Evolutionary dynamics has been widely used to characterize the evolution and formation of behavioral consensus. Governed by evolutionary dynamics, a network of agents reaches consensus at a selected state of all mutants or all residents. Of special interest is the question of how agents select the global consensus state through local state updating. This paper aims at establishing a link between local state updating and global consensus state selection. We develop a theoretical framework for analyzing the evolutionary dynamics on complex networks and derive some fundamental principles of consensus state selection. More specifically, if the probability that an agent adopts a mutant in one-step updating is monotonically increasing with the fitness of the mutant, monotonically increasing with the mutant set, and submodular or supermodular with the mutant set, then the probability that the network of agents converges to the all-mutant state is monotonically increasing with the fitness of the mutant, monotonically increasing with the initial mutant set, and submodular or supermodular with the initial mutant set, respectively. These results guarantee that the common decision of a group can well reflect the concerns of all group members with evolutionary dynamics. The theoretical results are applied to the well-known death-birth process on networks and validated by numerical simulation.