# Statistical Analysis and Modeling for Optical Networks
## 2 Literature Review on Statistical Analysis and Modeling for Networks
### Statistics and Network Characterization  
Complex networks are frequently observed across many sectors, such as biology, engineering, medical science, psychology, astronomy, and mathematics. The statistical analysis and modeling of complex networks has become a cornerstone in understanding diverse systems, from biological interactions to technological infrastructures, due to their inherent complexity and non-trivial topologies. These networks, composed of interconnected nodes and edges, exhibit emergent properties that necessitate statistical approaches for their characterization. Key statistical measures, such as degree distributions, clustering coefficients, and average path lengths, are employed to uncover fundamental structural properties, revealing phenomena like small-world effects and scale-free behavior, which deviate significantly from random network models [5]. Statistical modeling techniques are crucial for inferring network formation mechanisms and providing insights into the underlying processes that shape network evolution [6]. Further analysis includes the exploration of network dynamics, where temporal changes in network structure are modeled to understand network resilience, propagation phenomena, and adaptation to external stimuli [44]. Moreover, the application of statistical inference and machine learning techniques enables the prediction of network behavior and the identification of influential nodes, contributing to practical applications in network design and optimization [20; 21; 22; 23; 69]. The synthesis of these statistical methodologies allows for a comprehensive understanding of complex network behaviors, offering a powerful toolset for researchers across various disciplines to unravel the intricacies of interconnected systems [7]. Statistical distributions are fundamental tools for modeling and analyzing the complex behaviors, network requirements, and predictive modeling in optical networks [27; 28; 29; 30; 31; 32; 33]. The General Extreme Value (GEV) distribution in particular excels at characterizing link length variations, a critical aspect of network planning and optimization [28; 31]. Gaussian distributions are commonly employed to represent additive noise in optical signals, while  
Use of different areas for link-related parameter estimation,
* Use of convex area for optical networks,
* Estimation of convex area,
* Estimation of link-related parameters from the GEV distribution for link lengths,
* Finding the distribution of shortest path lengths,
* Estimation of the parameters of the Johnson \(\mathrm{S_{B}}\) distribution for shortest path lengths,
* Applications of Johnson \(\mathrm{S_{B}}\) distribution for shortest path lengths. We elaborate how these concepts and techniques have been utilized in this review. This paper adopts a systematic and comprehensive research methodology to synthesize existing knowledge and identify emerging trends in the field. The investigation commences with a foundational exploration into the classification of optical networks, delineating their diverse architectures, operational principles, and technological underpinnings to establish a clear definitional framework. This will be followed by a detailed characterization of optical networks, focusing on key performance indicators, architectural components, and their inherent complexities that necessitate statistical analysis. A core component of the methodology involves an in-depth review of nodal analysis and modeling of optical networks, examining various approaches to characterize node behavior, capacity, and resource allocation strategies within optical infrastructures. Concurrently, the paper undertakes a rigorous link analysis and modeling of optical networks, investigating methodologies for understanding link characteristics, impairments, and their impact on network performance. A significant focus is placed on the application of statistical distributions for modeling optical network parameters.