【作者】 郭龙；

【导师】 蔡勖；

【作者基本信息】 华中师范大学 ， 理论物理， 2010， 博士

【摘要】 自然科学和社会科学领域中的许多复杂系统都可以用复杂网络加以描述。大量的实证研究表明,真实网络中呈现出区别于规则网络和随机网络的特性,如小世界特性、无标度特性、层级结构特性等等。因此,对于复杂网络的拓扑特性的清晰认识将为研究复杂网络的动力学研究奠定基础。在本文中我们侧重研究复杂网络的拓扑结构和复杂网络上动力学行为之间的相互关系,从而由动力学角度探寻复杂网络拓扑特性的涌现机制。具有创新性研究的成果,有以下五个方面：1.分析了复杂网络拓扑结构对分形维数的影响。首先,采用Shanker定义的分形维数方法研究了真实网络的分形维数。然后,研究了小世界网络的捷径(Shortcuts)对分形维数的影响,通过非线性拟合给出了网络的分形维数df与捷径数(Np)之间的函数关系。最后,分析了规则格子尺寸对其分形维数的影响。这里,分形维数的定义可以推广到规则格子中,且分形维数的计算是精确解。2.首次提出利用列车运营情况的加权方法,研究了中国铁路网。首先,在L空间中分析了中国铁路网节点度分布以及节点度关联特性。然后,依据中国铁路网上实际列车运营情况,对中国铁路网中的节点和边进行加权。其中,每列列车对其停靠的每一个车站的贡献都是其停靠的所有车站数目的倒数。这里权重的定义弥补了抽象空间(L空间和P空间)中对网络拓扑分析的不足。3.复杂网络节点异质性对复杂网络上动力学行为的影响。分析了静态无标度网络上Bak-Sneppen模型进化动力学。其中,静态无标度网络构造模型中可调参数α反映了网络中节点异质性的强弱。α越大,网络的异质性越强。研究发现网络的临界平均适应度〈f〉*随着α的增加而减小。另一方面,网络的〈f〉0雪崩指数T随着可调参数α的变化而呈现阶梯下降,结合静态无标度网络上疾病传播动力学行为,将无标度网路分为三类：随机类、线性类和物理类。最后,分析了规则格子、小世界网络和无标度网络上改进Deffuant模型演化动力学,我们发现系统稳态opinion s*随着信任参数ε变化呈现出分叉现象。通过分析较小团簇的opinion及节点度与信任参数ε的关系,发现小世界网络和无标度网络上opinion达到一致态的形成过程具有明显差异,这在复杂网络的同步现象中也有所呈现。4.小世界网络捷径对opinion形成动力学的影响。类比于统计物理中的Ising模型,研究了小世界网络上二态opinion演化动力学。通过分析系统的opinion磁化强度、空间关联特性以及时间关联特性,发现小世界网络中的捷径具有增强系统长程关联的作用,从而促使系统演化到一致态。通过有限尺寸效应分析,发现从一维环开始生成的小世界网络上opinion相变是赝相变。5.自适应网络上社团结构的涌现。构造了一个简单的自适应网络演化模型。如果网络中两活动节点的opinion之差大于信任参数ε,连接这两节点的边将断开;反之,节点之间的连边促使两节点的opinion向彼此靠近,靠近强度依赖于收敛参数μ。网络中断边的重绕依据局域重绕连接和全局重绕连接两种方式。这反映了网络的拓扑结构演化动力学与网络上节点状态演化动力学之间的相互作用。通过模拟研究,发现全局重绕增强网络上opinion演化到一致态的相变能力；而局域重绕连接增强网络社团结构的涌现。局域重绕连接概率越强,网络中社团的模块强度越大。更多还原

【Abstract】 Many complex systems arising from nature and human society can be described as complex networks. A lot of experimental works show that real complex networks share some distinctive characteristic properties, such as the small-world effect, scale-free property and the hierarchical structure, which differ from the random network and the regular lattice. The topology structure of complex networks has been studied, which is helpful for us to search the dynamics of and on complex networks. In this paper, we focus on the interaction between the complex network structure and the dynamics on complex networks, and study the emergence of some distinctive characteristic properties in complex networks from the viewpoint of dynamics.The results of innovative research are shown as follows:1. To analyze the effects of complex network structure on fractal dimension. Firstly, the fractal dimensions of real complex networks are studied by means of Shanker’s defini-tion. Secondly, the effect of the shortcuts in small-world network on the fractal dimension is shown through the relation between the fractal dimension df and the shortcuts (Np), which is given by the method of nonlinear fitting. Finally, the effect of the regular lattice size N on its fractal dimension is analyzed through exact calculation.2. To study the China Railway Network(CRN) by proposing the weighted method based on the trains’real kinematics. Firstly, the degree distribution and the degree-degree correlation of CRN in space L are studied. Then, the weighted property of CRN is analyzed according to the trains’ real kinematics. The contribution of train j to terminal station i is 1/nj(i), where nj(i) is the number of stations that train j passes. Here, the definition of weight makes up the shortage of network topology analysis in the abstract space (spaces L and P).3. To analyze how the heterogeneity of complex networks affects the dynamics that occurs on complex networks. We analyze the Bak-Sneppen evolution model on static scale-free networks, where a tunable parameter a is defined to describe the heterogeneous prop-erty of complex network. The larger a is, the more heterogeneous the network is. Our study shows that the critical average fitness <f>* decreases with a increasing. On the other hand, the exponentτof the <f>0 avalanche decreases in a form of stair-case with a increas-ing. Combining epidemic spreading dynamics on the static scale-free networks and theτexponent, the scale free networks can be divided into three classes:random class, linear class and physical class. Then, extended Deffuant models on regular lattice, small-world networks and scale-free networks are studied, where the bifurcation phenomena is found. Through analyzing the relationship between the opinion, degree and theεin the minority clusters, we find the formation process of opinion in small-world networks is different from that in scale-free networks, which is also found in the synchronization phenomena in both networks.4. To analyze the effect of the shortcuts in small-world network on the opinion for-mation dynamics. According to the Ising model in statistical physics, we investigate the binary opinion formation dynamic on small-world network. The magnetization, spatial correlation and temporal correlation are studied. Our results show that the shortcuts en-hance the ability of long-range interactions which can drive the system towards reaching the consensus state. On the other hand, the phase transition in the small-world network which is built on one dimension substrate is a pseudo one.5. To analyze the emergence of community in the adaptive network. A simple adap-tive network model is proposed. The edge between active vertices is broken when the difference between two vertices’ opinion is greater than the confidence parameterε; Other-wise, each opinion moves partly in favor of the direction of the other. The rewiring of the broken edge follows the rules of the rewiring of local attachment (RLA) and the rewiring of global rewiring (RGA). All is mentioned above shows the interaction between the dynamics of complex network structure and the dynamics on complex network. Through simulations, we find that the RGA enhances the ability of the system reaching the phase transition, and the RLA enhances the formation of community structure in complex networks.更多还原