摘要
复杂网络是交叉学科中最为活跃的领域之一，受到了来自各个领域的广泛关注。本文内容主要研究复杂网络的动力学行为。其中包括：1)关于时滞复杂网络同步的局部稳定性及全局稳定性的若干研究；2)关于时滞耦合权重网络同步的扰动研究；3)关于切变复杂网络中相同步及同步过程研究；4)关于复杂网络连接的离散耦合映射同步分析方法的研究；5)关于移动无线网络上病毒传播的研究。
本文的主要创新成果如下：
1、时延复杂网络同步的局部稳定性及全局稳定性
由于信号受传输速度的限制或节点受自身处理能力的限制，复杂网络中通常存在时延。根据时延产生的原因不同，分别研究了由传输导致的时延耦合的复杂网络同步以及由节点自身引起的时延复杂网络同步。针对带时滞的时变耦合复杂网络同步，提出了用子空间分解和Lyapunov函数分析同步流形稳定性的方法。对由节点动力学引起的时滞复杂网络，分别获得了局部渐进同步和全局指数同步的判据。此外，还研究了时滞耦合的多个神经网络同步的问题。
2、时延耦合权重网络同步的扰动
实际生活中的网络大多数是动态的，网络中不断的有新的边加入或者旧的边移除，势必造成对网络同步性的影响。本文研究了拓扑结构随时间变化的时延权重网络同步的扰动现象，通过理论分析得到了与网络拓扑结构直接相关的同步性准则。根据同步性在网络拓扑受到微小扰动时的变化情况进一步分析了结构对动力学的影响。
3、切变复杂网络中相同步及同步过程
在一些分布式通信系统中，网络节点之间的连接往往在某个时刻是以概率随机存在的。本文研究了一组相位振荡器以切变方式耦合时产生的同步现象，发现了在较快的切换速度下，网络总是能够获得同步。另外，还研究了不同的网络拓扑结构在同步的形成过程中所起的作用，从切变网络模型中得到了均匀网络与非均匀网络不同的同步过程。
4、新的耦合映射复杂网络同步的分析
基于目前复杂网络同步稳定性分析方法中存在的一些局限性，提出了一种利用矩阵测度来研究同步的方法，并通过对离散耦合映射网络的分析，得到了较易验证的局部同步和全局同步的准则。
5、动态复杂网络上病毒传播
复杂网络拓扑结构对网络上动力学的影响也体现在传播行为上。本文提出了一种描述移动无线Ad hoc网络的动态复杂网络模型，该模型刻画了网络节点的移动性、通信信道的竞争机制以及物理连接的限制。借助于SIR模型研究了蠕虫在该模型上的传播行为，揭示了网络节点的移动性与病毒传播之间的关联关系。
As one of the most active areas in the interdisciplinary research field, complex networks attract extensive attentions from various fields of science and engineering. In this dissertation, we perform dynamics study on complex networks. The main contents of this dissertation include: (1) Local and global stability of synchronization in some delayed complex networks; (2) Synchronization and its perturbation of weighted networks with delayed couplings; (3) Phase synchronization and synchronization process in switching networks; (4) Approach to analyze synchronization of coupled map on complex networks; (5) Epidemic spreading on mobile wireless networks.
The main contribution in this dissertation can be summarized as follows:
1. Local and global synchronization study of delayed complex networks
Due to limited transmitting speed and process ability on nodes, usually there are time delays in complex networks. Considering the differences between the factors that induce time delays, we study the synchronization of time-delayed coupled complex networks where the delays are caused by limited transmitting speed and synchronization of coupled time-delayed dynamical systems where the delays are caused by properties of nodes, respectively. For the delayed complex networks with time-varying couplings, we present a new approach to analyze the stability of synchronization manifold by combining subspace decomposition and Lyapunov functions. As to the second case, we derive the criteria for locally asymptotical synchronization and globally exponential synchronization.
2. The study of synchronization and its perturbation in weighted complex networks with time delay
Many real-world networks are dynamically evolving, for example, new edges are introduced or old edges are removed continually, which certainly will have an effect on network synchronizability. In this dissertation we study the perturbation phenomenon of weighted complex networks with time delay when disturbing the coupling configurations. In the light of theoretical analysis, we obtained the criterion for synchronization which is directly related with network topology. We further investigated the impact of topology on dynamics under small coupling perturbations by applying our theoretical results.
3. Phase synchronization study on switching networks of coupled oscillators
It is often the case that due to the dynamic nature of each unit's states in some distributed communication systems, the existence of an information channel between a parr of units at each time instance is probabilistic and independent of other channels; hence, the topology of such networks varies over time. Taking this case into account, we study the onset of synchronization in arrays of phase oscillators with switching couplings. We found that the network always can achieve synchronization under fast switches. Moreover, we further studied the role of network topologies in the synchronization process and observed the different synchronization processes between homogeneous networks and heterogeneous networks from our switching network model.
4. Study on synchronization of coupled map networks by a new approach
Considering the limitation of existing stability analysis methods for complex network synchronization to some extent, we present a new approach to study the sychronizability by using matrix measure. Through theoretical analysis of coupled discrete maps, we obtained the local and global synchronization criteria which can be easily verified and applied in practice.
5. Spreading dynamics study on dynamical complex networks
The effect of network topology on system dynamics can also be manifested in epidemic spreading behaviors. In this dissertation, we present a dynamical complex network model to describe the mobile wireless Ad hoc networks. This model captures the mobility of nodes in networks, channel contention during communicating and limited physical connections. By means of basic SIR model, we studied the spreading dynamics of worms on this model, finding the relationship between the mobility of nodes and epidemic spreading.
引文
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