# A Spectral Gap-Based Topology Control Algorithm for Wireless Backhaul Networks
## 3 System Model and Problem Formulation
### System Model  
We use a graph \(G=(M\cup S,E)\) to model the topology of a wireless backhaul network composed of sets of Macro Base Stations (MBSs) \(M\), Small Cells (SCs) \(S\), and bidirectional wireless communication links \(E\subseteq V\times V\) between them, where \(V=M\cup S\) is the set of all the stations. The position of the stations in the plane is determined by a function \(l:V\rightarrow\mathbb{R}^{2}\) that assigns \(x\) and \(y\) coordinates to them. We use \(N(u)=\{v:(u,v)\in E\}\) to denote the one-hop neighborhood of \(u\). We assume that the SCs have a coverage range of \(R_{S}\) distance units, whereas the MBSs have a coverage range of \(R_{M}\) distance units. From the values of the coverage ranges, we have the following two definitions. Definition 1: A communication link \((u,v)\in E\) between stations \(u,v\in V\) is **feasible**: if \(dist(l(u),l(v))\leq R_{S}\) when \(u,v\in S\) or \(dist(l(u),l(v))\leq R_{M}\) when either \(u\) or \(v\) is in \(M\); where \(dist\) is the Euclidean distance. **Definition 2**. : _A network topology \(G=(M\cup S,E)\) is **feasible** if every link \((u,v)\in E\) is feasible._  
We further assume that:  
* All the stations are static; namely, they have fixed positions. * All the MBs have the same capabilities and coverage range. They are implemented by a Massive MIMO system with \(N_{M}\) antenna elements. * All the SCs have the same capabilities and coverage range. They are equipped with \(N_{S}\) antenna elements capable of supporting multiple parallel data streams. * MBSs and SCs can support all the possible links with the other MBs and SCs placed in their corresponding coverage area. More specifically, for an MBS \(u\in M\), the inequality \(N_{M}\geq|N(u)|\) holds. Similarly, for a SC \(v\in S\), the inequality \(N_{S}\geq|N(v)|\) also holds. * The energy cost of establishing a link from station \(u\) to station \(v\) is proportional to \(dist(l(u),l(v))\). * All the SCs have relay functionality; namely, they can communicate with any of their one-hop neighbors. For a given network topology \(G=(M\cup S,E)\), we use the following performance metrics:
* Network Diameter (\(D_{G}\)): Defined as the longest shortest path between any two stations in \(G\). Networks with a small diameter tend to induce lower delays and reduce the energy used to transport a packet from sources to destinations. * Network Robustness (\(R_{G}\)): Defined as the cardinality of the global minimum cut-set, which is equivalent to the minimum number of edge-disjoint paths connecting any two stations in the network. Networks with high robustness are more tolerant to link breaks due to interference or channel effects. This metric indicates the minimum number of edges that need to be removed to partition the network. * Network Cost (\(C_{G}\)): Defined as the sum of the length of the edges in the graph (see Equation (1)). This metric is related to the power used by the network to establish and maintain all the communication links. \[C_{G}=\sum_{e=(u,v)\in E}dist(l(u),l(v)) \tag{1}\]
* Network Entropy (\(S_{G}\)): Defined as the entropy (see Equation (2)) over the probability mass function (see Equation (3)) of using a particular link \(e\in E\) as part of the shortest path connecting an arbitrary source-destination pair. Network entropy measures the amount of information about how data traffic will be distributed across a particular network by a specific shortest-path routing protocol. High values of network entropy indicate that a given shortest-path routing protocol will tend to distribute data traffic evenly across the network. On the other hand, low values of network entropy indicate that this shortest-path routing protocol will tend to concentrate data traffic on a subset of the links (those with a high probability of being part of the shortest path). Similar to the betweenness centrality [39], \(Pr(e)\) (see Equation (2)) provides information about the influence of an edge over the data flows in a network.