This thesis targets at providing an algebraic method to indicate network behaviors. Furthermore, for a signed-average consensus problem of the system behaviors, event-triggering signed-average algorithms are designed to reduce the communication overheads. In Chapter 1, the background is introduced, and the problem is formulated. In Chapter 2, notations and basics of graph theory are presented. It is known that the terminal value of the system state is determined by the initial state, left eigenvector and right eigenvector associated with zero eigenvalue of the Laplacian matrix. Since there is no mathematical expression of right eigenvector, in Chapter 3, mathematical expression of right eigenvector is given. In Chapter 4, algebraic trait for structurally balanced property of a node is proposed. In Chapter 5, a method for characterization of collective behaviors under directed signed networks is developed. In Chapter 6, dynamic event-triggering signed-average algorithms are proposed and proved for the purpose of relieving the communication burden between agents. Chapter 7 summarizes the thesis and gives future directions.