Non-trivial consensus control on directed signed networks

This paper tackles the challenge of achieving non-trivial consensus control on directed signed networks. Although (interval) bipartite consensus and trivial consensus are convergence states on signed networks that have been extensively studied, non-trivial consensus represents a novel convergence state that has not been fully explored yet. In order to drive group states on a directed signed network to reach non-trivial consensus, a control technique based on linear system theory and our graph decomposition framework is proposed. Our technique operates under mild connectivity conditions and does not impose restrictions on whether the network is structurally balanced or unbalanced. Moreover, the consensus value can be preset arbitrarily as required. Based on matrix perturbation theory, the impact of the antagonistic interactions on the non-trivial consensus speed in signed networks with external control is further studied. Numerical simulations validate the effectiveness of our approach and theoretical framework. The work in this paper demonstrates that a group with coopetitive (cooperative-competitive) interactions under control can achieve consensus, which was previously considered exclusive to fully cooperative groups.