【作者】 王文旭；

【导师】 汪秉宏；

【作者基本信息】 中国科学技术大学 ， 理论物理， 2007， 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，遍及自然界和人类社会。其中颇具代表性且受到广泛研究的网络有互联网、万维网、铁路网、航空网、电力网、蛋白质相互作用网、新陈代谢网、基因调控网和各种合作性的网络等。研究这些网络不仅对人们的工作和生活至关重要，而且对了解自然界特别是生物系统的奥秘有深远的科学意义。另一方面，复杂网络研究关注个体之间的微观相互作用导致的系统的宏观现象。这种将系统行为作为一个整体的研究方式不受传统还原论方法的限制，从而能够预言复杂系统丰富的整体行为，包括自组织特性，涌现等。这使得以网络的方式研究复杂系统成为了必然趋势。同时，复杂网络的研究热潮促进了学科之间界限的打破，推动了统计物理、非线性动力学、应用数学、信息工程、社会学和生物学等多学科的交叉和发展。因此，复杂网络研究具有重大的理论价值。研究复杂网络的最终目标是理解网络上的各种动力学过程如何受到网络结构的影响，而网络的形成和演化机制决定网络的结构。因此研究网络结构的演化动力学成为了复杂网络研究的前提和热点之一。根据当前国内外复杂网络的研究动态和发展趋势，我们在网络结构的演化动力学机制和网络上的动力学过程方面做了比较系统的工作，涉及权重网络的演化、网络上的信息流、博弈过程、病毒传播和同步现象以及布尔动力学等。本文的主要工作如下。提出了交通流驱动机制、双向选择机制和双向吸引机制，建立了一系列权重网络演化模型，重现了实际权重网络中所观察到的节点权重、边权重和连接度的幂率特性，以及小世界特性、节点度和权重的非线性相关性等。基于这三种机制的模型分别得到了负的相配混合性，分层特性和正负可调的相配混合性，从而能够很好地刻画技术网络，社会网络和生物网络这三大类真实网络，并且回答了复杂网络研究十大问题之一的相配混合性问题。系统地研究了无标度网络上的交通动力学。我们提出了基于局域拓扑和动态信息的数据包路有规则，研究了系统中信息流从自由流到阻塞流的相变特性，并利用这一特性刻画网络的通讯能力。我们还发现无标度网络上交通动力学存在着迟滞回线和亚稳态的行为。我们的研究对于目前路由协议的优化以及新一代路由协议的设计有一定的指导意义。研究了无标度网络上的同步，提出通过去耦合过程来提高网络的同步能力。此外，研究了具有群落结构的无标度网络上的病毒传播，发现病毒传播中存在同步现象，并且同步现象随着群落结构强度的改变存在相变。我们运用有限尺度分析方法计算了相变指数并指出相变的普适类。探讨了网络上的演化博弈，提出基于个体历史记忆的演化机制、偏好学习机制和博弈与网络结构的共演化，能够很好地解释实际中普遍存在的和合作现象。而且我们发现了合作频率的分段和非单调行为、斑图相变、无标度网络的涌现、随机共振现象和双稳态行为等。提出运用动力学粗粒化的方法来研究无标度布尔网络在混沌区域的动力学特性，发现系统状态空间网络具有五种幂律分布特性。进一步将布尔动力学应用到真实的细胞及雹子基因调控网络，发现了动力学核心模块和控制细胞与雹子之间相互转化的两条关键通路，并运用动力学的方法对不同条件下表达的基因调控网络进行分类，得到很好的结果。通过研究不同条件下基因调控网络的Derrida曲线，我们发现正常细胞更加接近于混沌的边缘，这验证了Kauffman提出的混沌边缘假说。更多还原

