making use of all these networks exist in nature and human society, studying the epidemic dynamics of these networks, and realizing the controllability of the networks' dynamical behavior play significant role in promoting and enhancing the level of production and the quality of human life.
Nowadays, stability problem becomes the focus which attracts more and more attention in the study on the theory of complex networks. Based on the pinning control and the community structure, the paper makes a further study on Lyapunov V-Stability theory, presents the stability criterion and the pinning control strategy, and solves the stability problem of the complex nonlinear dynamical network finally.
The primary results of this thesis are described as follows.
1. A review is given on the research contents and current status of the theory of complex networks. The studies on networks' stability and the control strategy have also been summarized.
2. V-Stability theory is extended to the stability analysis of the stabilization of complex networks with the homogeneous orbit of an isolated node. The stability problem of the complex network which has uncertain couplings is simplified and converted into determining if the characteristic matrix of the network is negative semi-definite or not via replacing the influences on collective behavior caused by self-dynamics of the nodes by their passivity degrees.
3. According to the stability criterion, the pinning control strategy is designed to choose the controlled nodes. Hence the stability problem of uncertain complex network model to a homogeneous orbit of the isolated node is solved. The simulation verifies the results obtained.
4. Based on the V-stability theory, the stability of the aggregation network is analyzed for the complex networks with typical community structure. The main methods of finding community structure are introduced. Then a new strategy which designs pinning control strategy after dividing the original network into small communities is proposed. Compared to the common method, the efficiency and practicality of the new strategy are improved in the simulation.
5. Some complex network models with typical topologies are studied to determine under which situation the new strategy should be applied to solve the stability problem. A standard for choosing control strategy is given. The simulation verifies that the new pinning control method with the standard can reduce the number of the controlled nodes, avoid the meaningless partition and improve the effectiveness of the strategy.
The conclusion and perspective are given at the end of the dissertation.
目前,稳定性问题作为复杂网络理论中的研究热点,得到了研究者越来越多的关注和重视。为了更好地研究复杂网络的传播机理及其动力学特性,并解决含有不确定性耦合的复杂网络模型的稳定性问题,基于复杂网络的牵制控制思想和社团结构特征,本文进一步拓展了李亚普诺夫V稳定性理论,在复杂非线性动态网络的同步稳定性方面进行了理论研究,提出了稳定性判据以及牵制控制中的节点选择策略,实现了目标网络的稳定。
本文的研究工作主要包括:
1.对复杂网络理论的研究内容和研究现状进行了综述,并具体总结了本文在研究复杂网络理论中涉及的网络同步稳定特性和网络控制方面的应用情况。
2.拓展了李亚普诺夫V稳定性理论,通过用节点的无源度来描述各节点的自动态对网络整体行为的影响,将复杂网络的同步稳定性问题转化成为判断网络特征矩阵不等式是否成立,从而解决了含不确定性耦合的复杂网络各节点动态轨迹与孤立节点轨迹同步稳定的问题,大大简化了此类复杂网络稳定性问题的分析难度。
3.根据含不确定性耦合的复杂网络同步判据,针对复杂网络模型设计相应的牵制控制策略,使得受控后的网络特征矩阵不等式成立,从而实现了复杂网络各节点动态轨迹与孤立节点轨迹的同步稳定,解决了牵制控制中受控节点的选择问题。实验仿真验证了牵制控制策略的正确性和有效性。
4.针对具有典型社团结构特征的复杂网络,在李亚普诺夫V稳定性理论的基础上,分析了聚合网络的稳定性与各子群落稳定性之间的关系。简要介绍了寻找复杂网络社团结构的几种主要算法,提出了一种对大尺度网络先进行社团结构划分,再施加特定牵制控制的新方法。实验仿真验证了当网络社团结构明显时,该方法与原先直接施加特定牵制控制的方法相比能够提高算法的有效性。
5.研究了复杂网络中的几种典型拓扑模型,通过考察各网络结构的统计特性,将新的牵制控制方法与原有控制策略进行对比,提出了一个策略衡量标准,以便在实际模型的应用中更好地选择牵制控制策略。实验仿真验证了在该衡量标准下实施牵制控制策略,能够减少网络达到稳定所需牵制控制器的个数,并可以避免进行无意义的群落划分,提高了牵制控制算法的有效性。
最后是全文的总结与展望。
Since the 90's of the last century, the networks have been related to human beings more and more closely as the result of the highly development of Internet technology. These various kinds of networks, such as Internet, power grids, communication networks, neuron networks, metabolic networks, food webs, social networks, collaboration networks and citation networks have greatly infiltrated every corner of our life and affected the production and life of human society constantly. Therefore, comprehending and making use of all these networks exist in nature and human society, studying the epidemic dynamics of these networks, and realizing the controllability of the networks' dynamical behavior play significant role in promoting and enhancing the level of production and the quality of human life.
Nowadays, stability problem becomes the focus which attracts more and more attention in the study on the theory of complex networks. Based on the pinning control and the community structure, the paper makes a further study on Lyapunov V-Stability theory, presents the stability criterion and the pinning control strategy, and solves the stability problem of the complex nonlinear dynamical network finally.
The primary results of this thesis are described as follows.
1. A review is given on the research contents and current status of the theory of complex networks. The studies on networks' stability and the control strategy have also been summarized.
2. V-Stability theory is extended to the stability analysis of the stabilization of complex networks with the homogeneous orbit of an isolated node. The stability problem of the complex network which has uncertain couplings is simplified and converted into determining if the characteristic matrix of the network is negative semi-definite or not via replacing the influences on collective behavior caused by self-dynamics of the nodes by their passivity degrees.
3. According to the stability criterion, the pinning control strategy is designed to choose the controlled nodes. Hence the stability problem of uncertain complex network model to a homogeneous orbit of the isolated node is solved. The simulation verifies the results obtained.
4. Based on the V-stability theory, the stability of the aggregation network is analyzed for the complex networks with typical community structure. The main methods of finding community structure are introduced. Then a new strategy which designs pinning control strategy after dividing the original network into small communities is proposed. Compared to the common method, the efficiency and practicality of the new strategy are improved in the simulation.
5. Some complex network models with typical topologies are studied to determine under which situation the new strategy should be applied to solve the stability problem. A standard for choosing control strategy is given. The simulation verifies that the new pinning control method with the standard can reduce the number of the controlled nodes, avoid the meaningless partition and improve the effectiveness of the strategy.
The conclusion and perspective are given at the end of the dissertation.
引文
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