Community detection is a significant and challenging task in network science. Nowadays, plenty of attention has been paid on local methods for community detection. Greedy expanding is a popular and efficient class of local algorithms, which typically starts from some selected central nodes and expands those nodes to obtain provisional communities by optimizing a certain quality function. In this paper, we propose a novel index, called local superiority index (LSI), to identify central nodes. In the process of expansion, we use a fitness function to estimate the quality of provisional communities and ensure that all provisional communities must be weak communities. Evaluation based on the normalized mutual information suggests: (1) LSI is superior to the global maximal degree index and the local maximal degree index on most considered networks; (2) The proposed greedy algorithm based on LSI is better than some state-of-the-art algorithms on most considered networks.