# Complex Network Modeling and Analysis of Microfracture Activity in Rock Mechanics
## 3 Construction of the Rock Microfracture Network
### Space-Time Single-Link Cluster  
The single-link group method is commonly employed in seismology to evaluate the spatial clustering characteristics of vibrations. Currently, this method involves identifying the closest source point to each seismic starting point in space, connecting edges between them to form a single-link, and ultimately constructing a focal event sequence within a defined region comprising points and single-links. However, it is evident that earthquake occurrence is not solely determined by spatial proximity. For example, in the analysis of major earthquakes, despite the Diexi and Wenchuan earthquakes being about 100 km apart, their incidence spans over 74 years, with no established connection between them. Thus, when examining rock fracture properties, relying solely on spatial distance to calculate single-links inadequately captures the correlation between fracture points. According to space-time window theory [41], the occurrence of earthquake aftershocks is closely related to both space and time. Similarly, in the study of rock fractures, the occurrence of subsequent microfractures is affected by the spatiotemporal properties of the previous microfractures. Space-time single-link clustering is one such method for calculating the temporal and spatial clustering of earthquakes [42]. This method involves adding a time factor to the construction of a single-link group, determining the shortest space-time bond length between each break point by solving the time-space distance, and calculating the cluster attribute of the break point (Space-time single-link group theory (Figure 8)). A possible microrupture event i has a certain influence region in both time and space. For example, rupture point jis within the influence range of rupture i in both time and space, so microruptures i and j are correlated, whereas microrupture events m and n are not correlated.) The structure of the microfracture spatiotemporal single-link group is established. In the process of rock deformation and failure, the most basic properties of microfracture events are the occurrence time and spatial coordinates, both of which can be directly obtained through rock acoustic emission positioning tests. The spatiotemporal coordinates i (_xi_, _yi_, _zi_, _ti_) of any AE break point can be obtained according to the AE location test data. The time interval _tij_ and the space distance _dij_ (_i \(\
eq\) j_; \(i\), \(j\) \(\in\) [1, _n_], where \(n\) is the total number of break points) (Table 1):  
\[t_{ij}=t_{j}-t_{i} \tag{12}\]  
\[d_{ij}=\sqrt{\left(x_{j}-x_{i}\right)^{2}+\left(y_{j}-y_{i}\right)^{2}+\left( z_{j}-z_{i}\right)^{2}} \tag{13}\]  
The space-time bond length \(d_{ST}\) is the space-time distance between any two rupture events, which can be obtained by the time interval _tij_ and spatial distance _dij_ between the two rupture points [42]:  
\[d_{ST}=\sqrt{d_{ij}^{2}+C\cdot t_{ij}^{2}} \tag{14}\]  
where \(d_{ST}\) is the spatiotemporal distance between two acoustic emission events in mm _ST_; \(d_{ij}\) is the distance between two acoustic emission events, in mm; C is the spatiotemporal correlation constant, 1 mm-s-1, is taken according to the relevant literature on seismology; \(t_{ij}\) is the time interval between two acoustic emission events, in s.  
For any event in the microrupture event sequence [1, _n_], if the time series number of the event is m, then the spatiotemporal distance between the event and all other events can form a disordered finite set \(M=\{dm1\), \(dm2\), \(dm3\), \(\ldots\), \(dmm\}\). If the smallest element in the set \(M\) is d\(mm\), then event \(n\) is the event with the shortest space-time distance from event \(m\), and a space-time single-link is established between event \(m\) and event \(n\); that