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Community structure is one of the important topological characteristics of many complex networks. Detecting communities from networks has been intensively investigated in recent years. In most previous methods for community detection, the overlapping property of communities, which exists common in many real-world networks, is ignored. By combining commute-time kernel based distance measure and fuzzy affinity propagation, we present a new community detection algorithm CDKFAP for overlapping communities. Based on a new proposed index measures the fuzziness of nodes, the algorithm can rank and extract overlapping nodes of communities. The applications to computer-generated networks and real-world networks demonstrate the effectiveness of our algorithm.