Algebraic criteria for structure identification and behaviour analysis of signed networks

Characterization of Collective Behaviors for Directed Signed Networks

“…From ( 5), we know that the right eigenvector ξ and the left eigenvector η play an important role in investigating the collective behaviors of the system (4) when the quasi-strongly connected signed digraph G satisfies the condition C1) or C2). Based on [30], the mathematical expression of the left eigenvector η is given by…”

“…Remark 1. Theorem 1 provides an approach to calculating the right eigenvector ξ by exploiting the Laplacian matrix L. In comparison with the reference [30], the mathematical expression for the right eigenvector ξ is developed, which is convenient to explore the collective behaviors of signed networks (see Section V for more details).…”

“…From [6], [8], [30], the terminal value of all agents is determined by initial states, left eigenvector and right eigenvector that are corresponding to the zero eigenvalue of Laplacian matrix. The mathematical expression for the left eigenvector has been provided for the unsigned digraph and the signed digraph in [8] and [30], respectively. When considering the strongly connected and quasi-strongly connected unsigned digraphs, all entries of the right eigenvector are equal.…”