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 apply the 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 greedy algorithm based on LSI is better than the classical fast algorithm on most considered networks.