# Complex Network Modeling and Analysis of Microfracture Activity in Rock Mechanics
## 4 Results and Discussion
### Dynamic Evolution Characteristics of the Rock Microfracture Network  
Any real network is time-dependent. Combined with the theory of network dynamics, the evolving microfracture network can more accurately represent the change in rock microfracture activity. We choose two notations, the average degree \(<\)\(l\)\(>\) and the average clustering coefficient \(<\)\(<\)\(>\), to describe the properties of the microfracture network. Through the changes in \(<\)\(l\)\(>\) and \(<\)\(<\)\(>\), the specific process of rock fracture can be analyzed from the perspective of network evolution. \(<\)\(l\)\(>\) and \(<\)\(>\) can be solved via Equations (6) and (11). According to the definition of network notations, the node degree is the connected edges of the nodes, which indicates that one fracture event can affect other events in the network; the clustering coefficient describes the clustering degree of nodes, which indicates the aggregation property of microfractures in the network. Several simple graphs are used to illustrate the relationship between network evolution and changes in two characteristic parameters (Table 2). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Network Graph** & **Notations** & **Evolved Network Graph** & **Notations** \\ \cline{2-3}
**(2 Nodes)** & **Notations** & **Evolved Network Graph** & **Notations** \\ \hline
**O** & \(<\)\(l\)\(>\) = 0, \(<\)\(>\) = 0 & \(\quad\)\(<\)\(l\)\(>\) = 0, \(<\)\(>\) = 0 \\ \cline{2-3}
**O** & \(<\)\(l\)\(>\) = 1, \(<\)\(>\) = 0 & \(\quad\)\(<\)\(l\)\(>\) = 1.33, \(<\)\(>\) = 0 \\ \hline \end{tabular}
\end{table} 
Table 2: Change in notations when the two-node network becomes a three-node network (In the table, the black circle represents the fracturing point, and the short red segment represents the correlation between the fracturing points). Figure 14: **(a)** Microfracture network at \(\sigma_{\text{P}}\) and its simple properties; **(b)** degree distribution and cumulative degree distribution probability of **(a)**; **(b)** preliminarily proves that this network belongs to a complex network. The main characteristic of complex networks is that a large number of nodes have a small degree, while a few nodes have a large degree. In the fracture point network, there are only two types of relationships between any two fracture points: either they are related or they are unrelated. When a fracture point is isolated, its node degree is 0, resulting in an average degree of 0. Conversely, when two fracture points are related, the degree of both nodes is 1, yielding an average degree of 1. Upon the occurrence of a third fracture point, three possible relationships emerge between the third fracture point and the first two fracture points: (1) related to both, (2) unrelated to both, and (3) related to one of them. Hence, there are four possible outcomes in total. Notably, the increase in the average degree is associated with the interrelatedness of the fracture points, indicating the connection of microfractures or the propagation of microcracks within the network. On the other hand, the increase in the clustering coefficient is solely linked to the presence of closed loops, which is indicative of the rapid occurrence of continuous fracture events over a short time and distance or the aggregation of microfractures. As deduced from the preceding analysis, the alterations in notations effectively capture the evolution behavior of the microfracture network. Furthermore, the characteristic parameters of the network at any given time can be computed utilizing the parameters recorded by the acoustic emission sensor (Figure 15). During the early and middle stages of the loading test, microfractures in the rock manifested in a random manner, exhibiting strong discreteness, with no evident correlation between fracture points. Consequently, both the average degree and average clustering coefficient of the microfracture network remain at 0 during this period. This scenario persists until the middle stage of stable crack growth.