In social networks, the structural balance is a state of a group of individuals (nodes) with established mutual relationships (connection relationships) between them. It is easy to see that a social network can be described by a complex dynamical network model composed of the nodes subsystem (NS) and the connection relationships subsystem (CS), where the two subsystems are usually coupled with each other. It implies that the dynamic changes of nodes’ states may cause the structural balance in CS. However, few papers have discussed the relationship between the structural balance and the specific dynamic changes of the nodes’ states. This paper proposes a model of complex dynamical networks, and mainly focuses on the dynamic changes of states in NS which can lead to the structural balance in CS. It is proved that if each state in NS is doing a specific dynamic motion via the controller with the parameter adaptive law, then the CS can track a given structural balance matrix via the effective coupling and the structural balance can be achieved. Such a result can be regarded as an explanation of the relationship between the structural balance and the specific dynamic changes of the nodes’ states. Finally, the simulations verify the effectiveness of the proposed method.

中文翻译：

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一类复杂动力网络结构平衡的自适应控制

在社交网络中，结构平衡是一组个人（节点）之间具有已建立的相互关系（连接关系）的状态。不难发现，社交网络可以通过由节点子系统（NS）和连接关系子系统（CS）组成的复杂动态网络模型来描述，其中两个子系统通常相互耦合。这意味着节点状态的动态变化可能会导致CS的结构平衡。但是，很少有论文讨论结构平衡与节点状态的特定动态变化之间的关系。本文提出了一个复杂的动力学网络模型，主要集中在NS中状态的动态变化，这可能导致CS中的结构平衡。事实证明，如果NS中的每个状态都通过带有参数自适应律的控制器进行特定的动态运动，则CS可以通过有效的耦合来跟踪给定的结构平衡矩阵，从而可以实现结构平衡。这样的结果可以看作是结构平衡与节点状态特定动态变化之间关系的解释。最后，仿真验证了所提方法的有效性。