# A Cluster-Based Channel Assignment Technique in IEEE 802.11 Networks
## 2 Network Model
### Connectivity and Interferences Models Using Graphs  
Our models are based in the so-called multilayer graphs [28], which have recently emerged as a versatile tool with which to capture a high number of features of real situations into a single (multilayered) graph. Our multilayer graph is composed by two layers, being both geometric. Moreover, in both layers we have two different types of vertices, which are the two types of wireless devices that appear in Wi-Fi networks when operating in infrastructure mode: STAs and APs. Next, we provide a brief description of both layers. For a more in-depth description, please see [29, 30].  
The first one, called the connectivity layer, represents the connectivity between STAs and APs, so it represents the attachment between each STA and its associated AP. We have considered that each STA is associated with its closest AP (if it is in its coverage area that is defined by the sensitivity of the receiving antennas), as it is usually the AP with the maximum receiving power. In this case, edges are undirected because they represent the attachment between STAs and APs, which is bidirectional. The second layer, called the interference layer, includes all the possible interferences that a wireless device produces regarding other wireless devices. In other words, there is an edge between two vertices A and B provided that A produces interferences affecting B. In this case we consider a geometric weighted directed layer (or weighted digraph). It is a weighted graph because the weight of each edge represents the power of the interference signal. Moreover, it is directed as we consider that interferences received from APs are more harmful than the ones received by STAs, as APs are frequently more active and interferences will have layer impacts over time. For example, for a STA and an AP that interfere, we have two edges between them (with opposite directions), with different weights to represent that the interference produced from the AP to the STA is more harmful than the interference from the STA to the AP. Finally, it is important to note that two STAs attached to the same AP will not interfere with each other, as their communications with the AP are coordinated.  
Figure 1 shows an example of the connectivity layer of a Wi-Fi network with 49 APs and 245 STAs. From this figure, it is easy to explain a concept that will be used later on in this work--that is, the concept of a cluster. As can be seen in the figure, the whole network is composed by a set of APs, each one with a set of STAs attached to it. With this, we define a cluster as one of those sets of an AP and its attached STAs. On the other hand, Figure 2 shows, for the same example scenario, the resulting interference layer.  
As stated above, the weights of the edges in the interference layer will be determined by the power levels of the interference signals. More specifically, the weight between vertices \(V_{i}\) and \(V_{j}\) will be determined by:  
\[I_{V_{i}\to V_{j}}=P_{t}+G_{t}+G_{r}-L-P_{loss}+\psi+I, \tag{1}\]  
where, with all values expressed on the logarithmic scale, \(P_{t}\) represents the transmission power; \(G_{t}\) and \(G_{r}\) are the transmission and reception antenna gains, respectively; \(L\) includes the propagation losses because of obstacles, such as walls, doors, windows, etc.; and \(P_{loss}\) represents the signal loss due to the distance, and has been computed according to [31], considering that antennas are placed 1.5 m above the floor. Additionally, \(\psi\) represents the abovementioned asymmetry effect which makes interferences from APs more harmful that the ones received from STAs, as the interferences from APs will take place over far more time than the ones from STAs. On the linear scale, we have considered \(\psi=0.5\) for APs and \(\Psi=0.2\) for STAs. Finally, \(I\) is the indicator function and represents the collision