Recently, complex networks have attracted considerable research attention. Community detection is an important problem in the field of complex networks and is useful in a variety of applications such as information propagation, link prediction, recommendation, and marketing. In this study, we focus on discovering overlapping community structures by using link partitions. We propose a Latent Dirichlet Allocation (LDA)-Based Link Partition (LBLP) method, which can find communities with an adjustable range of overlapping. This method employs the LDA model to detect link partitions, which can calculate the community belonging factor for each link. On the basis of this factor, link partitions with bridge links can be found efficiently. We validate the effectiveness of the proposed solution by using both real-world and synthesized networks. The experimental results demonstrate that the approach can find a meaningful and relevant link community structure.