結構與功能—復雜網(wǎng)絡上的自組織爆炸同步以及強噪聲下網(wǎng)絡重構問題的研究
發(fā)布時間：2018-08-22 21:28
【摘要】：進入21世紀以來,復雜網(wǎng)絡科學在各個科學領域都受到了廣泛關注。復雜網(wǎng)絡的相關概念為人們認識客觀系統(tǒng)復雜性提供了一個切入點,并對復雜系統(tǒng)建模提供了堅實的基礎。同時,基于它發(fā)展起來的一系列方法也為大家分析和控制復雜系統(tǒng)提供了有力的工具。目前,復雜網(wǎng)絡領域中有兩個方面尤為受大家關注:其一是網(wǎng)絡結構對于系統(tǒng)的功能和動力學行為的影響,即從結構到動力學輸出的所謂“正問題”。特別是當簡單的動力學單元通過復雜的結構耦合起來以后,將會出現(xiàn)哪些新的集體行為和涌現(xiàn)現(xiàn)象,以及如何認識和控制這些現(xiàn)象,這一直是各領域研究的熱點內(nèi)容。再者就是如何通過系統(tǒng)輸出重構網(wǎng)絡的的結構,即從輸出數(shù)據(jù)到結構的所謂“逆問題”。如今,隨著觀測技術的進步和存儲、計算能力的極大提升,我們已然進入一個“大數(shù)據(jù)”的時代。在各個領域,特別是生物領域和社會領域,數(shù)據(jù)每時每刻都在積累。如何從這些數(shù)據(jù)中挖掘盡可能多的信息,特別是如何通過數(shù)據(jù)窺探隱藏其中的動力學機制和網(wǎng)絡結構一直是人們關注的焦點。在正問題中,一種所謂“爆炸同步”(Explosive synchronization)的現(xiàn)象近年受到了廣泛關注。爆炸同步是發(fā)生在耦合振子網(wǎng)絡中的一種一階同步相變過程,而大量的研究表明,在動力學參數(shù)與網(wǎng)絡結構參數(shù)之間預設特定的約束關系是實現(xiàn)該過程的一個十分重要的條件。該現(xiàn)象深刻的揭示了網(wǎng)絡結構對于系統(tǒng)行為的影響。在逆問題的相關研究中,噪聲在網(wǎng)絡重構中的作用在最近受到較多的討論和關注。在現(xiàn)實中很多網(wǎng)絡系統(tǒng)的觀測數(shù)據(jù),比如神經(jīng)系統(tǒng)、基因調(diào)控系統(tǒng)的生物數(shù)據(jù)等等,是系統(tǒng)的非線性結構和環(huán)境中噪聲相互作用的結果,而且這些非線性結構的具體形式和噪聲的關鍵統(tǒng)計特征往往是未知的,甚至還有部分節(jié)點(變量)的數(shù)據(jù)由于種種原因不能被直接測量到。在這種情形下,如何重構網(wǎng)絡結構是非常具有挑戰(zhàn)性和現(xiàn)實意義的問題。在這兩方面,我們的研究主要取得了如下進展:(Ⅰ)我們首先發(fā)現(xiàn)了爆炸同步和其所需的節(jié)點頻率-連接度正相關約束可以在振蕩網(wǎng)絡中自組織的產(chǎn)生,因此可以解除之前需要預設參數(shù)約束的限制。在擴散耦合的全同金茲堡-朗道振子網(wǎng)絡中,數(shù)值結果和解析分析都證實,先前人們研究中作為復雜網(wǎng)絡爆炸性同步的重要條件之一——節(jié)點頻率與連接度的線性正關聯(lián),可以在系統(tǒng)演化過程中自組織的形成,并產(chǎn)生爆炸同步的現(xiàn)象。鑒于金茲堡-朗道方程是動力學系統(tǒng)在Hopf分岔點附近的普適描述,類似的爆炸同步現(xiàn)象被證明在一般的反應-擴散動力學網(wǎng)絡中存在,且其中的參數(shù)和變量可以被一一映射到金茲堡-朗道振子網(wǎng)絡中。因此,我們的研究證明了爆炸同步是反應-擴散動力學系統(tǒng)中的一種涌現(xiàn)現(xiàn)象,廣泛存在于Hopf分岔點附近的振蕩系統(tǒng)之中。此外,我們還得到了爆炸同步發(fā)生的參數(shù)空間,這將為在實際系統(tǒng)中發(fā)現(xiàn)爆炸同步提供有益的指導。(Ⅱ)對于強噪聲作用下的非線性動力學網(wǎng)絡,我們提出高階關聯(lián)矩陣的方法(high-order correlation computations,簡稱 HOCC)重構網(wǎng)絡動力學結構。該方法可以處理加性白噪聲和乘性白噪聲下的非線性網(wǎng)絡重構問題,統(tǒng)一的推斷出網(wǎng)絡中非線性的節(jié)點動力學、相互作用結構以及噪聲的統(tǒng)計量。所有計算完全基于數(shù)據(jù),不要求已知系統(tǒng)和噪聲的其他信息,如節(jié)點的動力學形式或噪聲的統(tǒng)計特征。HOCC方法有以下特點:(i)通過計算數(shù)據(jù)的高階關聯(lián)矩陣,有效的處理系統(tǒng)中的非線性動力結構;(ii)通過差時關聯(lián)計算,將確定性動力學和噪聲的統(tǒng)計特征的重構退耦為兩個相互獨立的步驟;(iii)復雜網(wǎng)絡的逆問題最終被歸結為一個簡單的線性矩陣代數(shù)方程計算。經(jīng)過數(shù)值驗證和誤差分析,該算法的有效性和魯棒性得到了充分的證明。HOCC方法還被進一步推廣到有測量噪聲的情形,且在某些情況下對色噪聲也同樣適用。(Ⅲ)對于系統(tǒng)中不僅存在噪聲,還存在隱藏變量的情況,我們提出一種可以有效重構網(wǎng)絡動力學結構的方法。其關鍵思想在于,數(shù)據(jù)中不僅包含了已測量節(jié)點的信息,還包含了與它們發(fā)生作用的隱藏節(jié)點的信息。該方法的關鍵之處是數(shù)據(jù)的高階導數(shù)的相關計算可以有效的挖掘這些信息。此外,我們還特別關注了色噪聲下的網(wǎng)絡重構問題。一方面,我們可以將其作為有隱藏變量的逆問題的一個特例,用上方法將系統(tǒng)和色噪聲用統(tǒng)一的表達式重構出來;另一方面,我們又給出一種雙矩陣方程迭代算法,可以通過運算將網(wǎng)絡結構,噪聲強度以及噪聲關聯(lián)時間參數(shù)求出。這些方法的有效性在數(shù)值驗證中得到了充分的證明。
[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.
【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O157.5
本文編號：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.
【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O157.5
【參考文獻】
相關博士學位論文 前1條
1 肖井華;控制與同步時空混沌[D];北京師范大學;1999年
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