# Capacity and Delay Analysis for Large Social-Aware Mobile Ad Hoc Wireless Networks
## 7 Discussion and Conclusion  
As illustrated in Theorems 1-3, both network capacity and network delay are affected by the social characteristics and node mobility. In this section we discuss and analyze the results. Upper bound throughput of each node and achieved throughput of each node are listed in Table 2. From this table, we can see that the achieved throughput of each node differs from the upper bound throughput of each node only by a factor. Such discrepancies are consistent with the fact. A common phenomenon can be seen from Table 2. Both the upper bound throughput of each node and the achieved throughput of each node increase with social characteristics \(\delta\). Specifically, the value of achieved per-node throughput is between \(\frac{1}{n\sqrt{m}}\) and \(\frac{1}{\sqrt{n}\sqrt{m}}\). Minimum value of achieved per-node throughput is \(\frac{1}{n\sqrt{m}}\), when \(0\leq\delta<1\). And the maximum value of achieved per-node throughput is \(\frac{1}{\sqrt{n}\sqrt{m}}\) when \(\delta>3/2\). The upper bound of per-node throughput shows the same change tendency. From formula (8), we can seen that the greater value of social characteristics \(\delta\), the shorter distance between S-D pair. With the increase of \(\delta\), source node tends to select a closer destination node. This leads to traffic aggregation in large social-aware mobile ad hoc wireless networks. The parameter of social model depicts the degree of traffic aggregation. Traffic aggregation brings greater network capacity. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\delta\) & \(\delta<1\) & \(\delta=1\) & \(1<\delta<3/2\) & \(\delta=3/2\) & \(\delta>3/2\) \\ \hline \(\mathcal{T}\) & \(O\left(\frac{\sqrt{m}}{\sqrt{\log n}}\right)\) & \(O\left(\sqrt{\log n}\sqrt{m}\right)\) & \(O\left(\frac{n^{1-\delta}\sqrt{m}}{\sqrt{\log n}}\right)\) & \(O\left(\frac{\sqrt{n}\sqrt{m}}{\log^{2}n}\right)\) & \(O\left(\frac{\sqrt{n}\sqrt{m}}{\log n}\right)\) \\ \(\mu\) & \(\Omega\left(\frac{1}{n\sqrt{m}}\right)\) & \(\Omega\left(\frac{\log n}{n\sqrt{m}}\right)\) & \(\Omega\left(\frac{1}{n^{2-\delta}\sqrt{m}}\right)\) & \(\Omega\left(\frac{\sqrt{n}\sqrt{m}}{\log n}\right)\) & \(\Omega\left(\frac{1}{\sqrt{n}\sqrt{m}}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Throughput capacity. When the parameter of node mobility \(m\) is equal to \(b(n)\), make the upper bound throughput of each node minimum. The the upper bound throughput of each node is \(\frac{1}{\sqrt{n}}\), \(\frac{\log n}{\sqrt{n}}\), \(\frac{1}{n^{2-1/2}}\), \(\frac{1}{\log n}\) and \(\frac{1}{\sqrt{\log n}}\) respectively, when \(\delta<1\), \(\delta=1\), \(1<\delta<3/2\), \(\delta=3/2\) and \(\delta>3/2\). The parameters of node mobility \(m\) is equal to \(\mathcal{O}\), make the upper bound throughput of each node maximum. In order to obtain higher throughput capacity, we need to increase \(m\), which means a larger motion range. Figure 5 exhibits that different scaling behaviors of network delay result from different values of social model parameter \(\delta\). One can observe that for \(\delta>3/2\) the best network delay can be achieved \(\Theta(nm)\). Since \(m\in(b(n),1)\), it indicates that the network delay ranges from \(\Theta(\sqrt{n})\) to \(\Theta(n)\). As the parameter of social model becomes smaller, the network delay increases. The relationship between network delay and throughput capacity is mutual contradiction. Bigger throughput capacity requires as many nodes as possible to send or receive data at any time, which makes the waiting queue increase. On the other hand, reducing the network delay requires waiting queue as short as possible, which makes throughput capacity smaller. The throughput of each node is \(\Theta(\frac{1}{\sqrt{n\log n}})\) in static ad hoc wireless networks, meanwhile the network delay is negligible [5; 6]; the throughput of each node is \(\Theta(1)\) in mobile ad hoc wireless networks, with network delay \(\Theta(n)\)[7].