## 1 Introduction  
In recent years, network science research has attracted a great amount of attention from researchers in different fields including physics, mathematics, chemistry, medical science, biology, computer science, sociology and so on [1; 2; 3; 4; 5; 6; 7; 8; 9]. In particular, the vulnerability of complex networks is one of the most important directions due to its considerable effect on network cascade failure caused by random failure and deliberate attack [10; 11; 12; 13; 14; 15; 16; 17; 18]. Random failure can be regarded as a simple abstraction of successive errors in complex networks by destroying nodes or edges with uniform probability. Deliberate attack means that the network nodes or edges are attacked according to their importance in descending order under the premise of mastering global information of the network [19; 20; 21; 22; 23]. On 14 August 2003, a large-scale power cascade failure in the northeastern United States and eastern Canada caused global concern. Similarly, in early 2008, due to some damage to major transmission lines and key towers, severe ice sheet disasters in southern China caused large-scale blackouts. These examples indicate that important node failure may result in great damage on the whole network. One of the crucial questions in protecting networks from cascading failures is to design an efficient method to identify important nodes and take protective strategies. The important nodes in the network refer to a tiny fraction of special nodes that have great influence on the structure and function of the network [24; 25]. The more important the node, the greater influence the node failure causes. For example, researchers have found that, in networks such as email networks, the Internet, protein networks, food chain networks and peer-to-peer (P2P) networks, removing the nodes with the largest degree will cause the network to become very vulnerable. Many centrality measures are proposed to evaluate node importance in complex networks, including degree [26], closeness [27], betweenness [28; 29] and so on. Degree is a very simple and classic one which identifies the node importance by just calculating the number of nodes connected to the target node. However, the precision of its computation is not high enough in most applications and nodes with the same degree may play different important roles in a complex network. Betweenness and closeness collect global information of the network to compute the importance of nodes. The two centralities need to calculate the shortest path between any pair of nodes in the network. The computational complexity is too high and thus they are not suitable for large-scale networks. In [30], ego network betweenness was proposed to compute the betweenness centrality of ego in an ego network. Its scalability and ease of implementation makes it a good alternative for betweenness in large networks. A lot of metrics on the computation of complex network were designed for large complex networks later [31; 32; 33; 34]. Additionally, there are many other studies on this issue. [PERSON] et al. [35] proposed a novel network decomposition method to identify the most important nodes by continuously removing the peripheral nodes, and the simulation result showed that it is positive and efficient. [PERSON] et al. [36] considered that the importance of nodes is related to the degree of nodes and their neighbors, and proposed a new algorithm WL to rank the importance of network nodes. [PERSON] et al. [37] found that the number of connected subgraphs between neighboring nodes is the determining factor of node importance. [PERSON] et al. [38] proposed an entropic metric, Entropy Variation, defining the node importance as the variation of network entropy before and after its removal. By quantifying the structural similarity between nodes' neighborhoods, [PERSON] et al. [39] proposed a node importance ranking algorithm which only needs to obtain the neighborhood information within two hops of the node.