Characterization of Collective Behaviors for Directed Signed Networks

TL;DR

This paper investigates the mathematical properties of directed signed networks, focusing on the right eigenvector of the Laplacian matrix, and introduces criteria to characterize various collective behaviors.

Contribution

It provides a novel mathematical expression for the right eigenvector and establishes algebraic criteria for key collective behaviors in directed signed networks.

Findings

Right eigenvector is crucial for understanding collective behaviors.

Algebraic criteria for bipartite consensus and containment tracking.

Simulation confirms the theoretical results.

Abstract

This paper targets at exploring how to characterize collective behaviors of directed signed networks. The right eigenvector of the Laplacian matrix associated with zero eigenvalue is further investigated and its mathematical expression is proposed. It is shown that the right eigenvector plays an important role in determining the collective behaviors of directed signed networks. Furthermore, algebraic criteria are introduced for collective behaviors of directed signed networks, such as bipartite consensus, interval bipartite consensus and bipartite containment tracking. In addition, a simulation example is given to the correctness of our developed theoretical results.