## 6 Experimental Section and Analysis
### Experiment of Invulnerability Optimization Analysis  
To study the influence of the edge-increasing optimization method on the invulnerability of SBIN, this section mainly studies the comparative analysis of the invulnerability changes under four different edge-increasing strategies to the initial network, respectively. In this section, four kinds of edge-increasing ratio parameters, fa = 0.025, fa = 0.05, fa = 0.075, and fa = 0.1, are set. We proportionally delete the nodes through the improved tree attack strategy, and analyze the invulnerability changes under the four edge-increasing strategies RD, SMB, LBF and LDF. As shown in Figure 8a, the invulnerability of SBIN increases with a greater edge-increasing proportion in the random edge-increasing strategy. This indicates that, in SBIN, a greater proportion of random increased edges means a greater average connectivity of the network nodes. When a node is deleted under the improved tree attack strategy, due to the increase of the average connectivity, the connectivity of the other nodes connected to the node is less affected than in the situation where no edges are added. Therefore, under the same deletion ratio, a larger random edge-increasing proportion brings a higher invulnerability metric and stronger invulnerability of SBIN. As shown in Figure 8b, with a greater edge-increasing proportion in the SMB edge-increasing strategy, there are no significant changes in the invulnerability of SBIN. This is because SMB's edge-increasing strategy involves increasing the edge between adjacent nodes by a large degree, and the improved tree attack strategy deletes nodes according to the degree-betweenness centrality, precisely. Therefore, even if the number of connected nodes is increased for adjacent nodes, the invulnerability basically stays unchanged with the deletion of the nodes. Therefore, the SMB edge-increasing strategy is not suitable to be applied for invulnerability optimization under the improved tree attack strategy. Figure 8c,d shows a comparative analysis of the impact on invulnerability under the LBF and LDF edge-increasing strategies. It can be concluded from Figure 8c that the LBF edge-increasing strategy gets a better optimization result than the RD edge-increasing strategy. This is because adding edges between nodes with a low betweenness centrality increases the average betweenness centrality of the entire network, making the entire network have a higher number of shortest paths. When a node is attacked, the function of the network can be kept through the added edges. Therefore, the LBF edge-increasing strategy is better than RD. In Figure 8d, the LDF edge-increasing strategy is the most sensitive among the four edge-increasing strategies. The reason for this is that after the edge is added between the nodes with a low degree, the relevant nodes that are attacked in the improved tree attack strategy are not deleted because of the disconnection, and the function continues to be performed through the added edge. The LDF edge-increasing strategy theoretically increases the degree of the entire network, making the \begin{table}
\begin{tabular}{c c c} \hline \hline  & **TAS** & **ITAS** \\ \hline \(\alpha=0.05\) & 0.816 & 0.904 \\ \(\alpha=0.1\) & 0.65 & 0.58 \\ \(\alpha=0.15\) & 0.43 & 0.26 \\ \(\alpha=0.2\) & 0.18 & 0.19 \\ \(\alpha=0.25\) & 0.1365 & 0.171 \\ \(\alpha=0.3\) & 0.1357 & 0.16 \\ \(\alpha=0.35\) & 0.1311 & 0.149 \\ \(\alpha=0.4\) & 0.1144 & 0.1386 \\ \(\alpha=0.45\) & 0.1077 & 0.1278 \\ \(\alpha=0.5\) & 0.1009 & 0.117 \\ \(\alpha=0.55\) & 0.0941 & 0.1062 \\ \(\alpha=0.6\) & 0.0873 & 0.0954 \\ \(\alpha=0.65\) & 0.0805 & 0.0846 \\ \(\alpha=0.7\) & 0.0737 & 0.0738 \\ \(\alpha=0.75\) & 0.0669 & 0 \\ \(\alpha=0.8\) & 0.06 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Data statistics under different attack strategies \(\delta=10\). whole network flatten, and the gap between the large degree nodes and small degree nodes is further narrowed; hence, the entire network is more resistant to destruction.