# Multi-Attribute Decision Making Method for Node Importance Metric in Complex Network
## 4 Experimental Results and Analysis
### Network Attack Simulation Experiment  
#### 4.2.1 Attack Simulation to the USAir97 Network  
The USAir97 network is an infrastructure network; as a representative case of a real complex network, the network contains 332 nodes with 2126 edges, the number of k-shell indicators between nodes is 27 layers, and the relationship between nodes is very complex; its network topology is shown in Figure 6a. According to the above equations, we can obtain the weight distribution of the H-index, closeness centrality, k-shell indicator, and network constraint coefficients as:  
\[[H,COC,KS,NCC]=[0.27924,0.14960,0.27592,0.29524] \tag{17}\]  
Then, we will obtain the importance composite score MCNDI for 332 nodes.  
Figure 5: Monotonicity performance of different metrics on simulated networks.  
Usually, when a network is attacked, the change in network efficiency can be used to measure the degree of damage to the network.  
\[NE=\frac{1}{N(N-1)}\sum_{i\
eq j\in G}\frac{1}{d_{ij}} \tag{18}\]  
In Equation (18), \(G\) represents the set of nodes in the network, \(N\) represents the number of nodes in the network, and \(d_{ij}\) represents the length of the shortest path between the node \(i\) and the node \(j\). Here, we evaluate the accuracy of the node importance ranking algorithm by means of the network efficiency ratio.  
\[NEP=\frac{ne}{NE} \tag{19}\]  
In Equation (19), \(ne\) represents the network efficiency of \(G\) after deleting the set of nodes to be determined and \(NE\) is the initial network efficiency of the network. A smaller value of \(NEP\) means that the more important the deleted nodes are, the less secure the network is.  
On the one hand, we performed random attacks on the network, and on the other hand, based on different indicators that measure the importance of nodes, we performed deliberate attacks on the network by node ranking, and the network efficiency ratio was quantified as shown in Figure 7. To weaken the impact of network size, the deliberate attacks were scaled by network nodes. In Figure 7a, the horizontal axis represents the ratio of failed nodes to the total number of nodes, and the vertical axis represents the network efficiency ratio.  
Simulation results show that random attacks are less effective for real complex networks due to the scale-free nature of the real network, such that the network structure is strongly robust in the face of random attacks. In terms of deliberate attacks, the simulation results show that when simulating attacks on the USAir97 network according to the comprehensive index ranking proposed in this paper, the rate of decline in network efficiency showed the highest level of decline at the initial stage, and with the destruction of the network structure, although the rate fluctuated slightly, it eventually maintained the decline advantage in the rapid decline phase. The simulation results show that MCNDI is a metric that integrates many structural attributes of nodes and can more accurately reflect the real situation of nodes in the network.  
Figure 6: This figure shows the topological distribution of the real network in our network experiment. (**a**) Network topology of USAir97 network. (**b**) Network topology of technology routes network.