# The Effects of Topology on Throughput Capacity of Large Scale Wireless Networks
## 4 Transmission of Inter-Cluster  
In this section, we will conduct a study on inter-cluster transmission. For the inter-cluster transmission, it is hard to guarantee that each quasi-cube contains at least one node if we still adopt the transmission strategy of the intra-cluster. Thus, a new policy is proposed in this section for the inter-cluster traffic. Since the lowest node density is \(\underline{\Phi}\), according to percolation theory [2; 13], we can guarantee that each quasi-cube contains nodes with high probability if we partition the inter-cluster area into cubes with side lengths of \(c\sqrt[3]{1/\underline{\Phi}}\), where \(c\) is a constant. Similarly, we have the following Theorem.  
Theorem 4.2: _For the inter-cluster traffic, we can obtain an achievable rate of \(R_{inter}=\Omega\left(n^{\frac{2\gamma}{3}-1}\underline{\Phi}^{\frac{3}{3}}\right)\)._  
Proof: Since the node density in the space among the clusters is much lower, we propose an idea of \"information pipes\", as shown in Figure 6. In our network, we assume that \(\overline{\Phi}\) is the maximum node density and \(\underline{\Phi}\) is the minimum node density. The inter-cluster space is divided into cubelets with side lengths of \(c\sqrt[3]{1/\underline{\Phi}}\), where \(c\) is a constant.  
Therefore, the network consists of many cuboids of side length \(c\sqrt[3]{1/\underline{\Phi}}\times c\sqrt[3]{1/\underline{\Phi}}\times c \sqrt[3]{A}\), or \(c\sqrt[3]{1/\underline{\Phi}}\times\sqrt[3]{A}\times c\sqrt[3]{1/\underline{ \Phi}}\), or \(\sqrt[3]{A}\times c\sqrt[3]{1/\underline{\Phi}}\times c\sqrt[3]{1/\underline{ \Phi}}\). Then, the nodes outside the cluster region can build up \(\Omega\left((\sqrt[3]{A}\overline{\Phi})^{2}\right)\) \"information pipes\" among these clusters, and these \"information pipes\" need to serve \(\Theta(m)\) clusters. Correspondingly, each cluster can enjoy \(\Omega\left(\frac{(\sqrt[3]{A}\overline{\Phi})^{2}}{m}\right)\) \"information pipes\" on average. Similar to the proof of Theorem 4.1, let \(d=1\), and we get that the rate on the \"information pipes\" is also constant, while the difference is that we employ the same power to transmit in the \"information pipes\". The reason is that the size of cubelets in the \"information pipes\" is equal. Intuitively, the power of inter-cluster transmission would be larger than that of intra-cluster transmission. The reason is that the node density among the clusters is much lower, and we need to enlarge the size of cubelets, such that we can guarantee that each cubelet contains at least one node w.h.p.  
Based on the analysis above, there are \(\Omega\left((\sqrt[3]{A\Phi})^{2}\right)\) \"information pipes\" in the network, and these \"information pipes\" need to sever at most \(n\) nodes. Thus, we can get the achievable rate for each node in the inter-cluster transmission is:  
\[R_{inter}=\Omega\left(\frac{(\sqrt[3]{A\Phi})^{2}}{n}\right)=\Omega\left(n^{ \frac{2\gamma}{3}-1}\Phi^{\frac{2}{3}}\right). \tag{30}\]  
In particular, if the network model transforms to a homogeneous network, i.e., \(\underline{\Phi}=\Theta(\overline{\Phi})=\Theta(n^{1-\gamma})\), we can get that the achievable rate is  
\[R_{h}=\Omega\left(n^{\frac{2\gamma}{3}-1}\underline{\Phi}^{\frac{2}{3}}\right) =\Omega(\frac{1}{\sqrt[3]{n}}), \tag{31}\]  
where \(R_{h}\) is also the result of homogeneous three-dimensional wireless networks [17; 18; 19].