复杂网络上的动力学模型分析及随机影响

【作者】 刘茂省；

【导师】 阮炯；

【作者基本信息】 复旦大学 ， 应用数学， 2009， 博士

【摘要】 复杂网络是研究复杂系统的一门新兴学科,近几年受到国内外研究学者的广泛关注。任何复杂系统都可以抽象成为由相互作用的个体组成的网络,因而网络无处不在,遍及自然界、生物系统和人类社会。这些网络中比较为人所熟知的如神经网络、互联网、疾病传播网、蛋白质网、合作网等。为了理解这些复杂网络的结构和性质,人们首先就需要研究这些网络的动力学行为。通过对复杂网络动力学行为的研究,不仅可以更好地理解现实世界中的网络所呈现出来的各种动力学现象,而且还可以根据不同的网络特性采取不同的方法为我们的工作和生活服务,更重要的是在某些方面我们还可以设计出具有更好特性的网络为人类造福。传染病在人群中的流行、病毒在计算机网络上的蔓延、谣言在社会中的扩散等,都可以看成服从某种规律的传播行为。传统的传播动力学模型假定所研究范围的个体是充分混合的,即每个个体接触其他个体的数量基本相等,但近年来关于复杂网络的研究显示绝大部分的个体不是充分混合的,而是具有某些特殊结构。这些特殊结构导致了复杂网络上的传播行为与传统动力学模型有较大差异,这说明在考察传播行为时,必须结合特定的网络结构才能建立更加符合实际的动力学模型。本论文在前人工作基础上对复杂网络上的传播行为做了一些深入的探索和研究,主要考虑了非线性传播率的传播动力学,两种群的传播动力学。另外我们知道,随机因素会对复杂网络的动力学行为产生一定的影响,因此我们还研究了随机噪音对网络传播的影响,随机离散的神经网络模型,以及共同噪音引起的离散系统的同步问题。这些研究无论在理论上还是在实际应用中都具有重要意义。本文的主要内容可概述如下:第一章介绍复杂网络(包括神经网络)的研究背景与进展,给出本文所要用到的预备知识,并给出全文的结构。第二章研究复杂网络上带有非线性传播率的模型。研究了带有非线性传播率的均匀网络模型,给出了模型的平衡点的存在条件、稳定性以及Hopf分岔分析等,得出了在网络传播中出现周期性现象的原因;研究了带有非线性传播率的非均匀网络模型,给出了无病平衡点的稳定性分析。第三章研究复杂网络上两种群的传播动力学模型。主要介绍了均匀网络上的两种群的传播模型,研究了非均匀网络上两种群的传播模型,给出了疾病传播的阈值和平衡点的稳定性分析。另外还给出了两种群传播模型三种免疫策略,得出了均匀网络和非均匀网络上的有效免疫策略。第四章研究复杂网络上的随机传播模型。研究了均匀网络上的随机传播模型和非均匀网络上的随机传播模型,得出了随机稳定性和随机分岔的条件。研究表明随机噪音对不同拓扑结构的网络传播模型会起到不同的作用。第五章研究随机噪音对离散神经网络的影响。考虑了两类随机噪音影响的离散神经网络模型,利用随机动力系统中的鞅理论,得出了随机离散神经网络的平凡解是几乎必然渐近稳定的条件。第六章研究离散状态下共同噪音引起的同步问题。先讨论了一般的离散系统发生同步的条件,然后讨论了带有时滞的离散系统发生同步的条件。第七章对论文工作进行了总结,并对今后要研究的工作提出了展望。更多还原

【Abstract】 Complex networks have been considered as an important approach for describing and understanding complex systems by many researchers recently. Any complex system is composed of interacted individuals, which can be naturally represented by graphs with individuals denoted by nodes and interactions by links. From this point of view, complex networks are ubiquitous, ranging from nature and biological system to society. The well-known networks include neural networks, Internet, epidemic spreading networks, protein-protein interaction networks, collaboration networks, etc. To understand the structures and properties of networks, the study of dynamical behaviors on these complex networks are needed firstly. By studying the dynamic properties of complex networks, one can not only understand the dynamic properties presented in real-world networks but also take different measures for networks with different structures. Furthermore, these results can been applied to design real networks to achieve some desirable that benefit all over the world.The spread of epidemics, computer virus and rumors can be seen as same behaviors with some laws. In the traditional spreading models, the individuals are considered to be mixed adequately, that means each individual contacts other individuals with the same rate. However the recent study of complex networks shows that the distribution of individuals is not homogeneous, but has some special structures. These special structures make the spreading behavior on complex networks different from the traditional models. In order to make the model more practical, one must consider the special structures in studying the spreading behavior. Based on the work of many researchers this dissertation extend the previous results on complex networks, which are the spreading model with non-linear incidence rate and with two interacting species. Furthermore, stochastic factor will affect the dynamic behaviors of complex networks, so the effect of the stochastic noise on complex network, stochastic discrete-time neural networks, and the synchronization of discrete-time systems with common noise are also investigated. These results are significant not only in mathematical theory but also in many applied fields.The main work in this dissertation is listed as follows:In chapter 1, the research background and progress on complex networks, including neural networks, are introduced. Moreover, some preliminaries and the structure of dissertation are given.In chapter 2, the spreading models on complex networks with a generalized nonlinear incidence rate are presented. Firstly the model on homogeneous networks with nonlinear incidence rate is considered, and the existence, the stability of equilibria and the Hopf bifurcation of the model are given. Then the model on a heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. It is shown that the basic reproductive number is independent of the functional form of the nonlinear incidence rate, while the number of the equilibria and the behaviors are indeed different from the corresponding model with linear incidence rate.In chapter 3, models for the spread of two interacting species on complex networks are presented. The dynamic behaviors of the models on the homogeneous network and heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. Three immunization strategies are applied to models. The analytical and simulated results are given to show that the proportional immunization strategy is effective on heterogenous scale-free networks.In chapter 4, the general stochastic nonlinear models of spreading, describing the effect of random fluctuations on complex networks are proposed. It has been found that fluctuation noise would trigger a state of networks from instability to stability. The probability density function for the proportion of infected individuals are found explicitly, and the stochastic bifurcation is analyzed by probability density function. It is a better explanation of occurring the different phenomena with sensitive parameters in many real-world complex networks.In chapter 5, the effect of stochastic noise about the discrete-time stochastic neural networks are investigated. By using the martingale convergence theorem, the almost surely stability condition of two sub-classed of stochastic neural networks are analyzed. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical results.In chapter 6, the synchronization of discrete-time system and the discrete-time system with delay are discussed. These systems are all induced by common external noise. A set of sufficient conditions for the synchronization is found. These conditions show that the synchronization occurs in a wide class of discrete-time system and the discrete-time system with delay.At the end of this dissertation, the conclusions and some topics for future work which include complex networks and stochastic dynamical systems are given.更多还原