Positive semidefinite is not an inherent property of signed Laplacians, which renders the stability and consensus of multi-agent system on undirected signed networks intricate. Inspired by the correlation between diagonal dominance and spectrum of signed Laplacians, this paper proposes a self-loop compensation mechanism in the design of interaction protocol amongst agents and examines the stability/consensus of the compensated signed networks. It turns out that self-loop compensation acts as exerting a virtual leader on these agents that are incident to negative edges, steering whom towards origin. Analytical connections between self-loop compensation and the collective behavior of the compensated signed network are established. Necessary and/or sufficient conditions for predictable cluster consensus of signed networks via self-loop compensation 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. Simulation examples are provided to demonstrate the theoretical results.

翻译：积极的半限值不是已签字的Laplacians的固有属性,它使多试剂系统在未经签署的网络上的稳定性和共识错综复杂。由于对角支配地位和已签字的Laplacians的频谱之间的相互关系,本文件提议在设计代理人之间互动协议时采用自我循环补偿机制,并审查已获得补偿的已签字网络的稳定性/共识。结果显示,自我循环补偿对发生负边缘的这些代理商起到了虚拟领导作用,引导其走向源头。在自我循环补偿和已补偿的已签字网络的集体行为之间建立了分析联系。提供了通过自我循环补偿使已签字的网络的可预见集群共识的必要条件和(或)充分的条件。讨论了自我循环补偿的最佳性。此外,我们把结果推广到没有免费签名的Laplacian的对称的定向网络。我们提供了模拟实例,以证明理论结果。