Self-loop Compensation in Signed Networks

Stability of multi-agent systems on signed networks is intricate. To some extent, this is due to the associated signed Laplacian may lose its diagonal dominance property. This paper proposes a distributed self-loop compensation approach to rebuild the diagonal dominance of signed Laplacian, and subsequently, examine the stability and cluster consensus of the resultant compensated signed networks. Quantitative connections between the magnitude of self-loop compensation and the steady-state of the compensated signed network are analytically established, depending on the structural balance of signed networks. Some necessary and sufficient conditions for cluster consensus of compensated signed networks are provided as well as the explicit characterization of their steady-states. It turns out that structurally imbalanced networks need less self-loop compensation to be stable compared with the structurally balanced ones. The optimality of compensation magnitude is discussed. Both undirected and directed signed networks are examined. Simulation examples are provided to demonstrate the theoretical results.