# Complex Network Modeling and Analysis of Microfracture Activity in Rock Mechanics
## 1 Introduction  
The study of rock fracture mechanics has evolved significantly since [PERSON]'s pioneering work in the 1920s, which attributed brittle rock failure to pre-existing microcracks [1]. Subsequent research, particularly from the 1960s onward, categorized the rock failure process into five stages--crack closure, elastic deformation, stable growth, unstable propagation, and final failure--based on stress-strain curve analysis [2, 3, 4, 5, 6] These stages were later refined through the standardization of acoustic emission (AE) rates, offering deeper insights into microcrack evolution and cohesion loss [7, 8]. However, traditional methods like stress-strain analysis and AE signal monitoring have limitations in revealing the complex spatiotemporal evolution and correlation of microfractures during rock deformation and failure. While stress-strain analysis determines rock deformation stages, it does not reveal failure locations or microfracture activity. AE signals during rock deformation and failure offer valuable information for identifying failure locations and stages. The amplitude and frequency of AE signals correlate with natural earthquake magnitude and frequency, prompting the application of seismological methods to rock mechanics [9, 10, 11]. AE technology aids in predicting rock deformation and failure [12, 13, 14, 15, 16]. However, current research on rock microfractures primarily focuses on fracture description and statistical analysis of AE events, energy, and micromechanical properties, lacking correlation analysis of microfractures [17, 18, 19, 20, 21, 22]. Network science, a method for analyzing large datasets, captures inherent correlations and applies to various disciplines. The introduction of small-world networks by [PERSON] and [PERSON] and scale-free networks by [PERSON] and [PERSON] advanced complex network theory [23,24]. These networks exhibit small-world, self-organization, self-similarity, and scale-free properties, enabling topological analysis. Complex networks are widely used in studying seismic activity due to the clustering and correlation of earthquakes [25, 26, 27, 28, 29, 30]. Microfractures during rock deformation and failure are a form of microseismic activity, making complex network analysis relevant to rock mechanics. In rock mechanics, complex network theory has been applied to understand microfracture behavior. [PERSON] and [PERSON] used it to simulate microfracture propagation, while [PERSON] et al. analyzed rock fracture network parameters like degree distribution and clustering coefficients [31,32]. [PERSON] et al. further explored the dynamic failure characteristics of rock under different preloading conditions [33]. Recently, [PERSON] et al., [PERSON] et al. and [PERSON] et al. combined experiments and simulations to study rock fracture network evolution, linking microfracture nucleation and propagation to network topology [34, 35, 36]. Drawing on complex network applications in seismological research, this study innovatively models rock microfractures as a complex network, treating fracture points as nodes and their interactions as edges [37,38]. Our approach offers several distinct advantages over previous methods. First, by integrating complex network theory with spatiotemporal single-chain groups, fractal theory, and graph theory, we provide a more comprehensive and dynamic understanding of microfracture evolution. This integration allows us to capture the intricate spatiotemporal relationships between microfractures, which traditional methods often fail to reveal. Second, our dynamic network model can track the evolution of microfractures in real-time, offering a more accurate and timely prediction of rock failure. This is particularly crucial for ensuring the safety of engineering projects where early warning of rock instability is essential. Third, by analyzing network parameters such as the average degree and clustering coefficient, we can identify critical transitions from stable to unstable states in rocks.