# A Spectral Gap-Based Topology Control Algorithm for Wireless Backhaul Networks
## 2 Related Work  
In the past, topology control mechanisms have been studied mainly in Wireless Mesh Networks, Wireless Mobile Ad Hoc Networks, and Wireless Sensor Networks. The main objective of this type of mechanism is to modify the network topology to induce connectivity properties and improve the overall network performance [23; 24].  
Recently, topology-control mechanisms have also become relevant in wireless backhaul networks because they can help improve efficiency, promote traffic offloading from the core network, and avoid bottlenecks in the backhaul network. For instance, in [25], the authors propose planning schemes for microwave-based wireless backhaul networks based on tree topologies that jointly optimize multiple performance metrics, such as traffic hops (delay), long links, link crosses, small angles, and the sum of the link distances. In [26], the authors propose an iterative heuristic algorithm that constructs topologies that maximize the total backhaul flow in wireless multi-hop backhaul networks composed of SCs and networked flying platform (NFP) hubs. In [27], the authors show that if the topologies for Ultra-High-Dense Networks (UDN) combine MBSs and a dense tier of SCs, they can accomplish objectives of 5G networks such as meeting traffic demand while providing seamless coverage, high data rate for low-mobility users and spatial reuse. In [10], a comprehensive survey about the architecture, performance metrics, and guidelines of 5G networks is presented. The authors show that accommodating Massive MIMO with Small Cells improves performance metrics such as energy efficiency, coverage, capacity, resource efficiency, and spectral efficiency.  
In the context of topology control schemes, some works have proposed inducing expander-like topologies in networks [28; 29]. The rationale behind these works is that _expander_ graphs are sparse graphs with strong connectivity properties [20], and therefore, networks with expander-like topologies will tend to be robust even if they do not have a large number of links. In [30], the authors investigate the relationship between the properties of expander topologies implemented by wireless networks (called Wireless Expanders ) and the performance of such wireless networks in terms of enabling efficient broadcast solutions. In [31; 32], the authors model dynamic peer-to-peer (P2P) networks using expander graphs in which the spectral gap property is preserved under node insertions and deletions. In [33], the authors show that communication networks with expander-like topologies tend to be flexible and easy to reconfigure and optimize.  
Simulated annealing (SA) has been successfully used to solve complex problems in wireless networks. For example, in [34], the authors use SA for wireless backhaul planning to maximize system performance regarding network throughput and delay while reducing infrastructure utilization. Similarly, in [35], the authors propose an SA-based algorithm to solve a power consumption problem. The idea is to use SA to minimize the total power consumed by MBSs and SCs serving access and wireless backhaul links. Other works on the use of SA to optimize the performance of wireless networks are [36; 37; 38]. All these works show the effectiveness of simulated annealing as an optimization technique in the context of wireless networks.  
This literature review shows that, unlike our proposal that uses the LSG to find expander-like topologies that simultaneously optimize the network's energy cost, robustness, diameter, and entropy, previous topology control algorithms for backhaul networks either restrict their search to three-based topologies [25], or do not consider multiple objective functions [26; 27; 28; 29; 30; 31; 32]. Although SA has been used in the past for wireless backhaul planning [35], our algorithm is the first to incorporate the information provided by the LSG to find network topologies with low energy costs and strong connectivity properties.