# A Spectral Gap-Based Topology Control Algorithm for Wireless Backhaul Networks
## 4 SGTC: The Spectral Gap-Based Topology Control Algorithm  
This section introduces the proposed spectral gap-based topology control (SGTC) algorithm for wireless backhaul networks. The SGTC algorithm employs the information provided by the LSG to optimize the network topology with respect to the four performance metrics defined in Section 3.3. SGTC takes as input the set of network entities \(V=M\cup S\) (generically referred to as stations), an initial set \(E_{0}\) of links, functions \(l_{M}\) and \(L_{S}\) that map stations to positions in the plane, radio ranges for MBSs and SCs, and the value of \(Max\_Itera\) which determines the maximum number of iterations.  
The SGTC algorithm employs the simulated annealing (SA) meta-heuristic technique to explore the feasible solution space, seeking network topologies that yield a high LSG (\(\lambda_{2}\)). As we describe in more detail in the following paragraphs, the annealing process implements a random walk over the feasible solution space where a new candidate solution is accepted with a probability proportional to its LSG value and the current temperature of the annealing process. Initially, the temperature is set to a high value to promote a broad search of the solution space, and on each iteration, it is reduced to direct the random walk towards strictly better solutions.  
After the annealing process is completed, SGTC calculates the Pareto front of the best feasible network topologies that were visited during the process. This calculation is based on three criteria: minimizing the network diameter and maximizing network robustness and entropy. SGTC returns the Pareto efficient solution with the lowest energy cost as its output.  
\begin{table}
\begin{tabular}{l l} \hline \(G=(M\cup S,E)\) & A network. \\ \(V=M\cup S\) & Set of stations, including the set \(M\) of MBSs and the set \(S\) of SCs. \\ \(E\) & Set of unidirectional links connecting the stations. \\ \(E_{0}\) & Initial set of links in the annealing process. \\ \(l:V\rightarrow\mathbb{R}^{2}\) & Function that determines the position of the stations. \\ \(l_{S}:V\rightarrow\mathbb{R}^{2}\) & Function that determines the position of the SCs. \\ \(l_{M}:V\rightarrow\mathbb{R}^{2}\) & Function that determines the position of the MBSs. \\ \(N(u)\) & One-hop neighborhood of station \(u\). \\ \(N_{F}(G)\) & Set of one-hop feasible neighbors of network \(G\). \\ \(R_{M}\) & Radio range of the Macro Base Stations. \\ \(R_{S}\) & Radio range of the Small Cells. \\ \(N_{M}\) & Number of antenna elements in an MBS. \\ \(N_{S}\) & Number of antenna elements in an SC. \\ \(D_{G}\) & Diameter of network \(G\). \\ \(R_{G}\) & Robustness of network \(G\). \\ \(C_{G}\) & Cost of network \(G\). \\ \(S_{G}\) & Network entropy of \(G\). \\ \(f:E\rightarrow\mathbb{N}\) & Function that counts the number of shortest paths that include \(e\in E\). \\ \(A(G)=[a_{ij}]\) & Adjacency matrix of \(G\). \\ \(D(G)=[d_{ij}]\) & Degree matrix of \(G\). \\ \(L(G)\) & Laplacian matrix of \(G\). \\ \(\lambda_{2}\) & Second smallest eigenvalue of \(L(G)\) (Laplacian Spectral Gap). \\ \(Max\_Itera\) & Maximum number of iterations of the SGTC algorithm. \\ \(\alpha_{0}\) & Initial value of the temperature in the annealing process. \\ \(\alpha\) & Value of the temperature during the annealing process. \\ \(dist(l(u),l(v))\) & Euclidean distance between stations \(u\) and \(v\). \\ \(dist(l_{S}(u),M)\) & Euclidean distance between station \(u\) an its closest MBS in \(M\). \\ \(r(M)\) & Covering radius of a set \(M\) of MBS. \\ \hline \end{tabular}
\end{table}
Table 1: Summary of notation.