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Opinion Dynamic with agents immigration

We propose a strategy for achieving maximum cooperation in evolutionary games on complex networks. Each individual is assigned a weight that is proportional to the power of its degree, where the exponent alpha is an adjustable parameter that controls the level of diversity among individuals in the network. During the evolution, every individual chooses one of its neighbors as a reference with a probability proportional to the weight of the neighbor, and updates its strategy depending on their payoff difference. It is found that there exists an optimal value of alpha, for which the level of cooperation reaches maximum. This phenomenon indicates that, although high-degree individuals play a prominent role in maintaining the cooperation, too strong influences from the hubs may counterintuitively inhibit the diffusion of cooperation. We provide a physical theory, aided by numerical computations, to explain the emergence of the optimal cooperation. Other pertinent quantities such as the payoff, the cooperator density as a function of the degree, and the payoff distribution, are also investigated. Our results suggest that, in order to achieve strong cooperation on a complex network, individuals should learn more frequently from neighbors with higher degrees, but only to certain extent.