# Multi-Attribute Decision Making Method for Node Importance Metric in Complex Network
## 2 Related Theoretical Foundations  
Considering a graph \(G=(V,E)\) to represent complex networks, \(G\) is an undirected connected graph, where \(V\) is the set of nodes and \(V=\{1,2,3,\ldots,n\}\). \(E\) is the set of edges and \(E=\{1,2,3,\ldots,m\}\). Because the multi-attribute decision-making method can take advantage of all single indicators, in the process of selecting sub-indicators for multi-attribute decision making, it is important to first provide more comprehensive information about the importance of nodes, while homogeneity of selected indicators should be avoided as much as possible. The local information of nodes is the most basic importance measure of nodes, which characterizes the structural correlation between nodes and neighboring nodes and portrays the local importance of nodes; as the hub of information flow in the network, the shortest path of node association reflects the contribution of the node in the global information flow process, and the global location information of nodes reflects the distance of nodes from the center of the network and characterizes the core degree of nodes in the network. Meanwhile, focusing on the set of indirectly associated nodes, the nodes in the intermediary role tend to have more information. Therefore, in this paper, we select multi-attribute decision sub-metrics from four aspects: node local information, path centrality, node global location information, and node information quantity. Next, we present each single centrality indicator. **Definition 1**. : \(H\)_-index (abbreviated as \(H\))._  
Originally used extensively to assess scholarly communication among academics, the H-index has gradually been extended to apply to the study of a variety of complex networks. The definition for the H-index is as follows [16]: the degree of any node \(i\) is denoted by \(k_{i}\), and the degrees of its neighbors are \(k_{j1}\), \(k_{j2}\),..., \(k_{jk_{i}}\). For a finite number of real numbers \((x_{1},x_{2},\ldots,x_{n})\), we propose the operator \(H\), which will return an integer \(z=H(x_{1},x_{2},\ldots,x_{n})>0\), where \(z\) represents the largest possible integer value that matches the set of real numbers \((x_{1},x_{2},\ldots,x_{n})\) in which there exist at least \(z\) elements with values not lower than \(z\). Then, the H-index of node \(i\) is defined as:  
\[H_{i}=H(k_{j1},k_{j2},\ldots,k_{jk_{i}}) \tag{1}\]  
The H-index measures the importance of nodes from the overall quality of the neighbor node set. Compared with node degree, the H-index reinforces the importance of nodes with a large number of high neighbors and reduces the importance of other nodes, a feature that helps us better identify key nodes in real networks. **Definition 2**. : _Closeness centrality (abbreviated as COC)._  
Assuming that \(d_{ij}\) denotes the number of edges contained in the shortest path from node \(i\) to node \(j\), the closeness centrality of node \(i\) denotes the reciprocal of the sum of its distances from all other nodes in the network. The expression of closeness centrality is defined as:  
\[\text{COC}_{i}=\frac{N}{\sum_{j=1}^{N}d_{ij}} \tag{2}\]  
Closeness centrality expresses the distance between the node and the network center based on the standard of path centrality.The value of node closeness centrality represents the distance of the node from other nodes in the network. The closer the node is to the center of the network topology, the greater the importance. **Definition 3**. : \(K\)_-shell (abbreviated as KS)._  
The k-shell decomposition method divides the network into substructures directly related to degree centrality [28]. In this approach, the network is described as a hierarchical structure, revealing the complete hierarchy of its nodes. In Figure 1, the innermost nodes belong to a structure called the core, while the rest of the nodes are placed in more external layers (called k-shells).