## 4 Discussion
### Relationship between Systemic Risk and Community  
According to the results in Figure 8, we drew a scatter diagram of the number of community nodes and the systemic risk of the community, which is shown in Figure 9. It can be seen from Figure 9 that for these four states, communities with extremely high systemic risk often contain fewer banks. With the increase in community size, the fluctuation of systemic risk gradually decreases and is finally distributed around the value of the overall failure rate (see the horizontal dotted line in the figure). Moreover, in addition to the states TX, IL, MN, and MO, we also considered seven states with a high bank failure rate--Arizona (AZ), California (CA), Colorado (CO), Florida (FL), Georgia (GA), Nevada (NV), and Washington (WA)--and carried out the community division of the correlation networks of these states with a resolution of 0.8, resulting in a total of 69 communities. The number of nodes in these communities and the systemic risk of these communities were analyzed, and the results are plotted in Figure 10. It can be seen from this figure that, for small-scale community sub-networks, systemic risk may be extremely high or extremely low, with a wide range of values, and the sub-network is quite likely to suffer from high systemic risk. Meanwhile, for large-scale community sub-networks, the systemic risks of different communities are relatively close to a low level with a relatively narrow range of values, and the community is not prone to suffer from particularly high systemic risk. In order to deeply study the dependence of systemic risks on the size of community sub-networks, we carried out a regression analysis. According to the changing law of the nodes in Figure 10, we performed a power law distribution regression analysis on all the nodes, and the model was \(f(x)=ax^{b}\). Considering that the dependent variable in the model will not be equal to 0, we excluded 19 points in Figure 10 with a systemic risk value of 0 and left 50 sample points to perform the regression analysis. The results of the model are as follows:  
\[SR=a\times n^{b}, \tag{7}\]  
Figure 9: Relationship between systemic risk and the number of nodes in the community. where \(n\) represents the number of nodes in a community, \(SR\) represents the systemic risk of the community, and \(a\) and \(b\) are the coefficients of the model. If we take the logarithm of both sides of the model, a linear model can be obtained:  
\[log(SR)=log(a)+blog(n)=c_{0}+b\times log(n), \tag{8}\]  
At this point, we take logarithms of the horizontal and vertical values of the points in Figure 10. The linear regression analysis result gives \(c_{0}=-0.4506\) and \(b=-0.5499\), then the model is as follows:  
\[log(SR)=-0.4506-0.5499\times log(n), \tag{9}\]  
The significant levels of \(c_{0}\) and \(b\) are 0.067 and 0.000, respectively, showing that the regression effect is quite significant. With \(c_{0}\), we can derive \(a=e^{c_{0}}=0.6373\) in the original power law model. The scatter plot of the nodes in Figure 10 and the function curve of the original power law distribution are both depicted in Figure 11A. If the horizontal and vertical values of the points in Figure 10 are all logarithmic values, the function curve of the linear regression analysis function is shown in Figure 11B. Because the community samples with systemic risk to be 0 are removed in the regression analysis, the power law distribution model can only reflect the results when the community sub-network is at risk. This is to say that, when there are certain risks in the  
Figure 11: The results of regression analysis on the nodes in Figure 10. (**A**) shows the power law distribution of points in Figure 10. (**B**) shows the result after taking the logarithm to the horizontal and vertical values of points in (**A**). Figure 10.0 Relationship between systemic risk and the number of nodes in the community when the resolution parameter is 0.8.  
community sub-network, the systemic risk and the size of the community show a roughly decreasing trend.