# Complex Network Modeling and Analysis of Microfracture Activity in Rock Mechanics
## 3 Construction of the Rock Microfracture Network
### Construction Method of the Rock Microfracture Network  
The process of rock failure arises from the propagation and penetration of macroscopic cracks initiated by microfractures. Acoustic emission technology enables the monitoring of fundamental properties associated with all microfracture events in this process. By conceptualizing rock microfracture points as nodes within a network and considering the interactions between these fracture points as interconnected edges, all the acoustic emission points from rock fractures can form a network comprising numerous nodes and edges. Moreover, the network at various stages throughout the rock fracture process can be viewed as a dynamic complex network that undergoes temporal changes.  
By utilizing the temporal and spatial characteristics of acoustic emission events, the length of the spatiotemporal bond between fracture points is determined, and the correlation property of fracture points is inferred based on small spatiotemporal bond lengths. The rock fracture network model is then established through MATLAB matrix operations. The edge construction method involves the following steps: if the spatiotemporal distance between any two points is less than the characteristic bond length, it is concluded that the two fracture points are correlated; thus, an edge is added between them. Conversely, if there is no correlation between two points, no edge is added. The network construction process is outlined as follows.  
At the onset, there exists an initial isolated acoustic emission point. Upon adding a second point to the network, the spatiotemporal distance \(d_{12}=[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}+C(t_{2}-t_{1})^{ 2}]^{1/2}\) is computed to determine if \(d_{12}\) is less than or equal to the characteristic bond length \(d_{c}\). If true, an edge is added between the first point and the second point, and the value \(d_{12}=d_{21}=1\) is recorded in the \(2\times 2\) matrix. Conversely, if no edge is added, the value \(d_{12}=d_{21}=0\) is recorded in the matrix.  
Figure 11: Slope-time bond length curve.  
Once the number of nodes N \(\geq\) 2, each subsequent acoustic emission point m is added. The spatiotemporal distance dmn between this new point and all existing acoustic emission points is computed (1 \(\leq\) n \(<\) N, where N represents the total number of nodes). By assessing the relationship between dmn and the characteristic bond length \(d_{c}\), the connectivity of node m with any node n is determined. Similarly, when nodes m and n are connected, a\({}_{\text{mn}}\) = 1; otherwise, a\({}_{\text{mn}}\) = 0.  
With each addition of a node, the connection matrix A = {a\({}_{\text{ij}}\)} expands by one order, and correspondingly, the network composed of acoustic emission points and its physical attributes undergoes changes. The outcome is a network comprising all the sound emission points and edges, constituting the complex network model of rock fracture. The complex network of rock failure dynamically evolves with the progression of rock deformation and failure. The network flowchart illustrates the modeling process (Figure 12).  
Figure 12: Flowchart of the network model building program.  
There are 637 AE data points in sample GC1 before the peak, and a network model with all nodes and different numbers of edges (determined by the length of the characteristic key) can be constructed. By calculating whether there is an edge between any two nodes, the network model diagram corresponding to any loading time can be constructed with the help of the DataViewer3D data visualization tool (Figure 13). According to the solution of the network parameters, the dynamic evolution process of rock fracture development can be described.