## 3 A flexible Approach to a Reachable Network Topology with a Minimum Connection Degree
### Network Model Design
Connectivity is defined in relative to the directness of the route between nodes in a network topology. For example, for a network with \(\left/N\right/\) nodes, \(N\)={\(n_{1}\),\(n_{2}\),\(n_{3}\)....\(n_{\left|N\right|}\)}, where the degree of node (\(n_{j}\)) is \(deg(n_{i})\), the connectivity is given as:  
\[Connectivity(Net)=\sum_{i=1}^{\left|N\right|}\frac{deg[\oplus n_{i})}{2\binom{ \left|N\right|}{2}} \tag{1}\]  
Accordingly, a network has full connectivity when the network's _Connectivity (Net)_ equals to 1. The sufficient and necessary condition of full network connectivity are given as:  
\[\sum_{i=1}^{\left|N\right|}deg[\oplus n_{i})=\left|N\right|^{2}-\left|N\right| \tag{2}\]  
Full connectivity indicates that the transmission range of network is greater than or equal to the longest distance between any pair of nodes in the network and the degree of each node equals to \(\left/N\right/\)_-1_. The nodes in this network topology can directly exchange data. However, the sufficient transmission power is higher. In addition, this network topology will cause serious channel interference, which decreases the network QoS because the transmission range is too long. Reachability expresses the indirectness of the route between node pairs in a network topology. For example, in a network with \(\left/N\right/\) nodes where \(\left/N\right/\) nodes are formed of \(\left/G\right/\) partitions \(G\)={\(g_{1}\),\(g_{2}\),\(g_{3}\)....,\(g_{\left|G\right|}\)} and \(g_{j}\)={\(n_{1}\),\(n_{2}\),\(n_{3}\)....\(n_{\left|g\right|}\)}, reachability is given as:  
\[Reachability(Net)=\sum_{j=1}^{\left|C_{j}\right|}\frac{\left|C_{j}\right|}{2} \binom{\left|N\right|}{2} \tag{3}\]  
The requirement of full reachability in network topology is \(Reachability(Net)=1\). However, a network only possesses full reachability if a network has only one partition. The proof is as follows. If the network is fully reachable, we have:  
\[Reachability(Net)=1\]  
That is:  
\[\sum_{j=1}^{\left|C_{j}\right|}\binom{\left|C_{j}\right|}{2}=\binom{\left|N \right|}{2} \tag{4}\]  
Expanding both sides of the equation yields:  
\[\sum_{j}^{\left|C_{j}\right|}\left|C_{j}\right|^{2}-\sum_{j}^{\left|C\right|} \left|C_{j}\right|=\left|N\right|^{2}-\left|N\right| \tag{5}\]  
Because \(\left|N\right/\)=\(\sum_{j}^{\left|C\right|}\left|C_{j}\right|\), we have:  
Figure 2.0Then, the sufficient and necessary condition of full network reachability is:  
Figure 2.1  
There is at least an indirect or direct route between nodes. In this network topology, the degree of each node is at least greater than or equal to 1. Less power is consumed because of indirect delivery. Channel interference is decreased because the degree of the node is less than that of a node in a network with full connectivity. Therefore, the Qos of network is enhanced. The percentage of k-degree nodes (\(P_{c}^{k}\)) is defined as the proportion of nodes with the k degrees in a network topology. For example, in a network in which _Net=[N, L]_ with /_N_/ nodes and /_L_/ links, the node set \(N_{k}\)=[\(n_{i}\) / \(deg(n_{i})\)\(\geq\)\(\widehat{deg}(Net)\)]_ denotes that the node degree is equal to or greater than the expected node degree \(\widehat{deg}(Net)\). Thus, the percentage of k-degree nodes in a network is given as:  
Figure 2.2  
The full percentage of k-degree nodes indicates that all nodes in a network satisfy the expected node degree \(\overline{deg}(Net)\). This network will possess full connectivity if \(P_{c}^{k}\left(N_{k}\right)=\) _100%_ and \(k\)=\(|N|-\)_1_. Therefore, high percentage of k-degree nodes indicates that a relay node exists in any pair of nodes. The value estimated by Equation (8) will be adopted to verify the reliability of the adjustment model of transmission range. The transmission range is adjusted by node degree. This adjusted transmission range creates a network topology. In identical node density, the needed transmission range is longer and needs more power for higher node degree. And the number of connected nodes is higher.