Leader Group Selection for Herdability of Structurally Balanced Signed Networks

This work studies the herdability of networked systems. As an extension of classical controllability, herdability represents the ability of a system to drive its states to a specific subset in the state space. Particularly, a leader-follower signed network is considered, where leader nodes endowed with external controls are able to influence the states of the follower nodes. Weighted positive and negative edges are allowed to capture cooperative and competitive interactions among nodes. To enable network herdability, leader group selection is investigated in this work, i.e., identifying a small subset of nodes as leaders such that the resulting leader-follower network is herdable by the selected leaders. Focusing on structurally balanced signed networks, graph walks are leveraged to facilitate leader selection for network herdability. The cases of selecting leaders from the same partitioned set and different sets are considered, which are then extended to a special class of weakly balanced signed graphs. Examples are provided to illustrate the developed leader selection approaches.