## 3 Relative Importance Measure Based on Distance Distribution and Multi-index Fusion
### Introduction to Indicators  
Cosine similarity is a measurement method for the difference between two individuals and involves calculating the cosine value of the angle between two vectors in the vector space, mainly focusing on the measurement of the difference between two individuals from the dimension of direction. The basic idea is to covert the individual's index data into the vector space and then measure the difference between individuals by comparing the cosine values of the angle in the inner product space between different individual vectors [39]. In a \(M\)-dimensional space, assuming that \(A\) and \(B\) are \(M\)-dimensional vectors, namely \(A=[a_{1},a_{2},\cdots,a_{M}]\), \(B=[b_{1},b_{2},\cdots,b_{M}]\), then the cosine similarity \(\mathit{Cos}_{AB}\) can be expressed as:  
\[\mathit{Cos}_{AB}=\frac{\sum\limits_{i=1}^{M}{(A_{i}\times B_{i})}}{\sqrt{ \sum\limits_{i=1}^{M}{(A_{i})^{2}}}\times\sqrt{\sum\limits_{i=1}^{M}{(B_{i})^ {2}}}}=\frac{A\cdot B}{|A|\times|B|} \tag{6}\]  
where the value range of \(\mathit{Cos}_{AB}\) is [-1, 1], that is, \(\mathit{Cos}_{AB}\in[-1,1]\). In this work, the distance distribution of nodes in the network is converted into vector form; that is, in the network \(G\) with \(n\) nodes, the vectors of distance distribution of any node \(x\) and node \(y\) can be expressed as \(P_{x}\) and \(P_{y}\), respectively. Then, the formula for the cosine similarity between nodes can be represented as:  
\[D_{\mathit{Cos}}\big{(}P_{x}\|P_{y}\big{)}=\frac{P_{x}\cdot P_{y}}{|P_{x}| \times\big{|}P_{y}|} \tag{7}\]  
\[C_{xy}=\frac{1+D_{\mathit{Cos}}\big{(}P_{x}\|P_{y}\big{)}}{2} \tag{8}\]  
Normalization is performed for the cosine similarity between nodes \(D_{\mathit{Cos}}(P_{x}\|P_{y})\) based on Equation (8), and \(C_{xy}\) is obtained. Among them, \(C_{xy}\in[0,1]\). Euclidean distance, also called Euclidean metric, originates from the distance formula between two points in Euclidean geometry [40]. It is mainly used to measure the real distance between two points in \(M\)-dimension space; that is, focusing on the numerical difference between individuals. In a \(M\)-dimensional space, assuming that \(A\) and \(B\) are \(M\)-dimensional vectors, namely \(A=[a_{1},a_{2},\cdots,a_{M}]\), \(B=[b_{1},b_{2},\cdots,b_{M}]\), then the Euclidean distance \(\mathit{Euc}_{AB}\) can be expressed as:  
\[\mathit{Euc}_{AB}=\sqrt{\sum\limits_{i=1}^{M}{(a_{i}-b_{i})^{2}}} \tag{9}\]  
Similarly, the distance distribution of nodes in the network \(G\) is first converted into vector form. Then, the Euclidean distance between any node \(x\) and node \(y\) can be represented as \(\mathit{Euc}_{xy}\):  
\[E_{xy}=\frac{\mathit{Euc}_{xy}}{\mathit{Euc}_{\max}} \tag{10}\]  
Normalization is performed for Euclidean distance \(\mathit{Euc}_{xy}\) between node \(x\) and node \(y\) based on Equation (10), and \(E_{xy}\) is obtained. Among them, \(E_{xy}\in[0,1]\). From information theory, relative entropy, also called KL divergence or information divergence, is generally used to measure the difference between two probability distributions [41]. In this work, the difference in the distance distribution between different nodes in the network is calculated from the dimension of distribution to effectively find relatively important nodes in the network. For the network \(G\) with \(n\) nodes  
In addition, relative entropy is an asymmetric measure. Therefore, this study symmetrically converts the relative entropy between node distance distributions. The specific formula is as follows:  
\[Q_{xy}=\frac{D_{KL}\big{(}P_{x}\|P_{y}\big{)}+D_{KL}\big{(}P_{y}\|P_{x}\big{)}}{2} \tag{12}\]  
\[R_{xy}=1-\frac{Q_{xy}}{Q_{max}}=\frac{Q_{max}-Q_{xy}}{Q_{max}} \tag{13}\]  
The relative entropy in asymmetric form is converted into symmetric form \(Q_{xy}\) in Equation (12). On the basis of Equation (13), normalization processing is implemented for the relative entropy in symmetric form, then \(R_{xy}\) is obtained. Among them, \(R_{xy}\in[0,1]\).