## 2 Network Aspects and Survivability  
### Network and Topological Parameters Characterization  
Transparency has been a widely accepted assumption in the design of optical networks, particularly in backbone segments, since the advent of optical amplifiers and WDM technologies [16]. In line with this our analysis, assumes that optical networks are transparent, meaning that optical signals are transmitted from the source to the destination without optoelectronic conversion, maintaining their optical nature throughout the entire path. Therefore, in these networks, all node functionalities (such as multiplexing, switching, etc.) take place in the optical domain, and the node structure is based on reconfigurable optical add-drop multiplexers (ROADMs) [1]. A ROADM is responsible not only for locally adding and dropping optical channels but also for enabling optical bypass by switching optical channels from incoming to outgoing optical links. An optical link represents a physical interconnection between two nodes and is implemented using a pair optical fibers and optical amplifiers adequately spaced to compensate for the fiber losses. Each optical fiber supports WDM signals, meaning it carries a specific number of optical channels, with each channel being characterized by its own wavelength. As a result, the number of optical channels is determined by the number of wavelengths \(N_{\lambda}\) per link.  
In this work, we also assume that optical networks are static, which is a common assumption in optical backbone networks. In such networks, optical channels are pre-established offline according to a fixed traffic, matrix and remain unchanged during operation.  
The physical topology of an  
eigenvalue of matrix \(L\) and measures how well the graph is connected, while the latter corresponds to the largest eigenvalue of matrix \(L\) and quantifies how node degree imbalance is structured across the network.  
The non-spectral parameters considered in this work include the average node degree \(<\delta>\), the average number of hops per demand \(<h>\) and the robustness \(R_{G}\). The first parameter is defined as  
\[<\delta>=\frac{1}{N}{\sum_{i=1}^{N}}\delta_{i}. \tag{1}\]  
The number of hops per demand corresponds to the number of links traversed by a given traffic demand, after it has been mapped to a path using a specific routing strategy. Assuming a shortest path routing strategy, the hop matrix \(H=\left[h_{i,j}\right]\) is an \(N\times N\) matrix, where all diagonal elements are zero, and each elements \(h_{i,j}\) denotes the minimum number of hops traversed by the traffic demand between nodes \(v_{i}\), and \(v_{j}\). Considering a full-mesh logical topology the average minimum number of hops per demand is given by  
\[<h>=\frac{2}{N(N-1)}{\sum_{i=i}^{N-1}}\sum_{j=i+1}^{N}h_{i,j}. \tag{2}\]  
A full-mesh logical topology means that each node is logically connected to every other node in the network. This corresponds to a uniform traffic pattern, where there is one traffic demand between each pair nodes, a general assumption in this work.  
The starting point for evaluating network robustness is computing the redundancy matrix \(R=\left[r_{i,j}\right]\). This is a \(N\times N\) matrix, where each element is defined as the node disjoint pairwise connectivity and corresponds to the maximum number of nodes disjoint paths between nodes \(v_{i}\), and \(v_{j}\)[17]. This value is crucial for designing protection paths in optical networks [18], a topic that will be addressed in the following subsection. The robustness is defined as the average node-disjoint pairwise connectivity and is given by [10]  
\[R_{G}=<r>=\frac{1}{N(N-1)}{\sum_{i=i}^{N}}{\sum_{j=1}^{N}}r_{i,j}. \tag{3}\]  
To perform statistical analysis, large data sets are essential. One effective way of obtaining such data sets is to generate numerous network topologies, which can be done using random graphs designed to adequately describe the characteristics of real-world networks, as explained in Section 4.1.