## 4 Algorithm Analysis
This figure illustrates a comparative analysis of the PSD between the MCD-BP and other algorithms, such as LFC, NPC, U-turn, and MARA, applied to real-world topologies, namely, AttMpls, B2004, Cernet, NJLATA, USLD and V2008. Figure 7: Comparison of the path stretching degree of the algorithms. The X-axis delineates the names of the actual topologies, while the Y-axis corresponds to PSD values. The proximity of the PSD value to 1 indicates superior performance. This figure illustrates a comparative analysis of the PSD between the MCD-BP and other algorithms, such as LFC, NPC, U-turn, and MARA, applied to real-world topologies, namely, Abilene, Agis, Ans, A19719, A19723 and A19728. remains comparatively stable and lower as the topology graph increases in size. The PSD values obtained using the LFC algorithm closely resemble those derived from the U-turn algorithm. For instance, in the 20-4 topology, both algorithms yield a PSD value of 1.75. However, in the 60-4 topology, the PSD for the NPC algorithm is notably lower at 1.02, which is lower than that of other algorithms. In Figure 10, the PSD of the MARA algorithm is still higher than that of other algorithms and increases with the expansion of the topological scale. In the 200-2 topology, the differences among the five algorithms are more pronounced, with the highest PSD being 2.16 for the MARA algorithm. This is followed by 1.44 for the U-turn algorithm, 1.35 for the MCD-BP algorithm, 1.33 for the LFC algorithm, and the smallest PSD of 1.24 for the NPC algorithm. Through the above experimental data, it can be concluded that the MCD-BP algorithm proposed in this paper has a lower PSD. The subsequent Table 4 presents the specific experimental data pertaining to PSD. Figure 10: Comparison of the path stretching degree of the algorithms. The X-axis delineates the “node count-node degree” simulated topology, while the Y-axis showcases the PSD values. The proximity of the PSD value to 1 indicates superior performance. This figure depicts a comparative analysis of the PSD of the multi-node polygon simulation topologies, contrasting the MCD-BP algorithm against the LFC, NPC, U-turn, and MARA algorithms. Figure 9: Comparison of the path stretching degree of the algorithms. The X-axis delineates the “node count-node degree” simulated topology, while the Y-axis showcases the PSD values. The proximity of the PSD value to 1 indicates superior performance. This figure depicts a comparative analysis of PSD on simulated topologies with a small number of nodes, contrasting the MCD-BP algorithm against the LFC, NPC, U-turn, and MARA algorithms. ### Cross-Degree Means

The cross-degree mean refers to the average quantity of shared edges between the primary and backup paths of all nodes within a given topology. A lower cross-degree mean indicates a reduced number of common edges between the two paths, thereby enhancing the prevention of multiple path failures that could result from a single point of fault. The cross-degree mean correlates with algorithm performance; a lower cross-degree mean indicates superior algorithm performance, which is advantageous for improving network robustness. This can be observed in Figures 11 and 12. Notably, the cross-degree mean, as calculated by the MCD-BP algorithm, is predominantly 0 across various topologies, illustrating its exceptional performance relative to other algorithms. In Figure 11, the cross-degree means for the Agis topology by algorithms MCD-BP, LFC, NPC, U-turn, and MARA are 0, 2.40, 2.41, 2.34 and 3.39, respectively. Upon examining topologies A19719, A19723, and A19728, a discernible fluctuation in the cross-degree means of the algorithm becomes evident. This suggests that the intricacy of the network topology exerts a notable influence on algorithm efficiency.