Mixed evolutionary strategies imply coexisting opinions on networks

An evolutionary battle-of-the-sexes game is proposed to model the opinion formation on networks. The individuals of a network are partitioned into different classes according to their unaltered opinion preferences, and their factual opinions are considered as the evolutionary strategies, which are updated with the birth-death or death-birth rules to imitate the process of opinion formation. The individuals finally reach a consensus in the dominate opinion or fall into (quasi)stationary fractions of coexisting mixed opinions, presenting a phase transition at the critical modularity of the multiclass individuals' partitions on networks. The stability analysis on the coexistence of mixed strategies among multiclass individuals is given, and the analytical predictions agree well with the numerical simulations, indicating that the individuals of a community (or modular) structured network are prone to form coexisting opinions, and the coexistence of mixed evolutionary strategies implies the modularity of networks.