多重边复杂网络牵制同步控制节点集优化

Optimal Pinning Control Synchronization of Node Groups for Multi-Edges Complex Networks

摘要: 针对多重边复杂网络的牵制同步控制节点选取问题，提出一种基于图论的优化牵制节点组选择的策略。根据网络中边的性质不同，基于网络拆分的思想，通过引入时延将网络进行拆分。根据李雅普诺夫稳定性理论，使用自适应牵制控制器以及图论和矩阵分析等知识，得到可以优化多重边复杂网络牵制节点组选择的方案：当牵制节点个数固定时，从网络的对称拉普拉斯矩阵中删除受控节点所对应行和列后所得子矩阵的最大特征值，用此来衡量牵制方案的有效性。即：网络中单个节点的动力学系统与耦合强度给定时，最大特征值越大，牵制方案越有效。与此同时，对最大特征值的谱特性进行分析，以此对优化牵制方案提供理论支持。

Abstract: Aiming at pinning synchronization control node’s selection in multi-edges complex networks, a strategy based on graph theory to optimize the selection of pinning node groups is proposed. According to the edges’ properties that is different in the network, based on the network splitting’s idea, the network is split by introducing delay. According to Lyapunov stability theory, using adaptive pinning controller, graph theory and matrix analysis’s knowledge, a scheme that can optimize pinning node groups’ selection in multi-edges complex networks is obtained: When the pinning nodes’ number is fixed, the maximum eigenvalue of the sub matrix obtained by deleting the rows and columns corresponding to the controlled nodes from the symmetric Laplacian matrix of the network is used to measure the pinning scheme’s effectiveness. That is, when the dynamic system and nodes’ coupling strength in the network are given, the larger the maximum eigenvalue, the better the effectiveness of the pinning scheme. At the same time, the spectral characteristics of the maximum eigenvalue are analyzed, which provides a theoretical support for the optimization of the pinning scheme.