# Statistical Analysis and Modeling for Optical Networks
## 5 Statistical Analysis and Modeling for Optical Networks
### Dimensioning in Optical Networks
#### 5.3.3 Convex Area of Optical Networks  
Coverage areas in optical networks can be classified into three types: exact area (bounded by the topology's boundary links), convex area (the largest convex set formed by the network nodes), and geographical area (determined by the physical region where the network operates). Among these, the convex area is the most suitable for link-related estimations, as it enables accurate calculations of average link lengths. Using the probability density function of link lengths, various link-related parameters can be derived. While geometric measure theory provides a method for estimating a graph's approximate area, it requires numerous inputs and does not guarantee exact results. Additionally, such methods rely on complete knowledge of the network topology, making them impractical for optical networks in which exact areas fail to yield accurate estimations. Instead, coverage areas facilitate rapid estimation of optical network parameters, even without  
Figure 6: Exact area of a star shaped network (APBQCRDSETA) and its convex area (the pentagon shaped envelope, ABCDEA). The convex area of an optical network can be estimated from the circumferential ellipse of the optical network [32].  
full network details. The eccentricity of a network's ellipse (or ellipticity) is particularly useful in studying network properties, aiding in the determination of parameters such as diameter, radius, and shortest paths. These estimations play a crucial role in fields like networking, distributed computing, and graph theory, where rapid calculations are essential. Furthermore, rapid estimation methods assist in determining network diameters, shortest paths between nodes, and cost-related factors such as capital expenditure (CAPEX) during optical network planning.  
For regular structures like squares, rectangles, or regular polygons, a circumcircle can always be identified. However, in irregular graphs, which are common in optical networks, finding such a circle is not always feasible. It is very much the same for all large optical networks [32]. Instead, circumferential ellipses (CEs) are used, which is possible for all planar networks (meaning when a planar topology of a network is available, its CE can be estimated). The CE of an optical network is the smallest possible ellipse that encloses the entire network, touching at least three nodes (it is possible that the CE may touch more than 3 points). The only points of intersection between the CE and the optical network topology are these finite nodal points. If the CE is reduced in size, it will inevitably intersect at least one link (or at least two links in a survival optical network), ensuring that every network topology has a unique CE. These ellipses serve as valuable tools in analyzing network topology, optimizing link estimations, and improving network design efficiency. In Figure 6, we have shown the convex area (the regular pentagon ABCDEA) of a star-shaped network APBQCRDSETA. In Figure 7, we have shown the topology (in blue) and CE (in red) of the Bulgarian Research and Education Network (BREN) [32; 77]. It has been found that the convex area and the area of the CE of optical networks follow a linear trend.