## 4 Simulation-Based Performance Evaluation
### Comparison and Discussion  
First, we evaluate whether FTC is able to trace the \(ND_{ref}\) or \(k\). As illustrated in Figure 4, the average node degree is very close to the desired node degree when \(k=2\) as the number of nodes varies from 50 to 90. When \(k\) is bigger, the average node degree is slightly lower than expected, because it is less likely to get more neighbors. For instance, with the same communication range, the probability of having 4 neighbors is obviously lower than having only 2 neighbors. In addition, the maximum communication range is limited. We can conclude from Figure 4 that FTC is able to achieve desired average node degree \(k\) without knowing the locations of neighbors. Later, we will show that only k-Neighbor algorithm can achieve desired node degree as FTC, but k-Neighbor requires location information of neighbors. Second, we compare FTC with other algorithms. Figure 5 shows an instance of the topology after running different topology algorithms on the same network when \(k=3\) and \(n=65\). Figure 5 indicates that the network is very well connected, because each node runs at the maximum communication range, but it also causes severe signal inferences in the network, as well as too much energy consumption. LTRT in Figure 5d shows the second most connected network. FLSS in Figure 5c, k-Neighbor in Figure 5b and FTC in Figure 5e are quite similar. They preserve network connectivity with much lower amount of links between nodes, which means that much less signal inferences are introduced. Figure 5 implies that FLSS, k-Neighbor and FTC demonstrate a better tradeoff between network connectivity and link quality. FLSS may produce directional links. We only plot the undirected links, so the resulting network running FLSS shown in 5c is not a \(3-connected\) network, the original network shown in Figure 5a is at least a \(3-connected\) network though. Figure 4: Different desired node degree k.  
Third, we further study the average node degree. Figure 6 illustrates the average node degree for all algorithms when the number of nodes varies from 50 to 90. The average node degree increases when the number of nodes increases if running NONE algorithm, but other algorithms are able to keep the node degree at a much lower level. Particularly, FTC and k-Neighbor are not distinguishable in Figure 6, and they are the best algorithms that are able to trace \(k\). k-Neighbor is able to get the exact \(k\), because it  
Figure 5: Example of resultant topology (n = 65, k = 3). (**a**) NONE; (**b**) k-Neighbor; (**c**) FLSS; (**d**) LTRT; (**e**) FTC. calculates where the \(k-th\) nearest neighbor is, based on already known distance information. FTC, on the contrary, is able to reach the desired \(k\) (see also Figure 4), but without knowing the neighbor locations. LTRT's node degree is the second higher. As shown in previous Figure 5d, LTRT shows its superiority over FTC, FLSS, and k-Neighbors concerning the network connectivity. Moreover, it is worth noting that the communication range is proportional to the energy consumption of wireless sensor nodes. Therefore, a lower average communication range implies a lower energy consumption, which is a significant performance when designing WSNs. Figure 7 shows the average communication range. Recall that the maximum communication range of each node is 30 m in the simulation. On the one hand, apart from NONE algorithm (because the communication range of each node is fixed at maximum), other algorithms' average communication range decreases along with the number of nodes in the network increasing, because higher node density indicates that it is more likely to have the desired node degree with lower communication range. With the number of hops increasing, the energy consumption due to wireless transmission decreases, but the energy consumption due to forwarding messages increases. Overall, though, the energy is still saved, because the energy spent on wireless transmission is orders of magnitude higher than that spent on computation.