# Capacity Analysis of Incentive Schemes in Opportunistic Networks
## 4 Capacity Analysis for Incentive Schemes
### Upper Bound of Network Capacity  
Different from the other studies, because \(\Delta\) is deemed as a variable associated with incentive schemes by (21), it is not simplified during derivations. According to the interference restriction and the meeting probability among nodes and clusters, the amount of messages that can be transmitted in a time slot between clusters is \(O\left(\frac{mR^{2}}{\Delta^{2}}\right)\) if the transmission range is \(\Theta\left(R\sqrt{\frac{m}{n}}\right)\) and the amount of messages in a cluster is the similar. Therefore, we can propose the following theorem:  
**Theorem 1** (Throughput capacity of incentive schemes).: _In a sparse clustering regime with incentive schemes, the throughput capacity can achieve \(\Theta\left(\frac{mR^{2}}{\Delta^{2}}\right)\) and the upper bound can be reached if and only if the transmission range satisfies \(l_{i}=\Theta\left(R\sqrt{\frac{m}{n}}\right)\), where \(i\) is a duplication in a cluster, and \(l_{i}=\Theta(1)\), where \(i\) is a duplication between clusters. Each node can achieve \(\Theta\left(\frac{mR^{2}}{n\Delta^{2}}\right)\)._  
We dismiss the proof here for simplification, as it is similar to the proofs of Theorem 1 and Theorem 2 in [19].  
The above conclusion does not consider the limited ratio resources that messages cannot be sent out due to congested channels when many nodes send messages simultaneously in the same area. If the channel capacity can transmit \(W\) bits per second during the whole network alive time \(T\), each bit \(b\) at \(ht\)-hop satisfies  
\[\sum_{b=1}^{\lambda nT}\sum_{h=1}^{h(b)}1\leq\frac{WnT}{2}\leq WnT\,. \tag{22}\]  
According to (16), there are no other nodes in the disk radius \(\frac{\Delta}{2}\) times transmission range. Each node consumes \(\frac{1}{n}\) in area \(n\) averagely, thus, the summing area that source nodes consumes meets  
\[\sum_{b=1}^{\lambda nT}\sum_{h=1}^{h(b)}S_{b}\leq WT\,, \tag{23}\]  
where \(S_{b}\) is the area of bit \(b\) and \(h(b)\) is the hop of bit \(b\). The following lemma can be achieved with the above idea:  
**Lemma 1** (Throughput inequation considering ratio resources).: _Considering ratio resources, the throughput of bit b under sparse cluster regime should meet the following inequation:_  
\[\sum_{b=1}^{\lambda_{n}nT}\frac{\Delta^{2}}{4}\frac{\mathbb{E}[R_{d \bar{b}}]-1}{n}+\mathbb{E}\left[\sum_{b=1}^{\lambda_{n}nT}\sum_{h=1}^{h_{b}^{ \ast}\frac{nR_{b}^{\ast}}{mR^{2}}}\frac{\pi\Delta^{2}}{4}\frac{lh^{2}_{b}}{mR^ {2}}\right]\leq c_{1}^{s}WT\log n\,, \tag{24}\]  
_where \(c_{1}^{s}\) is a positive constant._  
The proof is omitted for simplification as it is similar to Proposition 3 in [20] and Lemma 3.3 in [2]. With the above lemma and theorem, we can deduce the following theorem:  
**Theorem 2** (throughput capacity under sparse clustering regime with incentive schemes).: _In a sparse clustering regime, the throughput capacity with incentive schemes is_  
\[O\left(\min\left\{\sqrt{\frac{n\log n}{mR^{2}\Delta^{2}}},\frac{mR^{2}}{n \Delta^{2}}\right\}\right) \tag{25}\]  
_and the upper bound can be achieved when_  
\[l=\Theta\bigg{(}R\sqrt{\frac{m}{n}}\bigg{)}.\]  
The proof is shown in Appendix A.