# Multi-Attribute Decision Making Method for Node Importance Metric in Complex Network
## 1 Introduction  
Complex network systems are widely distributed in the real world. With the discovery of scale-free distributions [1], small worlds [2], and community-based properties possessed by complex networks, complex network science has gained wide attention in computer science, biological science, transportation planning, social science [3; 4; 5; 6], and other fields. The node importance metric is one of the key research directions. The most important nodes can help us effectively defend against malicious network attacks [7], identify key genetic information [8], protect important transportation hubs [9], and strengthen the control of social opinion dissemination [10]. The importance of nodes in complex networks varies significantly depending on network topology and size. Current methods of node importance ranking include two main types: social network analysis methods and system science analysis methods [11]. In the social network analysis method, the main idea is that the importance of nodes is equal to the salience; the method does not destroy the connectivity of the network in the analysis process in order to ensure the topological integrity of the network. This is generally achieved through the use of a centrality indicator to measure the importance of nodes; the indicators that have been proposed include degree centrality, closeness centrality, betweenness centrality, clustering coefficient, Hirsch's index, and k-shell indicator [12; 13; 14; 15; 16; 17]. The main idea of the system science analysis method is that damage equals importance. Hence, after removing nodes from the network, the change of network properties isanalyzed to determine the importance of the removed nodes; common indicators include the shortest path, network efficiency and maximum connectivity subgraph [18; 19; 20]. Since the system science analysis method would destroy the topology of the network, causing changes in the connectivity state between nodes and the loss of information about the nodes themselves, we adopt the social network analysis method to study the node importance metric problem. Usually, most of these indicators measure the importance of nodes from a certain perspective, but since networks have multiple morphological characteristics, using a single indicator to assess the importance of nodes is often incomplete: local indicators such as degree centrality and clustering coefficient can be calculated only based on the local structure near the nodes, but they cannot evaluate the importance of nodes globally. The k-shell indicator considers the global hierarchical structure of the network, but a large number of nodes are distributed in the same layer, so they cannot effectively measure the importance of nodes in the same layer. At the same time, because of different evaluation perspectives, different centralities may produce different metric results or even conflict, causing difficulties in node ranking decisions. In order to assess the importance of nodes more rationally, two main research ideas are currently being investigated [21]. One is to combine the structural characteristics of the network and propose more reasonable evaluation indicators. [PERSON] [22] proposed an integral formula based on closeness centrality and betweenness centrality; [PERSON] [23] used a linear weighting approach to combine degree centers and proximity centers; [PERSON] [24] improved upon the structural hole method by considering the next nearest neighbor nodes. The above methods usually result in an optimized indicator that evaluates better, but the methods are hardly universal in terms of how to integrate the relevant indicators. In the second approach, multiple indicators are used to evaluate the same object, and the individual evaluation results of each indicator are combined with the information inherent in the data. This class of ideas introduces the multi-attribute decision-making (MADM) method to identify the importance of nodes. The main idea of MADM is to rank the options to be selected based on subjective or objective preferences.