Gao Zilin, Wang Yinhe
School of Automation, Guangdong University of Technology, Guangzhou, Guangdong Province, China.
School of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing, China.
PLoS One. 2018 Jan 31;13(1):e0191941. doi: 10.1371/journal.pone.0191941. eCollection 2018.
The nodes and their connection relationships are the two main bodies for dynamic complex networks. In existing theoretical researches, the phenomena of stabilization and synchronization for complex dynamical networks are generally regarded as the dynamic characteristic behaviors of the nodes, which are mainly caused by coupling effect of connection relationships between nodes. However, the connection relationships between nodes are also one main body of a time-varying dynamic complex network, and thus they may evolve with time and maybe show certain characteristic phenomena. For example, the structural balance in the social networks and the synaptic facilitation in the biological neural networks. Therefore, it is important to investigate theoretically the reasons in dynamics for the occurrence. Especially, from the angle of large-scale systems, how the dynamic behaviors of nodes (such as the individuals, neurons) contribute to the connection relationships is one of worthy research directions. In this paper, according to the structural balance theory of triad proposed by F. Heider, we mainly focus on the connection relationships body, which is regarded as one of the two subsystems (another is the nodes body), and try to find the dynamic mechanism of the structural balance with the internal state behaviors of the nodes. By using the Riccati linear matrix differential equation as the dynamic model of connection relationships subsystem, it is proved under some mathematic conditions that the connection relationships subsystem is asymptotical structural balance via the effects of the coupling roles with the internal state of nodes. Finally, the simulation example is given to show the validity of the method in this paper.
节点及其连接关系是动态复杂网络的两个主要部分。在现有的理论研究中，复杂动态网络的稳定和同步现象通常被视为节点的动态特征行为，这主要是由节点之间连接关系的耦合效应引起的。然而，节点之间的连接关系也是时变动态复杂网络的一个主要部分，因此它们可能随时间演化并可能呈现出某些特征现象。例如，社会网络中的结构平衡和生物神经网络中的突触易化。因此，从理论上研究其发生的动力学原因很重要。特别是，从大规模系统的角度来看，节点（如个体、神经元）的动态行为如何影响连接关系是一个值得研究的方向。在本文中，根据F. 海德提出的三元组结构平衡理论，我们主要关注连接关系部分，它被视为两个子系统之一（另一个是节点部分），并试图通过节点的内部状态行为找到结构平衡的动态机制。通过将Riccati线性矩阵微分方程用作连接关系子系统的动态模型，在一些数学条件下证明了连接关系子系统通过与节点内部状态的耦合作用实现渐近结构平衡。最后，给出了仿真示例以说明本文方法的有效性。
