Recently, the emergence of cooperation has become a central topic in the evolutionary game field, and coevolution of game dynamics and network topology structure can give us a fresh viewpoint of how the network evolves and cooperation arises. In this paper, we show in detail a picture of the co-evolutionary behaviors between the microscopic structure of the network and cooperation promotion in the public goods game (PGG). Based on a mechanism named after evolutionary preferential attachment (EPA), in which the growth of the network depends on the outcome of PGG dynamics, we explore the structural properties of networks and cooperative behaviors taking place on the networks created by EPA rules. Extensive simulation results indicate that the structure of the resulting networks displays a transition from homogeneous to heterogeneous properties as the selection strength increases, and the cooperative behaviors have a non-trivial state in which cooperators and defectors can simultaneously occupy the hub nodes in the network. Current results are of interest for us to further understand the cooperation persistence and structure evolution in many natural, social and economical systems.
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