## 4 Results and Discussion
### Wavelength Requirements Analysis  
Figures 1 and 2 show the average number of wavelengths \(N_{\lambda}\) required to satisfy a full-mesh logical topology as a function of different topological parameters. These are based on a set of 7000 random networks with 20 and 40 nodes, an average node degree of \(<\delta>\) = 3, 4 and 5, and uniform traffic. From these figures, one can see that the wavelength requirements, \(N_{\lambda}\), decrease as algebraic connectivity and the robustness increase. This behavior is expected since \(a_{G}\) is related to \(R_{G}\), in the sense that higher robustness values indicate the availability of more node-disjoint paths. Therefore, as the number of available paths increases, the wavelength requirements decrease. In contrast, the opposite trend is observed when the average number of hops per demand increases. This occurs becausea network with high \(<h>\) values implies that, on average, each traffic demand travels over more links, increasing link loads and potentially leading to wavelength conflicts on these links. Finally, from Figure 1b it appears that wavelength usage does not exhibit any correlations with the spectral radius, and no clear trend can be identified. By focusing on Figure 1a, one can see that the dependence of \(N_{\lambda}\) on algebraic connectivity approximately follows a power-law variation for each network size. Furthermore, the figure also shows that the slop of the power-law remains almost constant as the number of nodes increase from 20 to 40. This behavior was previously identified in [9] and allowed the authors to derive a formula to estimate the wavelength requirements in terms of algebraic connectivity. In our work, we aim to derive an alternative formula that may provide additional insights or improvements over the previous one. An additional trend, already identified in [8] and observable in Figure 1a, is the decrease in the wavelength usage as the average node degree increases. This is the consequence of the network becoming more connected, leading to shorter paths between node pairs. In order to derive an alternative formula, we obtained Figure 3 by generating a set of 27,000 random networks, with the number of nodes varying from 10 to 50. The linear trend lines indicate that the power-law slope remains practically unchanged as \(N\) increases, confirming the trend previously referred. Also, they suggest that as the number of nodes grows linearly, the fitted curves experience a linear upward translation along the y-axis. Figure 1: Wavelength requirements considering spectral parameters for varying number of network nodes and average node degrees: (**a**) Algebraic connectivity; (**b**) Spectral radius. Figure 2: Wavelength requirements considering non-spectral parameters for varying numbers of network nodes and average node degrees: (**a**) Robustness; (**b**) Average number of hops per demand. The observations above may form the basis for deriving a formula to predict the number of wavelengths in terms of algebraic connectivity. The first step in this derivation consists of obtained equations of the trend lines corresponding to different values of \(N\) using least squares (LS) curve fitting, a widely used regression technique. Using these equations, along with a new level of fitting, we arrive at the following prediction formula for the wavelength requirements:  
\[\hat{N}_{\lambda}=10^{nN^{2}+bN+c}\times a_{G}^{d} \tag{4}\]  
where the regression coefficients are \(a=-4.19\times 10^{-4}\), \(b=4.84\times 10^{-2}\), \(c=5.39\times 10^{-1}\) and the power law slope \(d=-0.57\). For comparison purposes, ref. [9] presents the same formal equation but with different parameter values:  
\[\hat{N}_{\lambda min}=10^{nN^{2}+bN+c}\times a_{G}^{d} \tag{5}\]  
where \(a=-2.93\times 10^{-4}\), \(b=3.15\times 10^{-2}\), \(c=5.72\times 10^{-1}\) and \(d=-0.8\). We used the notation \(\hat{N}_{\lambda min}\) in (5) because the number of wavelengths used to derived this equation was obtained using a lower bound.