# A Spectral Gap-Based Topology Control Algorithm for Wireless Backhaul Networks
## 6 Conclusions  
We introduced the spectral gap-based topology control (SGTC) algorithm for wireless backhaul networks that uses simulated annealing guided by the value of the Laplacian  
Figure 8: Network robustness \(R_{G}\) with increasing network size.  
Figure 9: Network cost \(C_{G}\) with increasing network size.  
Spectral Gap (LSG) to look for expander-like graphs, which are sparse graphs with strong connectivity properties such as small diameter, large value of the cardinality of its global minimum cut-set, and high shortest-path diversity. These are desirable properties for backhaul networks composed of Macro Base Stations and Small Cells because sparse topologies require less energy to establish and maintain their links, the diameter of a network impacts the upper bound of the end-to-end delay, the value of the cardinality of the global minimum cut-set determines the network robustness against link failures, and the shortest-path diversity promotes seamless traffic offloading from the MBSs to the Small Cells.  
We presented network entropy as a measure of shortest-path diversity, which determines the amount of information about how data traffic is distributed across a particular network, where high values of network entropy indicate that data traffic will be evenly distributed across the network.  
A series of experiments confirmed that the three connectivity metrics--diameter, robustness, and network entropy--improve as the value of the LSG increases, notably without requiring a significant increase in the number of links. This trend becomes more pronounced as the radio range of the stations increases because it increases the size of the space of feasible solutions. Our experimental results also showed that the topologies produced by SGTC are superior to those provided by a traditional Voronoi-based algorithm and that carefully placing the MBSs can improve the network robustness.  
Future work includes conducting analytical and simulation-based analyses to determine the impact of expander-like topologies on the performance of upper-layer network functions, such as channel access and routing. We also plan to extend our algorithm to dynamically adapt the network's topology to respond to the instantaneous traffic conditions. This is particularly important in the context of modern and future Internet applications where data traffic is bursty in nature, with large periods of low or no activity followed by sudden surges of high-intensity traffic.  
Conceptualization, S.J.G.-A., R.M.-M., S.A.P.-C. and M.E.R.-A.; data curation, S.J.G.-[PERSON] and [PERSON]-C.; formal analysis, [PERSON]-[PERSON], [PERSON]-[PERSON], [PERSON]-[PERSON] and [PERSON]-[PERSON]; investigation, [PERSON]-[PERSON] and [PERSON]-[PERSON]; methodology, [PERSON]-A., [PERSON]-M., [PERSON]-C. and [PERSON]-[PERSON]; software, [PERSON]-A. and [PERSON]-C.; supervision, [PERSON]-M.; validation, [PERSON]-A., [PERSON]-M. and [PERSON]-C.; visualization, [PERSON]-[PERSON], writing--original draft, [PERSON]-[PERSON], [PERSON]-[PERSON], [PERSON] and [PERSON]; writing--review and editing, [PERSON], [PERSON], [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the \"Consejo Nacional de Humanidades Ciencias y Tecnologias\" (CONAHCyT) of Mexico and the \"Instituto Politecnico Nacional\" under grant SIP 20240950.  
The source code used in Section 5 is available at [[https://github.com/Mysdafb/sgtc](https://github.com/Mysdafb/sgtc)]([https://github.com/Mysdafb/sgtc](https://github.com/Mysdafb/sgtc)) (accessed on 24 January 2024).  
We thank the technical support provided by the Data Science Lab of CITEDI-IPN.  
[PERSON] is employee of corteza.ai company. The authors declare no conflicts of interest.