# Complex Network Modeling and Analysis of Microfracture Activity in Rock Mechanics
## 3 Construction of the Rock Microfracture Network
### Solution of the Characteristic Bond Length \(d_{c}\)  
In the investigation of spatiotemporal single-link group architecture (ST-SLC), the presence of discrete break points led to the emergence of single-links with notably large spatiotemporal distance values within the spatiotemporal single-link group. Such single links, however, fail to accurately represent the true correlation properties between two points and should, therefore, be discarded. This paper introduces an eigenvalue to measure the correlation of microbreak points, termed the eigenbond length (\(d_{c}\)), which signifies the spatiotemporal distance of rock microbreak points. When the spatiotemporal single-link length (\(d_{ST}\)) exceeds the characteristic bond length (\(d_{c}\)), the \"time-space bond\" between the two break points lacks practical significance, suggesting either no correlation or a very weak correlation between the two points. Conversely, when the bond length (\(d_{ST}\)) between the microbreak points is less than the characteristic bond length (\(d_{c}\)), there is a meaningful \"space-time bond\" between the two break points, indicating a strong correlation between them. By calculating the spatiotemporal distance between any two adjacent points in the framework of the spatiotemporal single-link group, the spatiotemporal bond length-frequency relationship can be obtained, and the spatiotemporal bond length-\(d_{ST}\)-bond long frequency distribution relationship \(\rm N_{d}\) can be further obtained. Figure 9: Computational results of the spatiotemporal single-link group. According to fractal theory in mathematics, the relationship between the length and number of space-time bonds exhibits local scale-free characteristics. Within the scale-free region, the distribution of spatiotemporal bond lengths within the ST-SLC framework demonstrates a fractal structure, with its slope representing the fractal dimension (D\({}_{\rm N}\)). Similarly, the bond length-frequency relation also has the same fractal dimension (D\({}_{\rm N}\)), which can be divided into scale-free intervals with self-similarity. For example, in geotechnical or geological fields, the same D\({}_{\rm N}\) typically indicates the self-similarity of faults, geological structures, and other seismic events [43]. The fractal dimension can be determined by calculating the slope of the logarithmic curve of the spatiotemporal bond length and bond length frequency, as defined by Equation (15). The scale-free regions within the spatiotemporal single-link group framework can subsequently be delineated on the basis of various fractal dimensions. \[{\rm D}_{\rm N}=-\lim_{d\to 0}\frac{\ln{\rm N}_{\rm d}}{\ln d} \tag{15}\]  
where D\({}_{\rm N}\) is the frequency-key length fractal dimension; \(d\) is the spacetime distance between two break points, that is, the length of the spacetime bond, in ST-mm  
The N\({}_{\rm d}\)-ST-SLC architecture is the number of single-link frequencies in which the key is longer than \(d\). Following the processing of granite acoustic emission test data, the spatial and temporal single-link group framework for each sample is derived. In accordance with the aforementioned theory, the structure of rock single-link groups is analyzed, and the fractal dimension (D\({}_{\rm N}\)) is determined by fitting different scaling segments of the logarithmic curve of spatial and temporal bond length \(d_{\rm ST}\)-bond length frequency N\({}_{\rm d}\), as depicted in Figure 10. Each scale-free interval can then be categorized on the basis of different values of D\({}_{\rm N}\), enabling the resolution of feature key lengths. When \(d_{\rm ST}\leq 16\)ST mm, the fractal dimension D\({}_{\rm N}\approx 1.54\); conversely, when \(d_{\rm ST}>16\)ST mm, a significant change occurs, with the fractal dimension D\({}_{\rm N}\approx 4.96\). At this juncture, \(d_{\rm ST}=16\)ST mm serves as a pivotal point for discerning shifts in the frequency distribution of key lengths, denoting the characteristic key length (\(d_{c}\)) as 16 ST mm.