## 3 Relative Importance Measure Based on Distance Distribution and Multi-index Fusion
### Distance Distribution  
Distance distribution in complex networks is usually represented by the shortest path distribution between nodes. The node distance distribution in the network mainly considers the number of nodes with different shortest path lengths to the current node; thus, it can intuitively obtain the shortest path information of nodes in the network and reflect many important topological information in the network [38]. The distance distribution of each node \(v_{i}\) in the complex network can be represented as \(P_{i}=\{p_{i}(j)\}\); the calculation formula of \(p_{i}(j)\) is  
\[p_{i}(j)=\frac{N_{i}(j)}{n} \tag{1}\]  
where \(j\) represents the shortest path length with a value in the range of \(0\leq j\leq D(G)\). \(D(G)\) refers to the diameter of the network \(G\), and its value is the maximum distance between any two nodes in the network \(G\). \(N_{i}(j)\) represents the number of nodes with \(j\) of the shortest path length to node \(v_{i}\) in the network \(G\); \(n\) represents the number of nodes in the network \(G\). Take a network \(G_{example}\) as an example. The detailed calculation process of node distance distribution in \(G_{example}\) is introduced as follows. In Figure 1, the red nodes are the nodes in the current study while the yellow, light green, blue, and pink nodes represent the nodes that can be reached by taking one, two, three, and four steps consecutively, starting from the nodes studied currently. The number of nodes \(n\) in the sample network \(G_{example}\) is 20, and the diameter \(D(G)\) is 7. The distance distribution dimension mainly depends on the diameter of the  
Figure 1: The topology of the example network. network. Its value range is from 0 to \(D(G)\), with a total of \(D(G)+1\) cases. Therefore, the distance distribution dimension \(d\) of each node in \(G_{example}\) is 8. Provided that node 0 is used as the starting research node, the set of nodes \(N(i)=\left\{N_{i}(j)|0\leq j\leq D(G)\right\}\) that can be reached by node 0 in turn can be obtained by calculating the shortest path length between this node and other nodes in \(G_{example}\), that is, according to Formula (1) and \(N(i)\) obtained through the above analysis, the distance distribution \(P_{0}\) of node 0 can be obtained as: \(P_{0}=\left\{0.05,0.15,0.30,0.35,0.15,0,0,0\right\}\). Similarly, the distance distribution of any node in the sample network \(G_{example}\) can be obtained. For a network \(G\) with \(n\) nodes, if the distance distribution of \(n\) nodes is known \(P=\left\{P_{0},P_{1},\cdots,P_{n-1}\right\}\), then much important topology information in the network \(G\) can be obtained on the basis of the distance distribution of nodes. For example, the degree \(k_{i}\) of any node \(v_{i}\) in \(G\), the average degree \(\overline{k}\) of \(G\), the average path length \(APL\) of \(G\), and the closeness centrality \(CC_{i}\) corresponding to node \(v_{i}\). For the network \(G\) with \(n\) nodes, a distance distribution matrix \(X=[x_{ij}]\in R^{n\times d}\) is established on the basis of the distance distribution information of all nodes in \(G\); \(n\) refers to the total number of nodes in the network \(G\). 1. Degree \(k_{i}\) of node \(v_{i}\) \[k_{i}=nx_{i1}\] (2)
2. Average degree \(\overline{k}\) of network \(G\) \[\overline{k}=\frac{1}{n}\sum_{i=0}^{n-1}nx_{i1}\] (3)
3. Average path length \(APL\) of network \(G\) \[APL=\frac{2}{n(n-1)}\sum_{i=0}^{n-1}\sum_{j=1}^{D(G)}j\times nx_{ij}\] (4)
4. Closeness centrality \(CC_{i}\) of node \(v_{i}\) \[CC_{i}=\frac{n}{\sum\limits_{j=1}^{D(G)}j\times nx_{ij}}\] (5)  
The analysis indicates that the distance distribution of nodes contains abundant network topology information.