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Characterizing Herdability of Signed Networks via Graph Walks

Herdability, as a variant of controllability, indicates the capability of driving system states to a specific subset of the state space. This paper investigates the topological characterizations of the herdability of signed networked systems. Specifically, a dynamic signed leader-follower network is considered, where a small subset of the network nodes (i.e., the leaders) are endowed with exogenous control input and the rest nodes are influenced by the leaders via the underlying network connectivity. The considered network allows positive and negative edges to capture cooperative and competitive interactions, thus resulting a signed graph. Motivated by the practical applications, the systems states are required to be driven by the leaders to be element-wise larger than a positive threshold, i.e., a specific subset rather than the entire state space as in the classical controllability. To study the herdability of signed leader-follower networks, this work extends the results of [1] and characterizes topological structures of herdable networks. In particular, graph walk is exploited to develop sufficient conditions ensuring the herdability of signed networks via 1-walks and 2-walks. These results are then extended to acyclic graphs with multiple walks. Examples are provided to illustrate the developed topological characterizations.