## 4 Simulation Results and Analysis
### Two-DimensionalSimulation Parameters and Results  
The proposed algorithm was compared with classical DV-Hop, DV-Hop based on simulated annealing, DV-Hop based on particle swarm optimization (PSO), and DV-Hop based on the sparrow search algorithm (SSA). In the simulations, the default parameters were set as follows: 100 nodes, 20% beacon nodes, and a communication radius of 30 m. A total of 100 simulations were conducted, and the average value was calculated to minimize random errors. The normalized average localization error (ALE) is given by Equation (38):  
\[ALE=\frac{\sum_{i=1}^{N}\sqrt{\left(x_{ist}-x_{i}\right)^{2}+\left(y_{ist}-y_{ i}\right)^{2}}}{N\cdot R} \tag{38}\]  
where \(\left(x_{ist},\ y_{ist}\right)\) represents the estimated location of the unknown nodes, and \(\left(x_{i},\ y_{i}\right)\) represents the true location of unknown nodes. The simulation parameters were set as shown in Table 1.  
The simulation scenarios, respectively, used a 2D rectangular area of 100 m \(\times\) 100 m and anisotropic network areas with X-shape, H-shape, S-shape, C-shape, and O-shape topologies. The node topology distributions of the 2D scenarios are shown in Figure 4.  
The proportion of beacon nodes increased from 10% to 40%, and the simulation results of each algorithm are shown in Figure 5. The proposed algorithm outperformed the others across all scenarios with varying beacon node ratios, demonstrating better performance in scenarios with a higher proportion of beacon nodes. This improvement is attributed \begin{table}
\begin{tabular}{c c} \hline \hline
**Simulation Parameters** & **Value** \\ \hline Scenario size & 100 m \(\times\) 100 m \\ Communication radius & 20–50 m \\ Ratio of beacon & 10–40\% \\ Number of nodes & 50–150 \\ DOI & 0–0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Two-dimensional simulation parameters. Figure 4: Node distribution maps in 2D scenarios. (**a**) Rectangular area. (**b**) C-shaped area. (**c**) S-shaped area. (**d**) O-shaped area. (**e**) X-shaped area. (**f**) H-shaped area. to the more-connected topology, which enables greater node coverage and more detailed network information. The total number of nodes increased from 50 to 150, and the simulation results of each algorithm are shown in Figure 6. As the number of nodes increased, the proposed algorithm demonstrated robustness, remaining less affected by network topology and maintaining high accuracy even with complex node distributions. Figure 5: Comparison of localization error under different beacon ratios in 2D scenarios. (**a**) Rectangular area. (**b**) C-shaped area. (**c**) S-shaped area. (**d**) O-shaped area. (**e**) X-shaped area. (**f**) H-shaped area. Figure 6: Comparison of localization error under different numbers of nodes in 2D scenarios. (**a**) Rectangular area. (**b**) C-shaped area. (**c**) S-shaped area. (**d**) O-shaped area. (**e**) X-shaped area. (**f**) H-shaped area. The communication radius increased from 20 m to 50 m, and the comparative results of the algorithms under different topological distributions are shown in Figure 7. The proposed algorithm continues to exhibiting higher accuracy in small-scale anisotropic networks, demonstrating its high stability and resilience to changes in network size. The computational complexity of the proposed algorithm has been analyzed and compared, with the specific runtime of different algorithms in the 2D scenarios presented in Table 2. Compared to other algorithms, the proposed algorithm exhibits a slight increase in complexity. However, since the algorithm mainly operates on the beacon nodes, which are typically equipped with sufficient energy, its impact on the overall network remains minimal.