Minimal edge controllability of directed networks is investigated in this paper. A new edge dynamics model is first introduced with two nonzero parameters describing the linear relationship between the node states and the edge states. Three different digraphs as skeleton structures for minimal edge controllability are analyzed. The conditions ensuring both node controllability and edge controllability for these three digraphs are presented, respectively. It is found that cycles in these networks play an important role in edge controllability. The notion of minimal edge controllability is then extended to signed digraphs. It is shown that the minimal edge controllability of a signed cycle depends on the number of edges with negative weights, regardless of the placement of the negative weights on the edges. Some examples are presented for illustration and verification.

中文翻译：

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有向网络的最小边缘可控性

本文研究了有向网络的最小边缘可控性。首先引入了一个新的边缘动力学模型，其中两个非零参数描述了节点状态和边缘状态之间的线性关系。分析了三个不同的有向图作为最小边缘可控性的骨架结构。分别给出了保证这三个有向图的节点可控性和边可控性的条件。发现这些网络中的循环在边缘可控性中起着重要作用。然后将最小边可控性的概念扩展到有符号有向图。结果表明，有符号循环的最小边缘可控性取决于具有负权重的边缘的数量，而与负权重在边缘上的位置无关。