基于特征值谱的有向社团网络的同步过程

Synchronization is one of the hottest subjects of complex dynamical networks. Most of the present work focus on conditions to achieve synchronization, the stability of the fully synchronized state, without considering the information about what happens in the path to complete synchronization. Synchronization processes describe in detail the actual evolution mechanism of mesoscales in complex networks, reveal all different topological scales, and provide dynamical methods for the problem of the detection and hierarchy of communities. Researchers have begun to pay close attention to synchronization processes in undirected networks in recent years. The latest results uncover that processes and patterns of synchronization in random, small-world are far different form those in scale-free networks. However, there are few studies on synchronization processes in directed networks. We are concerned about how the directed property of edges in directed networks infulences the eigenvale spectra of coupling matrix and synchronization processes. Based on the above considerations, we plan to carry out our research from the following problems: the properties of Laplacian spectra of directed community networks; synchronization and desynchronization proocesses in directed community networks; the relationship between synchronization processes and the distribution of Laplacian spectra; detecting community structures in directed networks based on the synchronization processes. The research purpose of this project, is to understand the specific role of directed edges and community structure on the growth and formation of the dynamical behaviors of complex networks, to provide new theories and methods for the optimization of the network synchronization, and even to propose a new thinking for the research on mesoscales in complex networks.

同步是复杂网络的研究主题之一。目前人们主要研究网络的同步条件及同步态的稳定性等，忽视了网络还没有完全同步时的信息。而同步过程刻画了复杂网络的中尺度演化机制，揭示了复杂网络的拓扑尺度，为研究复杂网络的社团发现与分层问题提供动力学上的方法。无向网络同步过程近几年开始研究，最新成果表明，随机网络，小世界网络与无标度网络的同步过程和同步方式有着本质的区别。但是有向网络的同步过程还没有涉及。网络边的有向性对网络的谱分布和同步过程有着重要影响。基于以上考虑，我们拟对以下几个问题展开研究：有向社团网络的Laplace谱的分布规律；有向社团网络的同步过程及失同步过程；有向社团网络的同步过程与特征值谱分布之间的关系；基于同步过程探测有向网络的社团结构。本项目的研究目的是理解有向边及社团在复杂网络动力学行为的发展和形成中发挥的特定作用，为优化有向网络同步提供新的理论和方法，也为复杂网络的中尺度研究提供新思路。

复杂动态网络的同步过程是近年来复杂网络研究领域的前沿方向。本项目利用非线性动力学理论、复杂网络理论、随机微分方程理论和计算机仿真手段，根据耦合矩阵的特征值谱信息，深入系统的研究了有向社团网络的同步过程，并在此基础上探测社团网络的拓扑结构。本项目在Nonlinear Dynamics, Journal of Statistical Mechanics, IEEE Transactions on Automatic Control等国际著名期刊发表SCI论文3篇，EI论文1篇，中文核心期刊2篇，高等教育出版社出版学术著作1本，协助培养研究生2名。本项目的主要工作如下：.（1）研究复杂动态网络同步态、以耦合矩阵左特征向量加权平均态与孤立节点的解的关系，同步态与同步轨的关系；.（2）根据边的方向性把有向社团网络分为根块节点和叶块节点，研究这类有向社团网络的完全同步过程；.（3） 探讨辅助系统方法在研究复杂动态网络的广义同步的可适用性，并研究有向网络的广义同步过程；.（4） 研究在噪声作用下，根据动力学相关矩阵识别网络拓扑结构，分析噪声强度，耦合强度在识别过程中的作用；.（5）研究环、链、聚类环、聚类链的同步性质，以及聚类环和聚类链这两种特殊的社团网络的同步性质的尺度可变性。