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In overlay networks, the interplay between network structure and dynamics remains largely unexplored. In this paper, we study dynamic coevolution between individual rational strategies (cooperative or defect) and the overlay network structure, that is, the interaction between peer's local rational behaviors and the emergence of the whole network structure. We propose an evolutionary game theory (EGT)-based overlay topology evolution scheme to drive a given overlay into the small-world structure (high global network efficiency and average clustering coefficient). Our contributions are the following threefold: From the viewpoint of peers' local interactions, we explicitly consider the peer's rational behavior and introduce a link-formation game to characterize the social dilemma of forming links in an overlay network. Furthermore, in the evolutionary link-formation phase, we adopt a simple economic process: Each peer keeps one link to a cooperative neighbor in its neighborhood, which can slightly speed up the convergence of cooperation and increase network efficiency; from the viewpoint of the whole network structure, our simulation results show that the EGT-based scheme can drive an arbitrary overlay network into a fully cooperative and efficient small-world structure. Moreover, we compare our scheme with a search-based economic model of network formation and illustrate that our scheme can achieve the experimental and analytical results in the latter model. In addition, we also graphically illustrate the final overlay network structure; finally, based on the group selection model and evolutionary set theory, we theoretically obtain the approximate threshold of cost and draw the conclusion that the small value of the average degree and the large number of the total peers in an overlay network facilitate the evolution of cooperation.