In this paper, we proposed a link density clustering (LDC) method for overlapping community detection based on density peaks. We firstly use an extended cosine link distance metric to reflect the relationship of links. Then we introduce a clustering algorithm with fast search for solving the link clustering (LC) problem by density peaks with box plot strategy to determine the cluster centers automatically. Finally, we acquire both the link communities and the node communities. Our algorithm is compared with other representative algorithms through substantial experiments on real-world networks. The experimental results show that our algorithm consistently outperforms other algorithms in terms of modularity and coverage.