有色网络的同步与控制

This project studies synchronization and control of a class of complex network model based on colored graph theory. Firstly, we introduce colored network model corresponding to colored graph for decribing the real world better, in which, the color of each node denotes its dynamics and color of edge denotes not only the topological structure of colored network, but the inner mutual interaction between nodes. That is, the inner coupling matrices are nonidentical between different pairs of nodes can be denoted by the color of edge between them. Secondly, we study synchronization and control of general colored network and derive several synchronization creteria. Finally, we study the pinning control of colored network, especially edge-colored network. For the existing complex network models, many important results about optimal pinning control have been obtained, for example, the node with largest degree is prior to other nodes. However, most of the obtained results are invalid for colored network. This typical issue may be well studied by introducing the notation of colored degree.

本项目研究一类基于有色图理论的复杂网络模型的同步与控制问题。首先，为了更好的描述现实世界，基于有色图的概念提出有色网络模型：网络中结点的颜色表示其自身的动力学，连接边的颜色不仅表示网络的拓扑结构，还表示结点之间的内部耦合作用，不同颜色的边表示不同结点对之间的内部耦合矩阵是不一样的；其次，研究一般有色网络的同步与控制，并给出一些同步准则；最后，研究有色网络的牵制控制，对于现有的网络模型，已经得到了很多优化控制的结果，如优先控制最大度结点等，但这些结果对于有色网络而言，很多都不适用，通过引入有色度的概念将很好的研究这类典型问题。

近年来，对复杂网络的建模、分析与控制的研究取得了大量有价值的成果。本项目经过一年的深入研究，取得了如下成果：首先基于数学中有色图理论提出了有色网络模型，这类模型可以更好地描述现实复杂系统的复杂性；其次研究了有色边网络的牵制控制问题，以及有色网络的控制问题；再次提出了有色社团网络模型，并研究了聚类同步行为；最后对其他相关方向也进行了研究。