时间网络中的结构符号可牧性：通过 $\pi_p$-图的充分条件
Structural Sign Herdability in Temporal Networks: A Sufficient Condition via $π_p$-Graphs
- Indian Institute of Technology Kanpur（坎普尔印度理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究时间切换有向网络的可牧性，提出基于$\pi_p$-图的充分条件，证明其可保证结构符号可牧性，并揭示边权重幅度对可牧性的影响。
AI中文摘要：
在这封信中，我们研究了时间切换有向网络的可牧性。时间网络被建模为具有固定切换序列的切换系统，这比传统切换系统施加了更严格的可牧性条件。通过利用时间游走与可控性矩阵元素之间的关系，我们推导了可牧性的充分条件。我们进一步表明，边权重的幅度影响可控性矩阵的符号模式，从而影响可牧性。因此，时间网络中的可牧性不仅取决于网络拓扑和切换持续时间，还取决于边权重的幅度。受此观察启发，我们建立了时间网络中结构符号（$\mathcal{SS}$）可牧性的等价图论条件。特别地，我们引入了时间子系统的并多重图，并提出了$\pi$-图的概念。我们证明了$\pi_p$-图（一种时间演化的$\pi$-图）的存在是保证$\mathcal{SS}$可牧性的充分条件。提供了说明性示例以展示所提出的结果。
英文摘要：
In this letter, we study the herdability of temporally switching directed networks. A temporal network is modeled as a switched system with a fixed switching sequence, which imposes more restrictive herdability conditions than those of conventional switched systems. By exploiting the relationship between temporal walks and the entries of the controllability matrix, we derive sufficient conditions for herdability. We further show that the magnitude of edge weights influences the sign pattern of the controllability matrix, thereby affecting herdability. Consequently, herdability in temporal networks depends not only on the network topology and switching durations, but also on the magnitude of the edge weights. Motivated by this observation, we establish equivalent graph-theoretic conditions for structural sign ($\mathcal{SS}$) herdability in temporal networks. In particular, we introduce the union multigraph of temporal subsystems and propose the notion of a $π$-graph. We show that the existence of a $π_p$-graph, which is a temporally evolving $π$-graph, is sufficient to guarantee $\mathcal{SS}$ herdability. Illustrative examples are provided to demonstrate the proposed results.
