# Multi-Attribute Decision Making Method for Node Importance Metric in Complex Network
## 3 Multi-Attribute Composite Measure of Node Importance  
The node importance indicator based on multi-attribute decision making is fitted from four single indicators: H-index, closeness centrality, k-shell indicator, and network constraint coefficient. The combined node score can be expressed as:  
\[\text{Score }_{i}=\sum_{m}w_{i}\cdot\text{ Indicator }_{i} \tag{5}\]  
We can find that the weight of a single indicator affects the degree of its influence on the composite score, and whether the indicator weights can be reasonably determined will directly affect the performance of the node composite score. To solve the multi-criteria decision problem, [PERSON] [27] developed the CRITIC method, which derives objective weights for relevant indicators by quantifying the amount of information inherent in each evaluation criterion in order to rank targets without bias. The CRITIC method operates in the following process.  
For a set \(A\) of \(n\) chosen targets to be evaluated and a given system \(S_{j}\) of \(m\) evaluation criteria, the general form of the multi-criteria problem can be defined as follows:  
\[\text{Max}\{S_{1}(a),S_{2}(a),S_{3}(a),\ldots,S_{m}(a)\mid a\in A\} \tag{6}\]  
Define the membership function \(x_{i}\) by mapping the values of criterion \(S_{j}\) to the interval \([0,1]\). This transformation comes from the concept of an ideal point. We define \(x_{aj}\) as the degree to which alternative \(a\) is close to the ideal value \(S_{j}^{ideal}\). \(S_{j}^{ideal}\) represents the best performance of criterion \(S_{j}\). \(x_{aj}\) also represents the extent to which the alternative \(a\) is far from the negative ideal \(S_{j}^{anti-ideal}\). \(S_{j}^{anti-ideal}\) represents the worst performance of the criterion \(S_{j}\).  
\[x_{aj}=\frac{S_{j}(a)-S_{j}^{anti-ideal}}{S_{j}^{ideal}-S_{j}^{anti-ideal}} \tag{7}\]  
We can observe that when the evaluation system is a negative indicator, i.e., the indicator score is negatively correlated with the target performance, \(x_{aj}\) can still have practical significance. Using this method, the initial evaluation matrix can be transformed into a relative score matrix with generic elements \(x_{ij}\). Each assessment criterion corresponds to a score vector \(x_{j}\), which represents the scores of all elements to be assessed.  
\[x_{j}=\big{(}x_{j}(1),x_{j}(2),\ldots,x_{j}(n)\big{)} \tag{8}\]  
\[\sigma_{j}=\sqrt{\frac{\sum_{i=1}^{n}\big{(}x_{j}(i)-\bar{x}_{j}\big{)}^{2}}{n}} \tag{9}\]  
Here, the standard deviation \(\sigma_{j}\) quantifies the contrast intensity of the vector \(x_{j}\), reflecting the variation of scores within the evaluation indicator.  
Next, combining the vector \(x_{ij}\), we can construct the dimension \(m*m\) score matrix. Define per as the Pearson coefficient of \(x_{j}\) and \(x_{k}\). The value of per is lower if the scores of criterion \(j\) and criterion \(k\) are more inconsistent. Conflict\({}_{j}\) represents a measure of the conflicting differences between criterion \(j\) and other criteria.  
\[\text{Conflict}_{j}=\sum_{k=1}^{m}\Big{(}1-\text{ per }_{jk}\Big{)} \tag{10}\]  
Considering the two decision criteria of information contrast and inter-indicator conflict contained in the set of indicators, the amount of information \(C_{j}\) emitted by the \(j\)th criterion can be obtained by multiplicative aggregation:  
\[C_{j}=\sigma_{j}\cdot\text{ Conflict }_{j} \tag{11}\]The higher the \(C_{j}\) value, the greater the amount of information of the corresponding evaluation criteria, the greater the relative importance in the decision-making process, and the weight of the corresponding index can be obtained by standardizing the \(C_{j}\) value.  
\[w_{j}=\frac{C_{j}}{\sum_{k=1}^{m}C_{k}} \tag{12}\]  
Combining the indicator weights, we can define the multi-attribute CRITIC network decision indicator (MCNDI) as follows:  
\[MCNDI_{i}=w_{1}\cdot H_{i}+w_{2}\cdot COC_{i}+w_{3}\cdot KS_{i}+w_{4}\cdot NCC _{i} \tag{13}\]