Evolutionary dynamics of binary opinions with decision errors on social networks

In recent years, the evolutionary dynamics of opinions on social networks have attracted considerable attention. Evolutionary game theory has been employed to explore the mechanisms underlying opinion evolution. However, most prior works usually ignore the stochastic effects, such as decision-making errors, and thus far few studies have investigated how to promote the emergence of opinions within this context. In this study, we develop a game-theoretic model to investigate the dynamics of opinion evolution with decision errors on social networks. Inspired by previous research, we assume that individuals make decisions on opinion choices according to the global public knowledge and local interaction feedback. Using the method of coalescing random walks, we derive the average fraction of two opinions and determine the mathematical condition for the emergence of one opinion in the limit of rare errors. Our findings show that this critical condition depends on the network topology, the basic score of each opinion, and feedback scores. It is noteworthy that our theoretical analysis can be applied to any network structure. Moreover, we perform simulations on different types of networks, including regular, scale-free, and small-world networks, to validate our theoretical results. All simulation outcomes confirm the corresponding theoretical predictions. We further show that the emergence condition for one opinion, established under rare error limit, is robust across all levels of error. Specifically, when one opinion is favored over the other in the limit of rare errors, it remains favored for all error rate values, although its average fraction decreases as the error rate increases. Conversely, when the opinion is disfavored in the limit of rare errors, the average fraction increases with error rate, yet it never becomes favored even in the extreme case of high error rates.