Ranking node importance is of great significance for studying the robustness and vulnerability of complex network. Over the recent years, various centrality indices such as degree, semilocal, K-shell, betweenness and closeness centrality have been employed to measure node importance in the network. Among them, some well-known global measures such as betweenness centrality and closeness centrality can achieve generally higher accuracy in ranking nodes, while their computation complexity is relatively high, and also the global information is not readily available in a large-scaled network. In this paper, we propose a new local metric which only needs to obtain the neighborhood information within two hops of the node to rank node importance. Firstly, we calculate the similarity of node neighbors by quantifying the overlap of their topological structures with Jaccard index; secondly, the similarity between pairs of neighbor nodes is calculated synthetically, and the redundancy of the local link of nodes is obtained. Finally, by reducing the influence of densely local links on ranking node importance, a new local index named LLS that considers both neighborhood similarity and node degree is proposed. To check the effectiveness of the proposed method of ranking node importance, we carry out it on six real world networks and one artificial small-world network by static attacks and dynamic attacks. In the static attack mode, the ranking value of each node is the same as that in the original network. In the dynamic attack mode, once the nodes are removed, the centrality of each node needs recalculating. The relative size of the giant component and the network efficiency are used for network connectivity assessment during the attack. A faster decrease in the size of the giant component and a faster decay of network efficiency indicate a more effective attack strategy. By comparing the decline rates of these two indices to evaluate the connectedness of all networks, we find that the proposed method is more efficient than traditional local metrics such as degree centrality, semilocal centrality, K-shell decomposition method, no matter whether it is in the static or dynamic manner. And for a certain ranking method, the results of the dynamic attack are always better than those of the static attack. This work can shed some light on how the local densely connections affect the node centrality in maintaining network robustness.