Distributed Stabilization of Signed Networks via Self-loop Compensation

This paper examines the stability and distributed stabilization of signed multi-agent networks. Here, positive semidefiniteness is not inherent for signed Laplacians, which renders the stability and consensus of this category of networks intricate. First, we examine the stability of signed networks by introducing a novel graph-theoretic objective negative cut set, which implies that manipulating negative edge weights cannot change a unstable network into a stable one. Then, inspired by the diagonal dominance and stability of matrices, a local state damping mechanism is introduced using self-loop compensation. The self-loop compensation is only active for those agents who are incident to negative edges and can stabilize signed networks in a fully 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 network stability. Necessary and/or sufficient conditions for predictable cluster consensus of compensated signed networks are provided. 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. Novel insights into the stability of multi-agent systems on signed networks in terms of self-loop compensation are offered. Simulation examples are provided to demonstrate the theoretical results.