# The Effects of Topology on Throughput Capacity of Large Scale Wireless Networks
## 3 The Intra-Cluster Transmission
### Per-Node Rate of Intra-Cluster Traffic  
In this section, we will derive the rate of each phase. By comparing the rate of each phase, we obtain the rate bottleneck of intra-cluster transmission. First of all, we derive a general rate when the destination is located at \(d\) cubelets away. **Theorem 1**. : _In each quasi-cubelet \(s_{i}\), each node can transmit w.h.p. at a rate \(R(d)=\Omega(d^{-\alpha-3})\) to any destination located at distance \(d\), where the distance \(d\) is not Euclidean distance but the number of quasi-cubelets._  
Proof. : In each cluster, for a node located \(i\)th concentric sphere and \(r_{i}\leq\rho_{\max}\), it transmission power is  
\[P_{i}=P_{0}\cdot\left(\frac{2\pi r_{i}}{y}\right)^{\alpha}, \tag{13}\]  
where \(P_{0}\) is a constant denoting the power of the sphere with the minimum radius. Using the transmission model defined in Equation (4), we can derive the data rate if the destination is located \(d\) quasi-cubelets away. According to Equation (4), the key point is to handle the interference caused by simultaneous transmission. For the interference caused by simultaneous transmission, as shown in Figure 4, firstly, we can prove that the interferences from the opposite directions are equal. Let \(I_{1}\) and \(I_{2}\) denote the interferences caused by the node located in \(S_{i+l}\) and \(S_{i-l}\), respectively, where \(l\) is the number of quasi-cubelets from a given quasi-cubelets \(i\). Then, we have  
\[I_{1}(l)=P_{i+l}\frac{1}{(r_{i+l}-r_{i})^{\alpha}}, \tag{14}\]  
Figure 3: Routing strategy used for the intra-cluster transmission. It is obvious that the transmitter \(S\) transmits packets to the receiver \(D\) through five phases. The cross shows four directions that each node can transmit around the sphere. \[I_{2}(l)=P_{i-l}\frac{1}{(r_{i}-r_{i-l})^{\alpha}}, \tag{15}\]  
and  
\[\frac{I_{1}(l)}{I_{2}(l)}=\frac{P_{i+l}}{P_{i-l}}\cdot\frac{(r_{i}-r_{i-l})^{ \alpha}}{\left(r_{i+l}-r_{i}\right)^{\alpha}}=\left(\frac{r_{i+l}}{r_{i-l}} \cdot\frac{r_{i}-r_{i-l}}{r_{i+l}-r_{i}}\right)^{\alpha}=\left(1+\frac{2\pi}{y }\right)^{la}. \tag{16}\]  
Since \(y=\Theta\left(\left(\frac{n^{1-v}}{(1-v)\log n}\right)^{\frac{1}{3}}\right)\) and \(\alpha>3\), when \(l=o\left(\left(\frac{n^{1-v}}{(1-v)\log n}\right)^{\frac{1}{3}}\right)\), we can get \(\frac{I_{1}(l)}{I_{2}(l)}\to\ 1\). We can conclude that, using appropriate power control strategy, the interferences from other transmitters are equal if the transmitters are located the same number of quasi-cubelets away. Next, we will analyze the interference caused by the simultaneous transmitters in a cluster. For the intra-cluster transmission, we adopt Time Division Multiple Address (TDMA) scheduling and divide time into \(k^{3}\) successive time slots, where \(k=2(d+1)\). Then, we consider the disjoint set of subcubes that are allowed to transmit simultaneously. As shown in Figure 5, if the transmitter in a subcube \(s_{i}\) transmits toward a destination node which is located in another subcube and the distance is at most \(d\) subcubes away, we find an upper bound for the interference at the receiver. Notice that, in the \(j\)th tier, the transmitters belonging to \((2j+1)^{3}-(2j-1)^{3}\) closest subcubes are located at a distance of at least \(4 jd\) subcubes from \(s_{i}\). Figure 4: An illustration of interferences from two opposite directions. \(S_{i}\) receives interferences from \(S_{i+l}\) and \(S_{i-l}\). The distances of the two interferences are both \(l\) quasi-cubelets. Figure 5: An illustration of the TDMA schedule. Black subcubes can transmit simultaneously. Around each black subcube, there is a “silence” region of subcubes that are not allowed to transmit in the given time slot. Summing all the interferences, the upper bound of the interference could be given as follows:  
\[