结构与功能—复杂网络上的自组织爆炸同步以及强噪声下网络重构问题的研究
发布时间：2018-08-22 21:28
【摘要】：进入21世纪以来,复杂网络科学在各个科学领域都受到了广泛关注。复杂网络的相关概念为人们认识客观系统复杂性提供了一个切入点,并对复杂系统建模提供了坚实的基础。同时,基于它发展起来的一系列方法也为大家分析和控制复杂系统提供了有力的工具。目前,复杂网络领域中有两个方面尤为受大家关注:其一是网络结构对于系统的功能和动力学行为的影响,即从结构到动力学输出的所谓“正问题”。特别是当简单的动力学单元通过复杂的结构耦合起来以后,将会出现哪些新的集体行为和涌现现象,以及如何认识和控制这些现象,这一直是各领域研究的热点内容。再者就是如何通过系统输出重构网络的的结构,即从输出数据到结构的所谓“逆问题”。如今,随着观测技术的进步和存储、计算能力的极大提升,我们已然进入一个“大数据”的时代。在各个领域,特别是生物领域和社会领域,数据每时每刻都在积累。如何从这些数据中挖掘尽可能多的信息,特别是如何通过数据窥探隐藏其中的动力学机制和网络结构一直是人们关注的焦点。在正问题中,一种所谓“爆炸同步”(Explosive synchronization)的现象近年受到了广泛关注。爆炸同步是发生在耦合振子网络中的一种一阶同步相变过程,而大量的研究表明,在动力学参数与网络结构参数之间预设特定的约束关系是实现该过程的一个十分重要的条件。该现象深刻的揭示了网络结构对于系统行为的影响。在逆问题的相关研究中,噪声在网络重构中的作用在最近受到较多的讨论和关注。在现实中很多网络系统的观测数据,比如神经系统、基因调控系统的生物数据等等,是系统的非线性结构和环境中噪声相互作用的结果,而且这些非线性结构的具体形式和噪声的关键统计特征往往是未知的,甚至还有部分节点(变量)的数据由于种种原因不能被直接测量到。在这种情形下,如何重构网络结构是非常具有挑战性和现实意义的问题。在这两方面,我们的研究主要取得了如下进展:(Ⅰ)我们首先发现了爆炸同步和其所需的节点频率-连接度正相关约束可以在振荡网络中自组织的产生,因此可以解除之前需要预设参数约束的限制。在扩散耦合的全同金兹堡-朗道振子网络中,数值结果和解析分析都证实,先前人们研究中作为复杂网络爆炸性同步的重要条件之一——节点频率与连接度的线性正关联,可以在系统演化过程中自组织的形成,并产生爆炸同步的现象。鉴于金兹堡-朗道方程是动力学系统在Hopf分岔点附近的普适描述,类似的爆炸同步现象被证明在一般的反应-扩散动力学网络中存在,且其中的参数和变量可以被一一映射到金兹堡-朗道振子网络中。因此,我们的研究证明了爆炸同步是反应-扩散动力学系统中的一种涌现现象,广泛存在于Hopf分岔点附近的振荡系统之中。此外,我们还得到了爆炸同步发生的参数空间,这将为在实际系统中发现爆炸同步提供有益的指导。(Ⅱ)对于强噪声作用下的非线性动力学网络,我们提出高阶关联矩阵的方法(high-order correlation computations,简称 HOCC)重构网络动力学结构。该方法可以处理加性白噪声和乘性白噪声下的非线性网络重构问题,统一的推断出网络中非线性的节点动力学、相互作用结构以及噪声的统计量。所有计算完全基于数据,不要求已知系统和噪声的其他信息,如节点的动力学形式或噪声的统计特征。HOCC方法有以下特点:(i)通过计算数据的高阶关联矩阵,有效的处理系统中的非线性动力结构;(ii)通过差时关联计算,将确定性动力学和噪声的统计特征的重构退耦为两个相互独立的步骤;(iii)复杂网络的逆问题最终被归结为一个简单的线性矩阵代数方程计算。经过数值验证和误差分析,该算法的有效性和鲁棒性得到了充分的证明。HOCC方法还被进一步推广到有测量噪声的情形,且在某些情况下对色噪声也同样适用。(Ⅲ)对于系统中不仅存在噪声,还存在隐藏变量的情况,我们提出一种可以有效重构网络动力学结构的方法。其关键思想在于,数据中不仅包含了已测量节点的信息,还包含了与它们发生作用的隐藏节点的信息。该方法的关键之处是数据的高阶导数的相关计算可以有效的挖掘这些信息。此外,我们还特别关注了色噪声下的网络重构问题。一方面,我们可以将其作为有隐藏变量的逆问题的一个特例,用上方法将系统和色噪声用统一的表达式重构出来;另一方面,我们又给出一种双矩阵方程迭代算法,可以通过运算将网络结构,噪声强度以及噪声关联时间参数求出。这些方法的有效性在数值验证中得到了充分的证明。
[Abstract]:Since the beginning of the 21st century, complex network science has attracted wide attention in various scientific fields. The related concepts of complex network provide a breakthrough point for people to understand the complexity of objective systems, and provide a solid foundation for complex system modeling. Hybrid systems provide powerful tools. At present, two aspects of complex networks are of particular concern. One is the effect of network structure on the function and dynamic behavior of systems, i.e. the so-called "positive problem" from structure to dynamic output, especially when simple dynamic elements are coupled through complex structures. What new collective behavior and emerging phenomena will emerge, and how to recognize and control these phenomena have been the focus of research in various fields. What's more, how to reconstruct the network structure through system output, that is, the so-called "inverse problem" from output data to structure. We have entered an era of "big data" with tremendous improvements in capabilities. Data is accumulating all the time in all fields, especially in the biological and social fields. In the forward problem, a so-called "explosive synchronization" phenomenon has attracted wide attention in recent years. Explosive synchronization is a first-order synchronous phase transition process occurring in coupled oscillator networks. A large number of studies have shown that specific constraints are presupposed between dynamic parameters and network structure parameters. Relation is a very important condition to realize this process.This phenomenon reveals the effect of network structure on system behavior profoundly.In the study of inverse problems,the role of noise in network reconfiguration has received much discussion and attention recently.In