This paper addresses the issue that what is the optimal topology for the purely distributed peer-to-peer (P2P).It is usually assumed that unstructured network will form a Power-Law topology.However,a Power-Law structure is not always the best for all the applications.In this paper,the influence of the overlay topology on P2P search is firstly investigated and the precise relation among the distribution of the degree,the query rate and the success rate is attained.Then an optimal distribution model of degrees in terms of the popularity of data items is proposed to guide the overlay construction.Finally,the experimental results show that the proposed topology is effective to improve the success rate for the Random Walks.The fundamental result is significant for the optimal overlay structure and offers a new understanding of the replica deployment in unstructured P2P.