# A Spectral Gap-Based Topology Control Algorithm for Wireless Backhaul Networks
## 1 Introduction  
The demand for wireless bandwidth with Quality of Service (QoS) and ubiquitous network connectivity is continuously increasing due to the explosive growth in data traffic caused by a new generation of innovative high data rate services and applications accessed from a wide variety of wireless devices [1, 2]. To address this issue, wireless backhaul networks that connect Macro Base Stations (MBSs) and Small Cells (SCs) are becoming popular as they use multiple-input multiple-output (Massive MIMO) and millimeter-wave (mmWave) transmission technologies [3, 4, 5, 6] that use multiple antennas to provide significant improvements in wireless data transmission rates, link efficiency, network reliability, and energy efficiency [7, 8]. They achieve high spectral efficiency by serving many users simultaneously and enabling the placement of a dense number of SCs within the MBSs coverage area [9]. These SCs consist of low-power radio transmitter nodes that can handle the traffic of fixed and low-mobility users, which results in improved Quality of Service (QoS) in terms of throughput and outage as the distance between the access points and users is relatively short [10, 11]. Although current wireless backhaul networks utilizing Massive MIMO and mmWave technologies show great potential in meeting the demands of current and future mobileInternet users, they require innovative network topology-control mechanisms that surpass those implemented in conventional cellular networks such as LTE or LTE-Advanced [12]. This is essential for establishing a robust network backbone that fosters scalability and energy efficiency. Topology control (TC) mechanisms aim to adapt the topology of a network to optimize performance metrics such as energy consumption while maintaining connectivity requirements throughout the network [13; 14]. TC schemes were initially designed for Wireless Sensor Networks (WSNs), Mobile Ad Hoc Networks (MANETs), and Wireless Mesh Networks (WMNs) to improve overall performance [15; 16; 17]. In wireless backhaul networks, the topology is adjustable by modifying parameters such as transmission power and antenna radiation patterns [18; 19]. TC is generally challenging because there are several objective metrics to optimize, some potentially conflicting (e.g., energy consumption and robustness). To make the problem more tractable, we propose focusing the search for feasible topologies on _expander-like graphs_, which are sparse graphs with \"good\" connectivity properties [20; 21]. More specifically, we propose to search for topologies with high Laplacian Spectral Gap (LSG), which measures the strength of network connectivity and entanglement [22]. The proposed algorithm takes advantage of the directional beamforming capability of Massive MIMO systems to activate a set of feasible non-interfering communication links, which allows the efficient incorporation of a significant number of SCs within the MBS coverage, increasing the spatial reuse of bandwidth and increasing spectral efficiency. To the best of our knowledge, SGTC is the first topology control algorithm that effectively employs the information provided by the LSG to optimize the network topology by simultaneously minimizing the number of links and network diameter while maximizing the network entropy and the number of edge-disjoint paths connecting any two stations in the backhaul network. All these topological properties are desirable for backhaul networks because sparse topologies are energy-efficient, the network diameter impacts the upper bound of the end-to-end delay, the network entropy is related to the network's ability to distribute data traffic evenly across the network and promotes seamless traffic offloading from the MBSs to the SCs, and a higher number of edge-disjoint paths leads to better throughput and robustness. The rest of the paper is organized as follows. Section 2 presents and discusses related work on massive MIMO systems, Small Cells, topology control, and expander graphs in the context of our work.