## 3 Materials and Methodology
### Complex Network
#### 3.3.2 Network Topology Features  
Each network has specific topology features to characterize its connectivity, interactions and dynamic evolution process. Analyzing the topology features of a network could provide a comprehensive understanding of the network structure and identify the key nodes and their dynamic interaction characteristics in the network.  
(1) Degree  
Degree in a complex network is the most basic network topology feature. The degree of a node \(d(i)\) represents the number of edges that connect the node to other nodes. In a directed network, the degree of node \(i\) can be divided into out-degree (\(d_{out}(i)\)) and in-degree (\(d_{in}(i)\)). The out-degree of node \(i\) is calculated by the number of edges from node \(i\) to other nodes, while the in-degree of node \(i\) represents the number of edges from other nodes to node \(i\), which can be expressed as [30]:  
\[d(i)=\sum_{j\in V}a_{ji}+\sum_{j\in V}a_{ij} \tag{5}\]  
\[d_{in}(\mathrm{i})=\sum_{j\in V}a_{ji} \tag{6}\]  
\[d_{out}(\mathrm{i})=\sum_{j\in V}a_{ij} \tag{7}\]  
where \(V\) denotes the set of all nodes.  
The average of the degrees of all nodes in the network \(\langle d(i)\rangle\) is called the average degree of the network and is calculated as:  
\[\langle d(i)\rangle=\frac{1}{N}\sum_{i=1}^{N}d(i) \tag{8}\]  
Figure 3: Mapping association rules to directed weighted network.  
where \(N\) denotes the total number of nodes in the network.  
(2) Average path length  
Average path length \(L\) is the average of the shortest paths between all possible node pairs in the network, which can quantify the distance level of the network calculated as [33]:  
\[L=\frac{1}{N(N-1)}\sum_{i<j}l_{i,j} \tag{9}\]  
where \(N\) denotes the total number of nodes in the network; \(l_{i,j}\) is the number of edges of the shortest path between nodes \(i\) and \(j\).  
The network diameter is the maximum distance between any two nodes in the network, i.e., the maximum of all the \(l_{i,j}\). The network diameter is used to measure the density of the network, the smaller the diameter, the more densely connected the network is.  
(3) Clustering coefficient  
The clustering coefficient \(C\) measures the degree of aggregation between network nodes, which is defined as:  
\[C=2d^{\prime}{}_{i}/d_{i}(d_{i}-1) \tag{10}\]  
where \(d^{\prime}{}_{i}\) represents the number of edges among its neighbors; \(d(i)\) denotes the degree of the node. The higher the clustering coefficient of a node, the greater the likelihood that other nodes will cluster around it. The average clustering coefficient (ACC) of the network is the average of the CC of all nodes.  
In directed weighted networks, the clustering coefficients also need to take into account the weights of the edges. [PERSON] et al. [44] defined the formula for calculating the weighted clustering coefficients \(C_{i}^{w}\) as follows:  
\[C_{i}^{w}=\frac{1}{s_{i}(d_{i}-1)}\sum_{j,h}\frac{w_{ij}+w_{ih}}{2}a_{ij}a_{ih }a_{jh} \tag{11}\]  
where \(s_{i}\) is the sum of the weights of all adjacent edges of node \(i\); \(w_{ij}\) denotes the weights of the edges between adjacent nodes \(i\) and \(j\); \(w_{ih}\) denotes the weights of the edges between adjacent nodes \(i\) and \(h\).  
(4) Betweenness  
The betweenness of a node \(B_{i}\) is the number of times all shortest paths in the network pass through the node which can describes the influence of the node in the global network, and is calculated as [33]:  
\[B_{i}=\sum_{j\
eq i\
eq h\in N}\frac{N_{jh}(i)}{N_{jh}} \tag{12}\]  
where \(N_{jh}\) is the number of shortest paths connecting nodes \(j\) and \(h\); \(N_{jh}(i)\) is the number of shortest paths connecting nodes \(j\) and \(h\) and passing through node \(i\); \(N\) denotes the total number of nodes in the network.  
The betweenness centrality of a node \(C_{B(i)}\) is the normalized betweenness of the node, which is calculated as:  
\[C_{B(i)}=\frac{2B_{i}}{(N-1)(N-2)} \tag{13}\]