## 5 Invulnerability Optimization Model of SBIN
### Optimization Methods of Invulnerability in SBIN  
1. Random edge-increasing method (RD).
2. Intialize \(i=0\), and abstract the SBIN into the graph \(G=(V,E)\);
3. Randomly select two nodes with no edges connected in the graph \(G\), the graph after adding the edge is \(G^{\prime}\), set \(i=i+1\) and \(G=G^{\prime}\);
4. If \(i\leq N_{a}\), go back to step 2 and continue to increase the edge until the end of the edge increasing process;
5. Calculate the \(BC_{i}\) of all nodes and the maximum value \(BC_{\text{max}}\) in SBIN after adding the \(N_{a}\) edges.
2. Low nodes degree with high priority (LDF).
3. Initialize \(i=0\), and abstract the SBIN into the graph \(G=(V,E)\);
4. Arrange all nodes in SBIN from small to large according to the degree value;
5. Select two nodes with the smallest degree value and no edges connected in the graph \(G\), the graph after adding the edge is \(G^{\prime}\), set \(i=i+1\) and \(G=G^{\prime}\);
6. If \(i\leq N_{a}\), go back to step 2, or go to step 5;
7. Calculate the \(BC_{i}\) of all nodes and the maximum value \(BC_{\text{max}}\) in SBIN after adding the \(N_{a}\) edges.
8. Low nodes betweenness centrality with high priority (LBF).
9. Initialize \(i=0\), and abstract the SBIN into the graph \(G=(V,E)\);
10. Arrange all nodes in SBIN from small to large according to the betweenness centrality value;
11. Select two nodes with the smallest betweenness centrality value and no edges connected in the graph \(G\), the graph after adding the edge is \(G^{\prime}\), set \(i=i+1\) and \(G=G^{\prime}\);
12. If \(i\leq N_{a}\), go back to step 2, or go to step 5;
13. Calculate the \(BC_{i}\) of all nodes and the maximum value \(BC_{\text{max}}\) in SBIN after adding the \(N_{a}\) edges.
14. Adding shortcut in nodes with the maximum betweenness centrality method (SMB).  
As shown in Figure 5 below, node 1 is the central node, and its neighbor nodes are node 2, node 3, and node 4. The number next to the dotted line between the neighboring nodes indicates the numberof shortest paths passing through 3 nodes at the same time. If one selects two neighbor nodes with the largest betweenness centrality value among all the edges adjacent to the central node and add an edge between them, some of the shortest paths will no longer pass through the central node.  
1. Initialize \(i=0\), and abstract the SBIN into the graph \(G=(V,E)\);
2. Calculate the betweenness centrality value of all nodes in SBIN, find the node with the largest betweenness centrality value \(v_{BC_{\max}}\), and the edge set \(Ev_{BC_{\max}}\) connected to the node \(v_{BC_{\max}}\);
3. Sort the betweenness centrality value of all the edges in the order from largest to smallest, and record the corresponding node set as \(O_{v_{BC_{\max}}}\) at the other end of these edges;
4. Select the two nodes with no edges connected in the top from \(O_{v_{BC_{\max}}}\), add an edge between the nodes, the graph after adding the edge is \(G\)', set \(i=i+1\) and \(G=G\)';
5. If \(i\leq N_{a}\), go back to step 2, or go to step 6;
6. Calculate the \(BC_{i}\) of all nodes and the maximum value \(BC_{\max}\) in SBIN after adding the \(N_{a}\) edges.