## 2 Materials and Methods  
### The Proposed Method  
Considering an undirected and unweighted network \(G(N,E)\) composed of \(N\) nodes and \(E\) edges, the network is represented by \(A(a_{ij})_{N*N}\), where \(a_{ij}=1\) represents node \(i\) and node \(j\) are connected; otherwise, \(a_{ij}=0\). From the perspective of the network robustness and invulnerability, when a node does not work due to deliberate attack and the node is on the shortest path between a pair of its neighbor nodes, the connection of the neighbor node pair would be affected. As shown in Figure 1a, node \(c\) builds a bridge between node \(a\) and node \(b\). Therefore, removing node \(c\) will result in disconnection between nodes \(a\) and \(b\). In Figure 1b, nodes \(a\), \(b\), \(c\) are connected to each other, so node \(c\) no longer acts as an intermediary. In this situation, even if node \(c\) is removed, nodes \(a\) and \(b\) can still maintain effective connection. In addition, as shown in Figure 1c, nodes \(a\) and \(b\) can communicate with each other through \(c\), but other short communication paths such as \(a\leftrightarrow d\leftrightarrow b\) also exist in the neighbourhood. In this case, the bridging function of node \(c\) is weakened. Furthermore, we study the situation when there is a connection between the node's one-hop neighbors and two-hops neighbors. Figure 1d-f show several ways of direct or indirect contact between nodes \(a\) and \(b\). In Figure 1d, when there is only one shortest path between nodes \(a\) and \(b\), the bridging feature of node \(c\) is obvious. In Figure 1e and Figure 1f, when the shortest path length between nodes \(a\) and \(b\) is less than 3, even if node \(c\) is removed, nodes \(a\) and \(b\) still keep effective contact for there is absolutely at least one another path between nodes \(a\) and \(b\) in this case. Similar to Figure 1c, Figure 1f shows when there exist two shortest paths \(a\leftrightarrow c\leftrightarrow d\leftrightarrow b\) and \(a\leftrightarrow e\leftrightarrow f\leftrightarrow b\) of which the length is 3, the bridging function of node \(c\) is also weakened. In this paper, we just take into account the target node's neighbors within two-hops. Based on the above observations, we propose a novel method by taking into account the bridging feature of nodes to identify node importance in complex networks. For a node in the network, the greater the number of node pairs whose shortest paths pass through the target node, and the less  
Figure 1: Several different connections between neighbor nodes \(a\) and \(b\) of node \(c\). (**a**) node \(a\) and node \(b\) get connected by node \(c\); (**b**) nodes \(a\), \(b\), \(c\) get connected with each other; (**c**) node \(a\) and node \(b\) get connected by node \(c\) or node \(d\); (**d**) node \(a\) and node \(b\) get connected by node \(c\) and node \(d\); (**e**) node \(a\) and node \(b\) get connected directly; (**f**) node \(a\) and node \(b\) get connected by node \(d\); (**g**) node \(a\) and node \(b\) get connected by nodes \(c\), \(d\) or nodes \(e\), \(f\). the number of shortest paths between its neighbor node pairs, the more significant the bridge function and structure importance the node has. The evaluation value of the importance of node \(i\) calculated by the algorithm proposed in this paper can be expressed as  
\[NBF(i)=\sum_{m,n\in\Gamma_{1}(i)}\frac{1}{2P_{mn}^{(2)}}+\sum_{x\in\Gamma_{1}(i ),y\in\Gamma_{2}(i)}\frac{1}{3P_{xy}^{(3)}}\ \ \ \ m\
eq n,x\
eq y, \tag{1}\]  
where \(P_{mn}^{(2)}\) represents the number of the paths with the shortest path length two-hops between nodes \(m\) and \(n\), \(P_{xy}^{(3)}\) represents the number of the paths with the shortest path length three-hops between nodes \(x\) and \(y\), \(\Gamma_{1}(i)\) and \(\Gamma_{2}(i)\) separately represent the sets of neighbors one-hop and two-hops away from node \(i\).