# The Effects of Topology on Throughput Capacity of Large Scale Wireless Networks
## 3 The Intra-Cluster Transmission  
### The Spheroidal Percolation Model for Each Cluster  
For the three-dimensional network model, we consider each cluster as a sphere that is divided by cones. The cones share the same vertex, i.e., the center of the cluster. As shown in Figure 2, if we project the sphere to a horizontal plane, the cones will be the same size sectors. Let \(y\) be the number of projective sectors, and the angle of each sector is equal to \(\frac{2\pi}{y}\). According to the theory of solid geometry, it is easy to find that there are \(\Theta(y^{2})\) cones in a sphere. Moreover, we also partition a sphere into \(\frac{n/m}{y^{2}}\) concentric spheres. Since the node intensity is different in different areas, the radius of the \(i\)-th concentric sphere is defined as:  
\[r_{i}=\left(1+\frac{2\pi}{y}\right)^{i-1}\cdot\rho_{\min}, \tag{5}\]  
where \(\rho_{\min}\) denotes the minimum positive constant separation distance between the center and the other nodes in the cluster. Intuitively, according to the partition, each sphere is divided into quasi-cubelets, and the farther the quasi-cubelets are located from the cluster center, the bigger the quasi-cubelets are. In particular, the radius of the entire sphere is: \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Definition** \\ \hline \(n\) & The total number of nodes in the networks \\ \(L\) & Edge length of the network area \\ \(\gamma\) & Growth exponent of \(L\): \(L=\Theta(n^{\frac{1}{3}}),0\leq\gamma\leq 1\) \\ \(m\) & Average number of cluster centers \\ \(v\) & Growth exponent of \(m\): \(m=\Theta(n^{v}),0<v\leq 1\) \\ \(q_{j}\) & The number of nodes per cluster \\ \(d_{c}\) & Average distance between cluster centers: \(d_{c}=\Theta(n^{\frac{1-\pi}{3}})\) \\ \(s(\rho)\) & A dispersion density function \\ \(\delta\) & Decay exponent of \(s(\rho)\): \(s(\rho)=\min(1,\rho^{-\delta})\), \(\delta>3\) \\ \(\frac{P_{i}}{\Phi}\) & The transmission power of node \(i\) \\ \(\frac{\Phi}{\Phi}\) & The maximum node density \\ \(\frac{\Phi}{R(n)}\) & Per-node throughput \\ \hline \end{tabular}
\end{table} 
Table 1: Notations. Figure 2: The spheroidal percolation model for each cluster. We firstly divide the sphere into cones, and then use concentric sphere to partition the cones. Thus, each sphere is composed of many quasi-cubelets. The quasi-cubelet is open only if there is at least one node in the quasi-cubelet. The angle of each two neighboring radii is \(2\pi/y\) if we project the sphere to a plane. \[r_{\frac{\mu/m}{y^{2}}+1}=\left(1+\frac{2\pi}{y}\right)^{\frac{n/m}{y^{2}}}\cdot \rho_{\min},n\rightarrow\infty. \tag{6}\]  
Since the average distance between two neighboring clusters is \(d_{c}\), then take Equations (3) and (5) into consideration, we can derive the following equation:  
\[r_{\frac{\mu/m}{y^{2}}+1}=\left(1+\frac{2\pi}{y}\right)^{\frac{n/m}{y^{2}}} \cdot\rho_{\min}=e^{\frac{2\pi n^{1-v}}{y^{2}}}\cdot\rho_{\min}=\frac{d_{c}}{2}. \tag{7}\]  
Based on Equation (7), we have  
\[y=\Theta\left(\left(\frac{n^{1-v}}{\left(1-v\right)\log n}\right)^{\frac{1}{3} }\right). \tag{8}\]  
Let \(s_{i}\) denote the quasi-cubelet between the \(i\)th and \(\left(i+1\right)\)th concentric sphere and \(X(s_{i})\) as the number of nodes distributed inside \(s_{i}\). According to percolation theory [2], \(s_{i}\) is open if it contains at least one node and closed otherwise. By the SNCP model, the probability that a quasi-cubelet is open can be written as:  
\[p_{i}\equiv P\left(X(s_{i})\geq 1\right) \approx 1-e^{-r_{i}^{-\delta}\left(r_{i}\frac{2\pi}{y}\right)^{3}}\] \[=1-e^{-r_{i}^{3-\delta}\left(\frac{2\pi}{y}\right)^{3}}, \tag{9}\]  
where \(r_{i}^{-\delta}\) and \(r_{i}\frac{2\pi}{y}\) are the node density and arc length of quasi-cubelet \(s_{i}\), respectively. While \((r_{i}\frac{2\pi}{y})^{3}\) can be viewed as the volume of quasi-cubelet \(s_{i}\). It is easy to find that \(p_{i}\) decreases when \(i\) increases if \(\delta>3\). In addition, if \(i<j\), \(P\left(X\left(s_{i}\right)\geq 1\right)>P\left(X\left(s_{j}\right)\geq 1\right)\).