## 3 Relative Importance Measure Based on Distance Distribution and Multi-index Fusion
### Relative Importance Score Based on Multi-index Fusion  
To fully integrate the advantages of cosine similarity, Euclidean distance, and relative entropy in the direction, distance, and distribution dimensions, this study performs the multi-index fusion of cosine similarity, Euclidean distance, and relative entropy and calculates the weights of the different indexes by using the entropy weight method [42] to maximize the advantages of the different indexes. The entropy weight method is an objective weighting method that is widely used and often depends on the discreteness of data. It mainly weighs different indexes according to the amount of information of different evaluation indexes. Cosine similarity, Euclidean distance, and relative entropy are mainly considered in this work. Thus, weight allocation becomes necessary. A relative importance score matrix, \(Z=[z_{tg}]\in R^{|T|\times 3}\), is defined herein. \[z_{t1}=\frac{\sum\limits_{r=1}^{|R|}C_{tr}}{|R|},t=1,2,\cdots,|T| \tag{14}\]  
\[z_{t2}=\frac{\sum\limits_{r=1}^{|R|}E_{tr}}{|R|},t=1,2,\cdots,|T| \tag{15}\]  
\[z_{t3}=\frac{\sum\limits_{r=1}^{|R|}R_{tr}}{|R|},t=1,2,\cdots,|T| \tag{16}\]  
where \(z_{t1}\), \(z_{t2}\) and \(z_{t3}\) represent the arithmetic mean of cosine similarity, Euclidean distance and relative entropy between the \(t-th\) node in the target node set \(T\) and all nodes in the known important node set \(R\) respectively. \(|T|\) refers to the number of nodes in the target node set \(T\), \(|R|\) refers to the number of nodes in the known important node set \(R\), and \(g\) refers to the number of indexes, \(g=1,2,3\). Based on the relative importance score matrix, the entropy corresponding to cosine similarity, Euclidean distance, and relative entropy can be further calculated. The formulas are as follows:  
\[e_{g}=-\frac{1}{\ln|T|}\sum\limits_{t=1}^{|T|}p_{tg}\ln(p_{tg}) \tag{17}\]\[p_{tg}=\frac{z_{tg}}{\sum\limits_{t=1}^{|T|}z_{tg}} \tag{18}\]  
where \(e_{g}\) represents the entropy of the index in the \(g\) column and \(p_{tg}\) represents the proportion of the index in the \(g\) column of the \(t-th\) node in the target node set \(T\) in this column of indexes. After the entropies of different indexes are obtained, the weight coefficient \(\omega_{g}\) of each index can be further calculated. The weights corresponding to different indicators determine the relative importance scores of the target nodes in the network. The specific formula is:  
\[\omega_{g}=\frac{1-e_{g}}{\sum\limits_{g=1}^{3}{(1-e_{g})}} \tag{19}\]  
where \(1-e_{g}\) refers to information entropy redundancy. At the same time, \(\omega_{g}\) should meet the restrictive conditions of \(\sum\omega_{g}=1,g=1,2,3\). Therefore, the relative importance score of \(t-th\) node in the target node set \(T\) can be expressed as:  
\[s_{t}=\omega_{1}z_{t1}+\omega_{2}z_{t2}+\omega_{3}z_{t3} \tag{20}\]  
Finally, the relative importance scores of all nodes in the target node set \(T\) are sorted in descending order, and the nodes with high scores can be regarded as relatively important nodes. The calculation of the relative importance scores of the nodes in a network by using the DDMF method consists of the following steps:  
First, on the basis of the information of the shortest distance between nodes in the network \(G\), the distance distribution vectors of all nodes in the network \(G\) are calculated, along with all the nodes of the known important node set \(R\) and target node set \(T\). Second, the differences in the distance distribution of the nodes between the known important node set \(R\) and the target node set \(T\) are determined. The cosine similarity, Euclidean distance, and relative entropy of the distance distribution of the two node sets are then calculated and normalized. Finally, multi-index fusion is made for cosine similarity, Euclidean distance, and relative entropy, and the weights corresponding to different indexes are calculated using the entropy weight method.