## 2 The Network Model and Analytical Methods
### Community Division and Louvain Algorithm  
[PERSON] et al. (2003) defined a quantity called modularity to measure the quality of community structure [PERSON] et al. (2003). Modularity refers to the ratio of edges in the community to all edges in the network minus the expected value of such a ratio when the degree of all nodes in the network remains the same but the connections are uniformly randomly generated. The formula of modularity is as follows:  
\[Q=\frac{1}{2m}\sum_{i,j\in V}[W_{ij}-\frac{k_{i}k_{j}}{2m}]\delta(C_{i}=C_{j}), \tag{4}\]  
where \(m\) represents the sum of the weights of all edges in the network, \(k_{i}\) represents the sum of the weights of all edges of node \(i\), \(W_{ij}\) represents the weight of the edges between node \(i\) and node \(j\), and \(C_{i}\) and \(C_{j}\) represent the community including node \(i\) and \(j\). The function \(\delta(C_{i}=C_{j})\) indicates whether node \(i\) and node \(j\) are in the same community. If they are in the same community, \(\delta=1\); otherwise, it is \(\delta=0\). The module \(Q\) can reflect the closeness of the nodes within the community, and its value range is \([0,1]\). The larger the value of \(Q\) is, the closer the community is. Based on the concept of modularity, [PERSON] et al. (2008) proposed a fast community division algorithm, the Louvain algorithm [PERSON] et al. (2003), which can accelerate the running time and rapid convergence of a community merger. The idea of the algorithm is: first, each node is regarded as an independent community. At this time, the number of communities is the same as the number of nodes. Then, we allocate node \(i\) to the communities where its neighbor nodes are located in turn and calculate the modular change \(\Delta Q\) after the allocation. The neighbor node with the largest value of \(\Delta Q\) is recorded as \(k\). If the largest value of \(\Delta Q\) is greater than 0, then we allocate node \(i\) to the community where that neighbor node is located; otherwise, it remains unchanged. We repeat this process until the community structure to which all nodes belong is no longer changed, then the first iteration ends. In the process, the formula for calculating \(\Delta Q\) when moving an isolated node \(i\) to a community \(C\) is as follows:  
\[\Delta Q=\left[\frac{\sum_{in}+2k_{i,int}}{2m}-\left(\frac{\sum_{tot}+k_{i}}{2 m}\right)^{2}\right]-\left[\frac{\sum_{in}}{2m}-\left(\frac{\sum_{tot}}{2m} \right)^{2}-\left(\frac{k_{i}}{2m}\right)^{2}\right] \tag{5}\]  
where \(\sum_{in}\) is the sum of the weights of the edges inside the community \(C\), \(\sum_{tot}\) is the sum of the weights of the edges incident to nodes in \(C\), \(k_{i}\) is the sum of the weights of the edges incident to node \(i\), \(k_{i,in}\) represents the sum of the weights of the edges from \(i\) to the nodes in community \(C\), and \(m\) represents the sum of the weights of all the edges in the network. At the end of the first-stage iteration, the local modularity will reach its maximum value. Then, the second stage is opened, and all the nodes in the same community are regarded as a new node. This new node will have a closed loop pointing to itself, whose weight is sum of the weights of the edges in the community. The weight of the edges between any two new nodes is the sum of the weights of all the edges between the two communities. Then, we repeat the process of the first-stage iteration until the modularity of the entire network is no longer changed and the community structure is divided. In addition, [PERSON] (2003) introduced a parameter called \"Resolution\" to flexibly control the number and size of community division [PERSON] (2003). The resolution parameter can enlarge and reduce the number and scale of the community so as to realize the community division under different resolution levels and help to find the appropriate resolution level. When the resolution parameter approaches 1, the division is rough, the community scale is large, and the number of communities is small.