reality,many network system observations,such as nervous system and gene,have been made. The biological data of the control system and so on are the result of the interaction between the nonlinear structure of the system and the noise in the environment, and the concrete form of these nonlinear structures and the key statistical characteristics of the noise are often unknown, even some nodes (variables) of the data can not be directly measured for various reasons. How to reconstruct the network structure is a very challenging and practical problem. In these two aspects, our research has made the following progress: (1) We first found that explosive synchronization and its required frequency-connectivity positive correlation constraints can be self-organized in oscillatory networks, so it can be removed before the preset. Parametric constraints. In diffusion-coupled identical Ginzburg-Landau oscillator networks, numerical results and analytical analysis confirm that the linear positive correlation between node frequencies and connectivity, as one of the important conditions for explosive synchronization of complex networks, can form and produce self-organization during the evolution of systems. Since the Ginzburg-Landau equation is a universal description of dynamical systems near Hopf bifurcation points, similar explosive synchronization phenomena have been proved to exist in general reaction-diffusion dynamical networks, and the parameters and variables can be mapped one by one into the Ginzburg-Landau oscillator networks. It is proved that explosive synchronization is an emergent phenomenon in reaction-diffusion dynamics system, which exists widely in oscillatory systems near Hopf bifurcation points. In addition, the parameter space of explosive synchronization is obtained, which will provide useful guidance for discovering explosive synchronization in practical systems. (II) Nonlinear phenomena under strong noise. In this paper, we propose a high-order correlation computations (HOCC) method to reconstruct the dynamical structure of a dynamical network. The method can deal with the nonlinear network reconfiguration problem under additive white noise and multiplicative white noise. The nonlinear node dynamics in the network can be deduced uniformly. The interaction structure can be used to reconstruct the dynamical structure of the network. All calculations are based entirely on data and do not require other information about the system and noise, such as the dynamics of the nodes or the statistical characteristics of the noise. (iii) The inverse problem of a complex network is finally reduced to a simple linear matrix algebraic equation calculation. The validity and robustness of the algorithm are proved by numerical verification and error analysis. (iii) For the presence of not only noises but also hidden variables in the system, we propose an effective method to reconstruct the network dynamics structure. The key idea is that the data contains not only the information of the measured nodes, but also the information of the measured nodes. The key point of this method is that the correlation calculation of the high-order derivatives of the data can effectively mine the information. In addition, we also pay special attention to the problem of network reconfiguration in colored noise. On the one hand, we can regard it as a special case of the inverse problem with hidden variables. On the other hand, we present a Bi-matrix equation iterative algorithm, which can calculate the network structure, noise intensity and noise correlation time parameters by operation. The effectiveness of these methods has been fully demonstrated in numerical verification.
