# Statistical Analysis and Modeling for Optical Networks
## 2 Literature Review on Statistical Analysis and Modeling for Networks
### Statistics and Network Characterization
Specifically, the methodology will critically assess the widespread use of the GEV distribution for link length modeling, analyzing its theoretical underpinnings, empirical validation, and limitations across diverse network scenarios. This approach naturally leads to an examination of the use of different areas for link-related parameter estimation, exploring how geographical, topological, and population density factors influence the statistical distribution of link lengths and other relevant parameters. The review delves into the novel concept of the use of convex area for optical networks, investigating its relevance in network planning, resource optimization, and resilience. This involves an examination of methodologies for the estimation of convex area in various optical network configurations. Furthermore, the methodology also details the process of estimation of link-related parameters from the GEV distribution for link lengths, demonstrating how this distribution can be leveraged to infer critical network characteristics. Building upon the understanding of individual links, the research will proceed to explore methods for finding the distribution of shortest path lengths within optical networks, a critical aspect for routing and latency analysis. This involves a comprehensive review of statistical techniques employed for estimation of shortest path lengths in various network topologies. A dedicated segment of the methodology focuses on the estimation of the parameters of the Johnson \(\mathrm{S_{B}}\) distribution for shortest path lengths, highlighting its applicability and advantages in accurately characterizing the often skewed and bounded nature of shortest path distributions in optical networks. Finally, the review comprehensively discusses the applications of the Johnson \(\mathrm{S_{B}}\) distribution for shortest path lengths, demonstrating its utility in network design, performance prediction, and optimization strategies for future optical communication systems. The research primarily relies on a systematic literature review of peer-reviewed articles, conference proceedings, and reputable academic databases, employing keywords derived from the key phrases to ensure comprehensive coverage. The synthesis of findings involves a critical comparative analysis of different modeling approaches, their assumptions, limitations, and the practical implications of their findings for the design and operation of robust and efficient optical networks. The research methodology of this review paper critically examines the application of various statistical techniques and models to understand and predict network requirements. For each identified tool, a systematic analysis has been conducted, starting with its mathematical formulation and underlying assumptions. This involves delving into the probabilistic or statistical principles governing each model, such as the parameters of the GEV distribution for link lengths (e.g., location, scale, and shape parameters) and the Johnson S\({}_{\text{B}}\) distribution for shortest path lengths (e.g., shape, location, and scale parameters). Understanding these assumptions is crucial for evaluating the validity and limitations of applying a specific model to diverse optical network topologies and operational scenarios. Next, the applications in optical network analysis for each tool has been thoroughly investigated. This includes how these models are employed for tasks such as network planning, resource dimensioning, performance prediction, and cost estimation. For instance, the GEV distribution's utility in characterizing the variability of link lengths due to geographical constraints or deployment practices will be explored, while the Johnson S\({}_{\text{B}}\) distribution's role in understanding shortest path length distributions, which directly impacts latency and routing efficiency, will be detailed. Subsequently, the insights gained regarding network characteristics and performance from the application of these statistical models have been explained.