IEEE Trans Cybern. 2021 Dec;51(12):5659-5670. doi: 10.1109/TCYB.2019.2933478. Epub 2021 Dec 22.
The edge convergence problems have been explored for directed signed networks recently in 2019 by Du, Ma, and Meng, of which the analysis results, however, depend heavily on the strong connectivity of the network topologies. The question asked in this article is: whether and how can the edge convergence be achieved when the strong connectivity is not satisfied? The answer for the case of spanning tree is given. It is shown that if a signed network is either structurally balanced or r-structurally unbalanced, then the edge state can be ensured to converge to a constant vector. In contrast, if a signed network is both structurally unbalanced and r-structurally balanced, then its edge state does not converge to a constant vector any longer, but to a time-varying vector trajectory with a constant speed. Further, the dynamic behavior results of edges can be derived to address the node convergence problems of signed networks. The simulation examples are provided to illustrate the validity of the established edge convergence results.
最近在 2019 年，Du、Ma 和 Meng 对有向符号网络的边缘收敛问题进行了探讨，然而，其分析结果在很大程度上取决于网络拓扑的强连通性。本文提出的问题是：在不满足强连通性的情况下，边缘收敛是否可以实现，如果可以，应该如何实现？本文针对生成树的情况给出了答案。结果表明，如果一个符号网络是结构平衡的或者 r-结构不平衡的，那么边状态可以保证收敛到一个常数向量。相比之下，如果一个符号网络既结构不平衡又 r-结构平衡，那么它的边状态就不会再收敛到一个常数向量，而是收敛到一个具有恒定速度的时变向量轨迹。此外，还可以推导出边的动态行为结果，以解决符号网络的节点收敛问题。提供了仿真示例来说明所建立的边缘收敛结果的有效性。
