## 2 The Network Model and Analytical Methods
### Nodes Importance and PageRank Algorithm  
A node's importance reflects the importance of the node in the network. There are many methods to measure the importance of nodes, such as node centrality, K-shell decomposition, the PageRank algorithm, and the LeaderRank algorithm [27]. We will apply the PageRank algorithm [28] developed by [PERSON] to rank the importance of nodes, which has both speed and accuracy. The PageRank algorithm was originally developed to rank web pages by importance, and its basic assumption is that: (1) if a web page can be linked by many other web pages, the importance of the web page is higher; (2) if a highly important page links to another page, the importance of the page being linked to will also increase. The PageRank algorithm initially assigns the same importance score to each page, then designs an iterative algorithm based on these two assumptions to calculate the updated importance score of each page until the score is stable. The PageRank algorithm can be applied to any entity set that has the characteristic of mutual reference, and it is also suitable for ranking the importance of the nodes in a complex network. The PageRank value (PR) of node \(i\) can be expressed as follows:  
\[PR(i)=(1-q)+q\sum_{j\in In(i)}\frac{PR(j)}{k_{j}^{(out)}} \tag{6}\]  
where \(PR(i)\) is the PageRank value of the node \(i\) to be evaluated, \(q\in[0,1]\) is the damping factor, and \(1-q\) can be understood as the probability of a random jump to other nodes. Generally, \(q=0.85\), \(In(i)\) represents the collection of all nodes pointing to node \(i\), and \(k_{j}^{(out)}\) represents the sum of the weights of node \(j\) pointing to other nodes. The \(PR\) value is closely related to the in-degree of the node, and when the in-degree of a node is large, the \(PR\) value of that node tends to be larger.  
Some studies show that, in a real network, the degree of most nodes is often very low and a few nodes will have a higher degree (these are called the central nodes). Only these central nodes can have a strong impact on other nodes. For many nodes, their connection path to the central nodes may be quite long, so the influence of the central nodes on them may be slow. To find out which other nodes will affect these nodes faster and more directly, the communities in which these nodes are located should also be explored, and their status in the community and their connections with other nodes should be judged. Therefore, the risk of a node is also most likely to be related to the community in which it is affected, which gives us a certain theoretical basis for studying the financial systemic risk from the perspective of community.  
Since there are many methods of community division and the result of the division is not unique, it is necessary for us to use certain methods to test the reliability of the obtained community division result. In our previous paper [29], we used the Lovain algorithm to divide the communities of several types of networks with typical structures and analyzed the importance of nodes in the community sub-network. The results obtained show that there are obvious differences in the importance of nodes between different communities, and the importance ranking of all nodes in a community sub-network is highly consistent with their importance ranking in the original network, which fully demonstrates the significant effect of the community division. Based on this, we will first test the effectiveness of community structure division on bank correlation networks through the analysis of nodes importance, further analyze the differences in systemic risk between different community networks, and then explore the general law of financial systemic risk and community structure.