# A Spectral Gap-Based Topology Control Algorithm for Wireless Backhaul Networks
## 5 Experimental Results  
In this section, we present the results of a series of experiments that characterize the ability of the SGTC algorithm to find network topologies that optimize the performance metrics presented in Section 3.3, namely, that minimize the network diameter \(D_{G}\), maximize the network robustness \(R_{G}\), minimize the network cost \(C_{G}\), and maximize the network entropy \(S_{G}\). For a given network topology \(G=(V,E)\), and function \(l_{M}\) and \(l_{S}\) that assign the positions of the nodes in the plane, we compute its robustness \(R_{G}\) as the cardinality of the global minimum cut-set. To this end, we execute an instance of the Ford-Fulkerson algorithm [47] over the \(O(|V|^{2})\) pairs of nodes. To compute the network entropy, we run [PERSON]'s algorithm [48] for every pair of nodes and count the number of times (denoted by \(f(e)\)) that a given \(e\in E\) belongs to one of these shortest paths. Then, we use Equation (3) to compute the probability of using a particular edge \(e\) in the shortest path and compute \(S_{G}\) according to Equation (2). The network diameter is easily obtained from the lengths of the shortest paths of all the pairs of nodes, and the network cost is computed according to Equation (1). For the experiments presented in this section, we generate problem instances by assuming an uncorrelated position of the network entities (stations) within a normalized squared region of \(1\times 1\). Specifically, we compute each station's \(x\) and \(y\) coordinates by sampling the continuous uniform distribution with bounds \([0,1]\), which assumes an uncorrelated position of the nodes [49; 50]. As shown in Table 2, we vary the total number of stations \(|V|\) from 100 to 200, the normalized radio range of the MBSs from 0.3 to 0.5, and the normalized radio range of the SCs from 0.15 to 0.2. The number of MBSs equals \(\lfloor 0.1\times|V|\rfloor\) to have scenarios where approximately 10% of the stations are MBSs. We further assume that all the stations have enough antennas to establish all the links determined by the SGTC algorithm. The results presented in all the figures include the output of the proposed algorithm when using the following initial network topologies. 1. _Disconnected:_\(E_{0}=\emptyset\). The initial network has no links. 2. _Feasible fully connected:_\(E_{0}=\{e=(u,v)\in V\times V:e\text{ is }feasible\}\). The initial network contains all the feasible links. 3. _Voronoi topology_[51]:_\(E_{0}=\{e=(u,v)\in M\times M:e\text{ is }feasible\}\bigcup\{e=(u,v)\in S \times M:v=\arg\min_{v\in M}\{dist(l(u),l(v))\}\wedge e\text{ is }feasible\}\). The initial network contains all the feasible links connecting MBSs among themselves, and the SCs are connected to their closest MBSs if the corresponding link is feasible. In all the experiments, the maximum number of iterations of the SGTC algorithm equals 10,000, and the initial temperature \(\alpha_{0}\) of the simulated annealing process equals 1.0. Each experiment is run for 20 independent seeds. The experimental workflow composed of the instance generator, the SGTC algorithm, and a set of algorithms that compute the performance metrics is illustrated in Figure 1. The instance generator algorithm takes the number of stations \(|V|\), the percentage of MBSs, and the type of MBS placement (random or k-center) as input parameters. It returns functions \(I_{M}\) and \(I_{S}\) that determine the locations of MBSs and SCs, respectively. Next, the SGTC algorithm uses the functions \(I_{M}\), \(I_{S}\), the initial set of links \(E_{0}\), and the values of the radio ranges \(R_{M}\) and \(R_{S}\) as input parameters to compute the resulting topology \(G=(V,E)\) of the backhaul network. The initial temperature \(\alpha_{0}\) and the number of iterations \(Max\_Itera\) are the hyper-parameters of SGTC.