This paper proposes a new event-triggered scheme, in which each agent has various mechanisms to respectively determine when to exchange information with each of its different neighbors. This kind of event-triggered mechanism avoids two constraints in previous mechanisms: (1) simultaneous event detection; (2) synchronous information transmission to all of neighbors. Thus our proposed scheme can be applied into more general situations. In addition, our scheme is able to guarantee the asymptotic consensus and exclude the Zeno behavior. Furthermore, an alternative self-triggered algorithm is presented to thoroughly exclude continuous event detection. Finally, a numerical example is provided to verify the theoretical analysis.