This paper investigates the global behavior of controlled dynamical agents in directed networks. The agents are Lyapunov stable, are distributed in a line, and communicate through a directed network. The communication topology of the network is characterized by a directed graph and the control protocol is designed in simple linear decentralized feedback law. We study the different conditions under which agents will achieve aggregation, and critical and divergent trajectories, respectively. Our investigation on the dynamical agent system under network is extended to the time-delay network case. Furthermore, we study the case with two pre-specified virtual leaders in the system. Numerical simulations are given and demonstrate that our theoretical results are effective. The paper investigates the global behavior of controlled dynamical agents in directed networks. The agents are Lyapunov stable, are distributed in a line, and communicate through a directed network. The communication topology of the network is characterized by a directed graph and the control protocol is designed in simple linear decentralized feedback law. We study the different conditions under which agents will achieve aggregation, and critical and divergent trajectories, respectively. Our investigation on the dynamical agent system under network is extended to the time-delay network case. study the case with two pre-specified virtual leaders in the system. Numerical simulations are given and demonstrate that our theoretical results are effective.