【学位授予单位】：北京邮电大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O157.5
本文编号：2198346
[Abstract]:Since the beginning of the 21st century, complex network science has attracted wide attention in various scientific fields. The related concepts of complex network provide a breakthrough point for people to understand the complexity of objective systems, and provide a solid foundation for complex system modeling. Hybrid systems provide powerful tools. At present, two aspects of complex networks are of particular concern. One is the effect of network structure on the function and dynamic behavior of systems, i.e. the so-called "positive problem" from structure to dynamic output, especially when simple dynamic elements are coupled through complex structures. What new collective behavior and emerging phenomena will emerge, and how to recognize and control these phenomena have been the focus of research in various fields. What's more, how to reconstruct the network structure through system output, that is, the so-called "inverse problem" from output data to structure. We have entered an era of "big data" with tremendous improvements in capabilities. Data is accumulating all the time in all fields, especially in the biological and social fields. In the forward problem, a so-called "explosive synchronization" phenomenon has attracted wide attention in recent years. Explosive synchronization is a first-order synchronous phase transition process occurring in coupled oscillator networks. A large number of studies have shown that specific constraints are presupposed between dynamic parameters and network structure parameters. Relation is a very important condition to realize this process.This phenomenon reveals the effect of network structure on system behavior profoundly.In the study of inverse problems,the role of noise in network reconfiguration has received much discussion and attention recently.In reality,many network system observations,such as nervous system and gene,have been made. The biological data of the control system and so on are the result of the interaction between the nonlinear structure of the system and the noise in the environment, and the concrete form of these nonlinear structures and the key statistical characteristics of the noise are often unknown, even some nodes (variables) of the data can not be directly measured for various reasons. How to reconstruct the network structure is a very challenging and practical problem. In these two aspects, our research has made the following progress: (1) We first found that explosive synchronization and its required frequency-connectivity positive correlation constraints can be self-organized in oscillatory networks, so it can be removed before the preset. Parametric constraints. In diffusion-coupled identical Ginzburg-Landau oscillator networks, numerical results and analytical analysis confirm that the linear positive correlation between node frequencies and connectivity, as one of the important conditions for explosive synchronization of complex networks, can form and produce self-organization during the evolution of systems. Since the Ginzburg-Landau equation is a universal description of dynamical systems near Hopf bifurcation points, similar explosive synchronization phenomena have been proved to exist in general reaction-diffusion dynamical networks, and the parameters and variables can be mapped one by one into the Ginzburg-Landau oscillator networks. It is proved that explosive synchronization is an emergent phenomenon in reaction-diffusion dynamics system, which exists widely in oscillatory systems near Hopf bifurcation points. In addition, the parameter space of explosive synchronization is obtained, which will provide useful guidance for discovering explosive synchronization in practical systems. (II) Nonlinear phenomena under strong noise. In this paper, we propose a high-order correlation computations (HOCC) method to reconstruct the dynamical structure of a dynamical network. The method can deal with the nonlinear network reconfiguration problem under additive white noise and multiplicative white noise. The nonlinear node dynamics in the network can be deduced uniformly. The interaction structure can be used to reconstruct the dynamical structure of the network. All calculations are based entirely on data and do not require other information about the system and noise, such as the dynamics of the nodes or the statistical characteristics of the noise. (iii) The inverse problem of a complex network is finally reduced to a simple linear matrix algebraic equation calculation. The validity and robustness of the algorithm are proved by numerical verification and error analysis. (iii) For the presence of not only noises but also hidden variables in the system, we propose an effective method to reconstruct the network dynamics structure. The key idea is that the data contains not only the information of the measured nodes, but also the information of the measured nodes. The key point of this method is that the correlation calculation of the high-order derivatives of the data can effectively mine the information. In addition, we also pay special attention to the problem of network reconfiguration in colored noise. On the one hand, we can regard it as a special case of the inverse problem with hidden variables. On the other hand, we present a Bi-matrix equation iterative algorithm, which can calculate the network structure, noise intensity and noise correlation time parameters by operation. The effectiveness of these methods has been fully demonstrated in numerical verification.
【学位授予单位】：北京邮电大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O157.5
【参考文献】
相关博士学位论文 前1条
1 肖井华;控制与同步时空混沌[D];北京师范大学;1999年
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