## 3 System Model
### Network Model  
In this chapter, the network model is introduced, which models the connectivity properties among nodes. Networks just keep connection among nodes throughout the lifetime, and there are various operating cases, according to their communication modes.  
In the omni-directional case, a wireless network is often represented by a connected graph, where vertices correspond to the network nodes, and a directed edge from one vertex \(u\) to another vertex \(v\) indicates that data from the node \(u\) can reach directly to the latter node \(v\)[28]; the relationship can be expressed as a directed arrow from \(u\) to \(v\) as follows:  
\[u\to v:\vec{d}(u,v)<r_{u} \tag{13}\]  
The vector \(\vec{d}(u,v)\) denotes the Euclidean distance between nodes \(u\) and \(v\). The directed edge converts to an undirected one if the communication link is symmetrical, i.e., node \(v\) locates within the transmission range of node \(u\), and vice versa, which can be expressed as follows:  
\[u\leftrightarrow v:\left\{\begin{array}{l}\vec{d}(u,v)<r_{u}\\ \vec{d}(v,u)<r_{v}\end{array}\right. \tag{14}\]  
If the nodes share a same transmission range \(r\), the equation is simplified to the following:  
\[u\overset{r}{\leftrightarrow}v:d(u,v)<r \tag{15}\]  
The equation above implies that \(u\) and \(v\) are symmetric, connected with transmission range \(r\); the relationship of the three nodes sharing a unified transmission range is shown in Figure 3.  
Figure 2: Optimal beamwidth directivity diagram.  
In the directional case, it is much more complicated to take into account the situation where two nodes interact because a node can transmit/receive either directionally or omni-directionally. Given a communication mode pair \((X,Y)\), the symbols in the brackets individually represent the transmission and reception modes of a node, so if directivity is considered, there are altogether two modes that the node may work in, as listed below [29]:  
* \((D,O)\) ((respectively, \((O,D)\)) mode: A node \(u\) transmits directionally and receives omni-directionally, or conversely. In this case, \(u\) can reach any node \(v\) as long as its beam covers the latter.
* \((D,D)\) mode: A symmetric link sets up if both the sender and receiver are orientated toward each other and they all lay within each other's transmission range.  
The diverse link connection feasibility of different operating modes are illustrated in Figure 4, where the solid lines with arrows represent effective communication links, and the dashed ones denote invalid links. As can be seen, Figure 3 is actually a \((O,O)\) mode, where two nodes communicate without any additional limit, apart from the transmission range, while in Figure 4a of the \((D,O)\) mode, a sending node \(u\) must orientate toward its intended node \(v\). If two sending nodes transmit to a same destination simultaneously, a collision then occurs. In the \((D,D)\) mode shown in Figure 4b, two communicating nodes can set up a physical link only if they orientate toward each other at the same time, so node \(w\) does not influence the ongoing communication between \(u\) and \(v\).  
Although it is more convenient to analyze the condition \((D,O)\) or \((O,D)\), the actual network performance may not be as good as that of \((D,D)\), as it is not realistic to reduce the interference between nodes and improve the spatial reusage as desired (see Figure 4b). Thus, the \((D,D)\) mode with beamwidth \(\pi/3\) is limited in the research.  
Figure 3: Omni-directional connection between three nodes.