# A Complex Network Important Node Identification Based on the KPDN Method
## 5 Results and Analysis  
### Analysis of Real Networks  
#### 5.1.1 Size of Real Network from \(10^{3}\) to \(10^{4}\)  
As shown in Figure 4, this section compares the performance of the proposed method with other methods in actual networks. The robustness of networks under node removal was studied. In most cases, the proposed method outperforms other methods, and the relative size of the largest connected subgraph decreases faster. For example, Figure 4b shows the comparison of the decrease rate of the largest connected subgraph after deleting 20% of nodes in the source network by different methods. The results are: KPDN: 36.75%, PageRank: 37.82%, DWT: 54.31%, Degree: 58.88%, KPD: 62.06%, WL: 63.88%, CI: 64.16%, CC: 67.06%, K-shell: 68.31%, INCC: 67.44%, Random: 75.06%. Similarly, as shown in Figure 4a-e, the effect of deleting nodes on the connectivity coefficient of subnets is the most obvious. And, the effect is slightly better than other methods. Therefore, the method in this paper (the red curve in Figure 4) surpasses other methods in most cases, and the relative scale of the largest connected subgraph decreases more significantly. At the same time, it should be noted that in the comparison curves of each method, the relative scale of the largest connected subgraph does not decrease with the removal of nodes. For example, the power-eris1176 and bio-grid-worm datasets in Figure 4e,f. This is mainly because the initial node sorting method continues to change with the removal of the first batch of sorted nodes. The remaining network structure is different from the initial node importance sorting. We not only intuitively compare the size of the largest connected subgraph of the remaining subgraph but also introduce three indicators: subgraph sensitivity index, subgraph network connectivity efficiency, and number of remaining edges in the subgraph for supplementation. The threshold set in this paper is 100% of the original network size. When the susceptibility index peaks after removing a certain percentage of nodes, it means that each method has maximally decomposed the original network into fragments below the threshold size. Then, as the fragments continue to shrink, the susceptibility enters a downward stage. It can be observed from Figure 5 that the proposed method in this paper has an obvious effect on the change in the susceptibility index of the subgraph. To further demonstrate the sensitivity results on these six real-world networks of scale from \(10^{3}\) to \(10^{4}\), we have summarized the node consumption ratio when each real-world network reaches the extreme point under different removal strategies. Taking Figure 5b as an example, when the peak appears, the proportion of nodes removed is: KPDN: 11.46%, PageRank: 12.94%, DWT: 18.81%, Degree: 18.83%, KPD: 22.05%, INCC: 22.16%, CI: 22.46%, WL: 22.91%, K-shell: 24.33%, HC: 24.20%, CC: 24.43%, Random: 24.81%. Similarly, in Figure 5b,c,f, it can be seen that the network can be quickly fragmented by deleting fewer nodes. At the same time, it can also be observed from Figure 5a,d,e that the proposed method is in a leading position. Therefore, the proposed method in this paper has an obvious effect on the change in the susceptibility index of the subgraph. Figure 4: Robustness comparison of 10 methods for node removal in real networks (results of maximum connected coefficient. The horizontal axis represents the proportion of nodes removed, and the vertical axis represents the subnetwork maximum connectivity coefficient). (**a–f**) correspond to: (**a**) email-univ, (**b**) p2p-Gnutella08, (**c**) fb-pages-government, (**d**) US-airports, (**e**) power-eris1176, and (**f**) bio-grid-worm. It can be observed from Figure 6 that the impact of network node removal on the remaining number of network edges varies with different networks. For example, in air transport networks, as some airports are gradually removed due to weather and other factors, the air routes related to the removed nodes will also be affected and canceled.