# A Spectral Gap-Based Topology Control Algorithm for Wireless Backhaul Networks
## 5 Experimental Results
### Performance Evaluation  
Figures 6-9 show the performance attained by SGTC when increasing the network size from 100 to 200 stations; MSBs are placed either at random or by solving the related k-center problem to minimize their covering radius; and for values of the stations' radio ranges \(R_{M}\in\{0.3,0.5\}\) and \(R_{S}\in\{0.15,0.2\}\). In order to have a performance baseline, we include the results of a traditional Voronoi-based algorithm that adds to the network all the feasible links connecting two MBSs and the feasible links connecting every SC to its closest MBS. In the figures, each point corresponds to the average result computed over 20 independent experiments, and the error bars correspond to the standard deviation value. Missing points in the graphs indicate that the corresponding algorithms could not connect the network in at least one of the twenty independent experiments. Figure 4: Network robustness with increasing Laplacian Spectral Gap (LSG, \(\lambda_{2}\)). Figure 5: Network cost with increasing Laplacian Spectral Gap (LSG, \(\lambda_{2}\)). From Figure 6, we can observe that SGTC outperforms the traditional Voronoi-based algorithm by consistently achieving a higher value of network entropy \(S_{G}\). The figure also shows that the network entropy benefits from having higher values of radio ranges. The reason is that the number of feasible links increases as the radio range increases, which also increases the space of feasible topologies that SGTC can investigate. From the figure, we can also notice that the MBS placement has no noticeable effect on this metric. Lastly, this figure does not include results for the Voronoi-based algorithm when \(R_{M}=0.3\) and \(R_{S}=0.15\) because the algorithm did not produce connected networks. Figure 7 shows the network diameter \(D_{G}\) attained by the algorithms under the different scenarios. Similar to the case of the network entropy, SGTC produces the networks with the smallest diameters when \(R_{M}=0.5\) and \(R_{S}=0.2\), which indicates that the algorithm can take advantage of a large space of feasible topologies to find well-connected networks. For \(R_{M}=0.3\) and \(R_{S}=0.15\), the Voronoi-based algorithm achieves smaller values of network diameters by predominantly using the sub-network consisting of links between MBSs. This, however, has the disadvantage of concentrating the traffic in the MBSs. As for the previous metric, this figure shows that the MBS placement has no noticeable effect on the network diameter. Figure 6: Network entropy \(S_{G}\) with increasing network size. Figure 7: Network diameter \(D_{G}\) with increasing network size. Figure 8 shows that the network robustness \(R_{G}\) follows a similar trend to that of the previous metrics. SGTC produces the most robust networks when the space of feasible topologies is larger, namely, when \(R_{M}=0.5,R_{S}=0.2\), and there are many stations. However, unlike the case of the previous metrics, SGTC finds more robust networks for the case of \(R_{M}=0.3,R_{S}=0.15\) when the MBSs are placed using the k-center algorithm. The reason is that by reducing the covering radius of the MBS, the probability that all the SCs are within the radio range of an MBS is increased, which increases the probability of having more paths connecting the SCs to the rest of the network. Lastly, Figure 9 shows that the network cost \(C_{G}\) attained by the traditional Voronoi-based algorithm is smaller than that of SGTC. This result was expected because the Voronoi-based algorithm's topologies include fewer links; the SCs are only connected to their closest MBS. As we saw in the previous metrics, this reduced cost comes at the expense of fragile, weakly connected networks that tend to concentrate traffic in the MBSs, increasing the probability of contention hot spots. This behavior becomes apparent when \(R_{M}=0.3\), \(R_{S}=0.15\) because the links selected by the Voronoi-based algorithm are not enough to connect the network.