## 3 Results
### Analysis of Bank Correlation Network Model
#### 3.2.1 Community Division and Nodes Importance Analysis  
We focused on a comparative study of states with a large number of banks, including Texas (TX), Minnesota (MN), Missouri (MO), Iowa (IA), and Illinois (IL). First, the Louvain algorithm was used to divide the five correlation networks into communities, and the resolution parameter of the division was set to 1.0 by default. Then, the PageRank algorithm was used to quantify the importance of all nodes, and scatter diagrams of the PR (PageRank) values of all the nodes in different communities were drawn for five networks, which are shown in Figure 5. The figure shows that the most important several nodes always tend to be clustered in the same community. Furthermore, we performed a statistical analysis of the PR values of the nodes in each community, as shown in Figure 5. The results are shown in Table 1, which shows that the distribution of node importance between different communities is obviously different. All the phenomena show the significance of the community division method for the bank correlation networks. Figure 5: Scatter diagrams of PR values for communities in the five networks of TX, MO, MN, IL, and IA. Furthermore, for each divided community, we retained the nodes and connecting edges inside this community and removed the nodes outside this community and the connections between this community and other communities to form an independent community sub-network. Then, the new PR values of these nodes in the sub-network were recalculated. The new PR value ranking of these nodes in the sub-network was compared with their PR value ranking in the original network. All the comparison results are shown in Figure 6. It can be seen that, for a group of nodes in the same community, their new PR value rank in the sub-network is highly consistent with the rank of the PR value of this group of nodes in the original network. Therefore, it can be asserted that the order trend of the two groups of PR values has a high similarity. Based on that a node's importance reflects the important position of the node in the network and the degree of its influence on the other nodes, Figure 6 shows that, after the community division of a bank correlation network, removing the edges between communities will have little impact on the interaction between the nodes within a community. Namely, the connections between communities have little effect on the interactions between nodes within communities. Thus, after the community division, each community can basically retain the nature of these nodes in the original network. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Community \({}^{a}\)** & **N \({}^{b}\)** & **Mean of PR (\(\times\)10\({}^{3}\)) Values** & **Variance of PR (\(\times\)10\({}^{3}\)) Values** \\ \hline TX\({}_{0}\) & 248 & 2.22523743548387 & 0.0215978777464978 \\ TX\({}_{1}\) & 233 & 1.92335242060086 & 0.0659583239182102 \\ MO\({}_{0}\) & 120 & 2.81006100453333 & 0.107091500416574 \\ MO\({}_{1}\) & 121 & 2.51621886913223 & 0.0255431313667735 \\ MO\({}_{2}\) & 113 & 3.17106368377876 & 0.0049496146085455 \\ MN\({}_{0}\) & 118 & 2.71631342961864 & 0.0469145350950722 \\ MN\({}_{1}\) & 125 & 2.624615255376 & 0.0338409023282173 \\ MN\({}_{2}\) & 125 & 2.811184867392 & 0.0160391232523291 \\ IL\({}_{0}\) & 212 & 2.3452891415849 & 0.0132360279522334 \\ IL\({}_{1}\) & 226 & 2.22477301762389 & 0.0512349427066714 \\ IA\({}_{0}\) & 122 & 3.85486265188524 & 0.0200009060100606 \\ IA\({}_{1}\) & 17 & 3.31266943647059 & 0.0154961123730195 \\ IA\({}_{2}\) & 123 & 3.84871037436585 & 0.130251077681873 \\ \hline \hline \end{tabular} \({}^{a}\) The subscripts indicate the different communities in the state’s banks correlation network. \({}^{b}\) N respresents the number of nodes in each community.  
\end{table} 
Table 1: Statistical analysis of the PR values of nodes in each community shown in Figure 5.  
Through the community division, some nodes that are closely connected can be divided into the same community, while the connection between two communities is lower.