# Capacity and Delay Analysis for Large Social-Aware Mobile Ad Hoc Wireless Networks
## 4 The Upper Bound of Throughput Capacity  
In this section, we use the network models and assumptions above to derive an upper bound of capacity for social-aware mobile ad hoc wireless networks. For simplicity, we use S-D pairs instead of source and destination node pairs. Since pairs are selected by the social model, it is of great importance to find the relation between rank and geographical position. We show the relation between rank and geographical position in following lemma 1, which is different from the destination node randomly selected pattern. **Lemma 1** ([26], Lemma 3). : _In large social-aware mobile ad hoc wireless networks, if the rank of node \(j\) to source node \(i\) is \(Rank_{i}(j)=r\), the distance between node \(i\) and node \(j\) satisfies \(x=\Theta(\sqrt{\frac{r}{n}})\) with probability 1, when \(n\rightarrow\infty\)._  
The probability can be calculated as  
\[P\left(x\leq\sqrt{\frac{cr}{\pi n}}\right) =\sum_{k=r}^{n}\frac{n!}{k!(n-k)!}\left(\frac{cr}{n}\right)^{k} \left(1-\frac{cr}{n}\right)^{n-k}\] \[<2c_{1}(ce^{1-c})^{r}. \tag{3}\]  
Similarly,  
\[P\left(x\geq\sqrt{\frac{cr}{\pi n}}\right)<2c_{2}(ce^{1-c})^{r}. \tag{4}\]  
Then, the probability \(x=\Theta(\frac{r}{n})\) can be calculated as  
\[1-P\left(x=\Theta\left(\sqrt{\frac{r}{n}}\right)\right) <2c_{1}(ce^{1-c})^{r}+2c_{2}(ce^{1-c})^{r}\] \[=o(1). \tag{5}\]  
The whole proof of this lemma is demonstrated in Reference [26]. **Lemma 2**. : _In large social-aware mobile ad hoc wireless networks, the distance of data cumulative transmission can be represented as_  
\[\mathfrak{D}=\left\{ \begin{array}{ll}\Theta\left(\frac{\sqrt{\log n}}{\sqrt {n}\sqrt{m}}\right)&0\leq\delta<1\\ \Theta\left(\frac{1}{\sqrt{n\log n\sqrt{m}}}\right)&\delta=1\\ \Theta\left(\frac{\sqrt{\log n}}{n^{n-1/2}\sqrt{m}}\right)&1<\delta<3/2\\ \Theta\left(\frac{\log^{3/2}n}{n\sqrt{m}}\right)&\delta=3/2\\ \Theta\left(\frac{\sqrt{\log n}}{n\sqrt{m}}\right)&\delta>3/2.\end{array} \right.\]  
Take back the normalizing factor in (2). According to the asymptotic sum of p-series, \(F_{n}\) can be represented as  
\[F_{n}=\sum_{i=1}^{n}\frac{1}{i^{\delta}}=\left\{ \begin{array}{ll}\Theta(1)& \delta>1\\ \Theta(\log n)&\delta=1\\ \Theta(n^{1-\delta})&0\leq\delta<1.\end{array} \right. \tag{6}\]  
\(\mathcal{L}(S,D)\) denotes Euclidean distance of S-D pair under rank-based social model. It is clear that  
\[E\{\mathcal{L}(S,D)\}=\frac{1}{F_{n}\sqrt{n}}\sum_{r=1}^{n}\frac{\sqrt{r}}{r^ {\delta}}. \tag{7}\]Then,  
\[E\{\mathcal{L}(S,D)\}=\left\{ \begin{array}{ll}1&0\leq\delta<1\\ 1/\log n&\delta=1\\ n^{1-\delta}&1<\delta<3/2\\ \log n/\sqrt{n}&\delta=3/2\\ 1/\sqrt{n}&\delta>3/2.\end{array} \right. \tag{8}\]  
The number of super-cells that data packets go through is \(\frac{E\{\mathcal{L}(S,D)\}}{\sqrt{2m}}\), and the average transmission distance in the cell is at least \(b(n)\). The average cumulative transmitted distance \(\mathfrak{D}\) can be calculated as  
\[\mathfrak{D}\geq\frac{E\{\mathcal{L}(S,D)\}}{\sqrt{2m}}\sqrt{b(n)}.\]  
Substituting \(E\{\mathcal{L}(S,D)\}\) and \(b(n)\), we get Lemma 2. Let \(\mathcal{L}(X_{i},X_{j})\) denote the distance between node \(X_{i}\) and node \(X_{j}\). When the equation \(\mathcal{L}(X_{k},X_{i})\geq(1+\Delta)\mathcal{L}(X_{j},X_{i})\) is satisfied, data from node \(i\) to node \(j\) can be transmitted successfully. \(\Delta\) is a given threshold to avoid interference. Euclidean distance \(\mathcal{L}(X_{j},X_{k})\) and \(\mathcal{L}(X_{i},X_{l})\) hold the protocol interference model, node pairs \((X_{i},X_{k})\) and \((X_{j},X_{l})\) can transmit data simultaneously.