Distributed Stabilization of Signed Networks via Self-Loop Compensation

The positive semidefiniteness of Laplacian matrices is a critical guarantee of the consensus of unsigned multi-agent networks, which is not valid for signed Laplacian matrices. In this paper, we first analyze the stability of signed networks by introducing a novel graph-theoretic concept called negative cut set, which indicates that the existence of negative edges in signed networks can easily render the network unstable. This fact hinders the applicability of consensus protocol on signed networks. To this end, inspired by the interplay between diagonal dominance and matrix stability, a local state damping mechanism is introduced using self-loop compensation, which is activated only for those agents who are incident to negative edges and can stabilize signed networks in a distributed manner. Quantitative connections between self-loop compensation and the stability of the compensated signed network are established for a tradeoff between compensation efforts and the stability/consensus of signed networks. The optimality of self-loop compensation is discussed. Furthermore, we extend our results to directed signed networks where the symmetry of signed Laplacian is not free. The correlation between the stability of the compensated dynamics obtained by self-loop compensation and eventually positivity is further discussed. Simulation examples are given to demonstrate the theoretical results.