# The Effects of Topology on Throughput Capacity of Large Scale Wireless Networks
## 3 The Intra-Cluster Transmission
### Per-Node Rate of Intra-Cluster Traffic
: _The nodes on the SH can achieve a rate of \(\Theta\left(\left(\frac{(1-v)\log n}{n^{1-v}}\right)^{\frac{2}{3}}\cdot f( \delta)\right)\) w.h.p., where \(f(\delta)\) is called \"heterogenous factor\", which is only decided by the heterogeneous topology of the node's distribution._  
Proof. : Compared with the case of RH, the transmission around the cluster becomes more complicated since the number of nodes in each sphere-tier are not equal. If the number of nodes in every sphere-tier are identical, we can get a corresponding rate of \(\Theta\left(\left(\frac{(1-v)\log n}{n^{1-v}}\right)^{\frac{2}{3}}\right)\). Furthermore, taking the effect of heterogeneity into account, the achievable rate can be written as:  
\[R_{2} =\Theta\left(\left(\frac{(1-v)\log n}{n^{1-v}}\right)^{\frac{2}{3}} \cdot f(\delta)\right). \tag{25}\]  
\(\square\)  
In the following Lemma, we analyze the impact of \(f(\delta)\) and compare the rate on RH and SH. **Lemma 3**. : _For the intra-cluster traffic of each independent cluster, the nodes within radius \(\rho_{max}\) can achieve a rate of \(\Omega\left(1/(n^{1-v}((1-v)\log n)^{2})^{\frac{1}{3}}\right)\) w.h.p._  
Proof. : The routing strategy based on percolation theory has been deeply researched. Here, we give a rigorous derivation of the achievable rate during the highway phase in the intra-cluster transmission. To get the rate bottleneck, we need to compare the rates on the RH and SH. The rate on RH has been derived in Lemma 1. We will calculate the rate of SH. As the number of nodes on different paths is not in the same level, the condition along the cluster is more sophisticated with respect to the radial direction. According to the analysis above, we use \(E(N_{i})\) to denote the mathematical expectation number of nodes in each concentric sphere. Then, we can achieve the expectation from our model and simplify it as follows:  
\[E(N_{i})=\frac{n}{m}\left(r_{i}\frac{2\pi}{y}\right)^{3}\frac{1}{r_{i}^{3}}y, \tag{26}\]  
where \(\frac{n}{m}\) is the number of nodes in a cluster, \(\frac{1}{r_{i}^{3}}\) is the node density of concentric sphere \(i\) and \((r_{i}\frac{2\pi}{y})^{3}y\) is the volume of concentric sphere \(i\). Using (5) to substitute \(r_{i}\), we can get:  
\[ \begin{split} E(N_{i})&=\frac{n}{m}\left(r_{i}\frac{2 \pi}{y}\right)^{3}\frac{1}{r_{i}^{\delta}}y\\ &=8\pi^{3}\frac{n}{m}y^{-2}r_{i}^{3-\delta}\\ &=8\pi^{3}\frac{n}{m}y^{-2}\left(1+\frac{2\pi}{y}\right)^{i(3- \delta)}\\ &=8\pi^{3}\frac{n}{m}\left(\frac{(1-v)\log n}{n^{1-v}}\right)^{ \frac{3}{3}}\left(1+\frac{2\pi}{\left(\frac{(1-v)\log n}{n^{1-v}}\right)^{ \frac{3}{3}}}\right)^{i(3-\delta)}.\end{split}  \tag{27}\]  
As for \(\delta>3\), we can find that \(E(N_{i})\) decreases with the increasing of parameter \(i\). Thus, the SH closest to the cluster center needs to service the most nodes. Then, we can get the achievable rate on the SH:  
\[R_{2}>R_{2}(i=1)=\Theta\left(\left(\frac{(1-v)\log n}{n^{1-v}}\right)^{\frac{ 2}{3}}\left(1+\frac{2\pi}{\left(\frac{(1-v)\log n}{n^{1-v}}\right)^{\frac{2}{ 3}}}\right)^{3-\delta}\right). \tag{28}\]  
In the end, comparing the \(R_{1}\) and \(R_{2}\), we can find that, for any \(i\), \(R_{1}<R_{2}\), i.e., the data rate on any SH larger than that of RH. Correspondingly, the rate bottleneck is on the RH. Thus, the data rate of intra-cluster is  
\[R_{intra}=R_{1}\!=\!\Omega\left(1/(n^{1-v}((1-v)\log n)^{2})^{\frac{1}{3}} \right). \tag{29}\]