## 4 Discussion
### Robustness of Community Division  
We used the Louvain algorithm to perform the community division of bank correlation networks. The realization of the Louvain algorithm was also based on the idea of modularity proposed by [PERSON] et al. (2003) [PERSON] et al. (2003). It is acknowledged that modularity may suffer from a resolution limit when the number of links increases. Although bank correlation networks are weighted networks and the resolution limit is less preponderant, it is necessary to test the robustness of the results of our community division. Next, we take the network of IL as an example and check the robustness of the community division results when the resolution parameter is 0.8 by retaining only the subset of links with statistically significant correlations in the bank correlation network of IL. As for how to calculate the statistical correlation of edges, the work of [PERSON] et al. (2013) gives us some inspiration. In this work, a weighted projected network is established based on a bipartite network. Then, some edges are deleted from the projected network based on some of the connection characteristics of the bipartite network itself, resulting in a statistically correct network. Since a bank correlation network is also established from the bank-asset network, it is similar to the projected network. However, the bipartite network in [PERSON] et al. is an unweighted network and just considers whether  
Figure 8: Comparison of the systemic risk of different communities of four networks. there is a connection between two nodes, while the bank-asset bipartite network we study is a weighted and nearly fully connected network that each bank invested in almost all six types of assets. Thus, the approach we take is to simply remove some edges in the bank correlation network whose weights are too small and at the same time ensure that these is at least one edge for any node. In this way, through simple calculation, we find that 25.79% of the edges with a small weight will be deleted and the remaining 74.21% edges can be retained in the bank correlation network of IL, thus obtaining a new bank association network. Now, we will compare the community division of the new bank correlation network when the resolution parameter is set to a default of 1 with the previous community division of the IL network when the resolution parameter is 0.8. The work [31] proposed some widely used indicators to compare the accuracy and precision of the detection of pairs of nodes in a given partition. The Rand index is essentially the accuracy of the pair classification and is defined as:  
\[R=\frac{TP+TN}{TP+FP+TN+FN}.\]  
In the formula, \(TP\) is the number of true positive pairs, which represents the number of pairs of nodes in the same community both in the considered and reference division. \(FP\) is the number of false positive pairs, which represents the number of pairs of nodes in the same community in the considered partition but in different communities in the reference division. (\(TN\)) is the number of true negative pairs, which represents the number of pairs when each node of the pair does not belong to the same community both in the considered division and in the reference division. Lastly, \(FN\) is the number of false negative pairs, which represents the number of pairs when each node of the pair does not belong to the same community in the considered partition while both nodes belong to the same community in the reference division. Furthermore, there is a adjusted Rand index (\(ARI\)), which is defined as  
\[ARI=\frac{TP+TN-E(TP+TN)}{TP+FP+TN+FN-E(TP+TN)}\]  
where \(E(TP+TN)\) is the expected value of the true pair classifications estimated between a random partition and the reference partition.