# Capacity and Delay Analysis for Large Social-Aware Mobile Ad Hoc Wireless Networks
## 4 The Upper Bound of Throughput Capacity
\[\mathcal{L}(X_{j},X_{k}) \leq(1+\Delta)\mathcal{L}(X_{i},X_{k});\] \[\mathcal{L}(X_{i},X_{l}) \leq(1+\Delta)\mathcal{L}(X_{j},X_{l}). \tag{9}\]  
From the triangle inequality, we have  
\[\mathcal{L}(X_{j},X_{i}) \geq\mathcal{L}(X_{j},X_{k})-\mathcal{L}(X_{i},X_{k})\] \[\geq\Delta\mathcal{L}(X_{i},X_{k}). \tag{10}\]  
Likewise, we have  
\[\mathcal{L}(X_{i},X_{j})\geq\Delta\mathcal{L}(X_{j},X_{l}).\]  
Therefore,  
\[\mathcal{L}(X_{i},X_{j})\geq\frac{\Delta}{2}\left(\mathcal{L}(X_{i},X_{k})+ \mathcal{L}(X_{j},X_{l})\right). \tag{11}\]  
Hence, there is an essential disjoint disks zone centered transmitters in the same slot, in which the radius is \(\frac{\Delta}{2}\) times the length of this hop. Square region \(\mathcal{O}\) is 1, then we have  
\[\sum_{(T_{i}^{t},R_{i}^{t})\in S(t)}S_{E}\leq 1,\]  
where \(S(t)\) is the set of transmitter-receiver pairs transmitted simultaneously. \(S_{E}\) is the area of disjoint disks - consumed area that a single hop's transmission occupies. We have  
\[S_{E}=\frac{\pi\Delta^{2}}{4}\mathcal{L}(T_{i}^{t},R_{i}^{t})^{2},\]  
where \(\mathcal{L}(T_{i}^{t},R_{i}^{t})\) is the distance between a transmitter-receiver pair. Then, we can get  
\[\sum_{(T_{i}^{t},R_{i}^{t})\in\mathcal{S}(t)}\mathcal{L}(T_{i}^{t},R_{i}^{t})^{2} \leq\frac{4}{\pi\Delta^{2}}. \tag{12}\]  
According to Cauchy-Schwartz inequality, we have  
\[\mathcal{L}(X_{j},X_{k}) \leq(1+\Delta)\mathcal{L}(X_{i},X_{k});\]0  
Assuming that all nodes are semi-duplex. \(\frac{n}{2}\) transmitter-receiver node pairs can transmit at the same time, then  
\[\mathcal{L}(X_{j},X_{k}) \leq(1+\Delta)\mathcal{L}(X_{i},X_{k});\]1  
During a large enough duration of time \(t\), the network transport capacity is \(\mathcal{T}\cdot t\cdot\frac{n}{2}\cdot\mathfrak{D}\). The amount of data transferred during a given time \(t\) is at most \(W\cdot t\). The network transport capacity is lesser than the product of \(W\cdot t\) (the amount of data transferred) and \(\sum_{(T_{i}^{t},R_{i}^{t})\in\mathcal{S}(t)}\mathcal{L}(T_{i}^{t},R_{i}^{t})\) (the cumulative distance of all hops). Thus, the following inequality holds  
\[\mathcal{L}(X_{j},X_{k}) \leq(1+\Delta)\mathcal{L}(X_{i},X_{k});\]2 \[\mathcal{L}(X_{j},X_{k}) \leq(1+\Delta)\mathcal{L}(X_{i},X_{k});\]3  
We can obtain the following inequality  
\[\mathcal{L}(X_{j},X_{k}) \leq(1+\Delta)\mathcal{L}(X_{i},X_{k});\]4  
Thus, an upper bound throughput of each node can be represented as  
\[\mathcal{L}(X_{j},X_{k}) \leq(1+\Delta)\mathcal{L}(X_{i},X_{k});\]5  
**Theorem 1**. : _Under restricted mobility model and rank-based social model, the upper bound throughput of each node in large social-aware mobile ad hoc wireless networks is_  
\[\mathcal{T}=\left\{ \begin{array}{ll}\mathcal{O}\left(\frac{\sqrt{m}}{\sqrt{ \log n}}\right)&0\leq\delta<1\\ \mathcal{O}\left(\sqrt{\log n}\sqrt{m}\right)&\delta=1\\ \mathcal{O}\left(\frac{n^{1-\delta}\sqrt{m}}{\sqrt{\log n}}\right)&1<\delta< 3/2\\ \mathcal{O}\left(\frac{\sqrt{n}\sqrt{m}}{\log^{3/2}n}\right)&\delta=3/2\\ \mathcal{O}\left(\frac{\sqrt{n}\sqrt{m}}{\log n}\right)&\delta>3/2.\end{array} \right.\]