Community structure is one of the most important topological characteristics in the complex network, being a hot research area in different fields. A novel community detection algorithm is proposed based on edges rank and modularity optimization. Local graph is sparsificated and edges are ranked according to the similarity. Therefore, a method called the fast rank based community detection (FRCD) by maximizing modularity and fast mergement of edges is achieved. Meanwhile the method is also extended to dynamic and real time community detection on the basis of initial community structure, and a fast and robust dynamic community detection algorithm called the incremental dynamic community detection (IDCD) is presented. Theoretical analysis exhibit that FRCD has linear complexity for network edges. Experimental results in real world and artificial networks demonstrate the high accuracy and good erformance of the algorithm on static community detection and tracking dynamic structure of networks.