【Abstract】 Complex network modeling has been considered as an important approach for describing and understanding complex systems. A complex system is composed of many interacted individuals, which can be naturally represented by a graph with individuals denoted by nodes and interactions by links. From this point of view, complex networks are ubiquitous, ranging from nature to society. In the past few years, we have witnessed a great devotion of scientists to understanding the underlying mechanisms of complex networks together with the dynamical processes taking place on them. The advances of complex networks are propelled by several parallel developments, including the computerization of data acquisition, increase of computing power, breakdown of boundaries between disciplines and increasing needs for understanding the behavior of an integrated complex system as a whole. Since the groundwork laid by Barabasi, Albert, Watts and strogatz, more and more attention has been given to this emerging field. The widely observed "small-world" and "scale-free" properties shared by many real networks have now completely changed the traditional view and even knowledge of many people on the real-world networks and spurred the rapid development of the interdisciplinary scientific field.So far, the investigation of complex networks has covered many fields, including physics, chemistry, technology and biology, where intensively studied networks cover as diverse as the Internet, World Wide Web, point-to-point networks, collaboration networks, airport networks, power grids, protein-protein interaction networks, genetic regulatory networks and epidemic spreading networks, etc. These networks are of high technological and intellectual importance and the desire to understand such interwoven complex systems has encountered significant challenges as well. The ultimate goal of studying complex networks is to understand how topological properties affect the dynamical processes taking place on them. However, what should be first done is to deeply understand how complex networks possess those common topological features in a self-organized way. Inspired by the current international research interests, we focused on the evolutionary dynamics of the network structures and the dynamical processes occurring over complex networks. Our work contains the evolution of weighted networks, information traffics on scale-free networks, synchronization on networks, evolutionary games on networks, Boolean dynamics and genetic regulatory networks.We have proposed several evolutionary models for weighted networks, including the traffic driven model, the mutual selection model and the mutual attraction model. The weighted networks generated by such models possess power-law distributions of strengths, weights and degrees with the exponents tuned by model parameters between 2—3, which are well consistent with the empirical evidence. In particular, the weighted network generated by the traffic driven model has a disassortative mixing property and a nonlinear correlation between strength and degree, which are in good agreement with real weighted technological networks. For the mutual selection model, the obtained statistical hierarchical property by the model can well reproduce the real observations about weighted social networks. The mutual attraction model can generate networks with both assorta-tive and disassortative mixing properties, indicating that the model can well mimic social networks, technological networks and biological networks. Especially, our work has provided a good answer to one of the ten leading questions in the study of complex networks: why all social networks are assortative, while technological and biological networks are disassortative?We have systematically investigated the dynamics of information traffics over scale-free networks. Several routing strategies of managing data packets have been proposed, including the local routing strategy and the mixed routing strategy based on local static and dynamic information. We can quantify the capacity of a network by the phase transition from free flow state to congestion state, and we have found the optimal parameter values of each model, resulting in the highest efficiency of scale-free networked traffic systems. Moreover, we have found hysteresis loop in networked traffic systems with a finite packet-delivering ability. Such hysteresis loop indicates the existence of bistable states in the traffic dynamics over scale-free networks.We have studied the collective synchronization behavior over scale-free networks and proposed a decoupling process to enhance the synchronizability of scale-free networks. Because of the low cost in performing this task, the decoupling process method may have very good potential applications. Furthermore, we have investigated the epidemic spreading over scale-free networks with community structures and found their synchronization behavior. Interestingly, there exist phase transitions of the synchronizability depending on the strengths of the community structures in the networks. On the basis of the finite size scaling, we have obtained the values of phase transition exponents, indicating that the phase transitions belong to the mean-field type.We have systematically studied the evolutionary games on networks. A memory-based snowdrift game has been proposed. We found that in regular lattices, the cooperation level versus payoff parameter shows a step structure with each step corresponding to a typical spatial pattern. There is a sharp pattern transition between any two different patterns. This interesting phenomenon has not been reported previously. For scale-free networks, the cooperation level versus payoff parameter in the memory-based snowdrift game displays non-monotonous behavior with cooperation level peaking at some specific parameter values, which suggests that suitable encouragement of selfish behavior can lead to the optimal cooperation level.Whereafter, we proposed an evolutionary model by coupling the evolution of games and the network. It is found that the cooperative behavior can be considerably promoted and the scale-free as well as assortative mixing topological properties can emerge from the coevolution. Furthermore, we have presented a preferential learning mechanism and investigated the evolutionary games on weighted adaptive networks for better mimicking the widely observed cooperation behavior. Besides, by adopting the homogenous small-world networks, we have explored the effect of randomness in evolutionary games and found coherence resonance phenomena therein.In order to understand the dynamical properties in the chaotic range of scale-free Boolean networks, we proposed a dynamical coarse graining method performed in the state space. It is found that the weighted and directed state graph possesses five power-law distributions, independent of the strength of the coarse graining. Furthermore, we have explored the dynamics of real genetic regulatory networks of cells by using the Boolean network model. Interestingly, a universal core modular is found. Subsequently, we can classify the cellular regulatory networks under different conditions in terms of the dynamical properties of the core modular. Through the study of the Derrida curve under different conditions, we found that the normal cell is closer to the edge of chaos, which supports the hypothesis of Kauffman.更多还原