We consider the problem of reaching consensus in communication networks that are modeled by directed graphs. We assume the existence of a message authentication mechanism (such as digital signatures) to verify the integrity of messages. We identify the necessary and sufficient conditions on the directed communication graph for the following problems to be solvable: (i) exact consensus in synchronous systems; and (ii) approximate consensus in asynchronous systems.

翻译：我们考虑在由有向图建模的通信网络中达成共识的问题。假设存在消息认证机制（如数字签名）以验证消息的完整性。我们确定了有向通信图上的必要充分条件，使得以下问题可解：(i) 同步系统中的精确共识；以及 (ii) 异步系统中的近似共识。