Focusing on the instability of Label Propagation Algorithm (LPA), an advanced label propagation algorithm for community detection was proposed. It introduced the concept of LeaderRank score to quantify the importance of nodes, and chose some core nodes according to the node importance in descending order, then updated labels layer by layer outward centered on every core node respectively, until no node changed its label any more. Thus the instability caused by the random ranking of nodes was solved. Compared with several existing label propagation algorithms on LFR benchmark networks and real networks, both of the Normalized Mutual Information (NMI) and modularity of community detection result of the proposed algorithm were higher. The theoretical analysis and experimental results demonstrate that the proposed algorithm not only improves the stability effectively, but also increases the accuracy.