Invariant Kernel-Based Synchronization for Certain Edge-Colored Networks

In this article, the synchronization issue of certain edge-colored networks is investigated. We start with a linear subspace determined by the colored edges and show that the invariant kernel of this linear subspace is precisely consists of all the synchronous states. Moreover, this invariant kernel is characterized by a cluster of algebraic equations. Especially, for network described by a polynomial vector field, due to the property of Noetherian rings, this invariant kernel can be determined by a finite number of algebraic equations. Furthermore, the equivalence between network synchronization and the asymptotic behavior of the aforementioned invariant kernel are proved. Based on the colored edges and the invariant kennel, we decompose the original network twice, arriving at a three-layer network: the first two layers, referred to as the external part, can only synchronize to an equilibrium point, while the third layer, known as the internal part, possesses the same colored edges. For this three-layer network, we construct two Lyapunov-type functions for the external part and the internal part, respectively, to establish the synchronization criteria. In particular, our criteria involve linear matrix inequalities and polynomial inequalities of smaller scale, which can be solved by the existing semi-definite programming tools. Finally, this article provides three examples to illustrate the effectiveness and advantages of the theoretical results presented.