Herdability is a variant of controllability, and is an indicator of the ability to drive system states to a specific subset of the state space. This paper characterizes the controllable subspace and herdability of signed weighted networks. Specifically, a dynamic signed leader–follower network is considered, in which a small subset of the network nodes (i.e., the leaders) is endowed with exogenous control input and the remaining nodes are influenced by the leaders via the underlying network connectivity. The considered network permits positive and negative edges to capture cooperative and competitive interactions, resulting in a signed graph. Motivated by practical application, the system states are required to be driven by the leaders to be element-wise above a positive threshold, i.e., a specific subset rather than the entire state space as in classical controllability. Graph partitions are exploited to characterize the controllable subspace of the system, from which sufficient conditions are derived to render the system herdable. It is revealed that the quotient graph can be used to infer the herdability of the original graph, wherein criteria of the herdability of quotient graphs are developed based on positive systems. Examples are provided to illustrate the developed topological characterizations.

遗传性是可控制性的变体，并且是将系统状态驱动到状态空间的特定子集的能力的指标。本文描述了带符号加权网络的可控子空间和可遗传性。具体来说，考虑动态签名的领导者跟随者网络，其中网络节点的一小部分（即，领导者）被赋予外源控制输入，其余节点受领导者通过基础网络连接的影响。所考虑的网络允许正向和负向边缘捕获合作和竞争性交互作用，从而生成带符号的图形。出于实际应用的动机，系统状态需要由领导者驱动，以使其在正阈值以上逐个元素地设置，即，一个特定的子集，而不是像传统的可控性那样整个状态空间。利用图形分区来表征系统的可控制子空间，从中可以衍生出足够的条件以使系统可管理。揭示商图可以用来推断原始图的可遗传性，其中商图的可遗传性的标准是基于正系统建立的。提供了一些示例来说明已开发的拓扑特征。其中商图的可遗传性的标准是基于正系统制定的。提供了一些示例来说明已开发的拓扑特征。其中商图的可遗传性标准是基于正系统制定的。提供了一些示例来说明已开发的拓扑特征。