# Capacity Analysis of Incentive Schemes in Opportunistic Networks
###### Abstract  
Many incentive schemes address the selfishness issue in opportunistic networks and show performance improvement by simulations. However, the insights of incentive schemes that affect network performance are not clear. Network capacity analysis can reveal how factors affect performance, which is a guideline for new designs. To analyze incentive schemes, a well-defined mathematical model is necessary, which cannot be achieved by existing analytical models based on empirical formulas or types of incentive schemes. First, this paper proposes such a model to show the incentive degree with the incentive scheme, cooperation degree, energy usage, buffer usage, and security based on a quantum game model. Verification compares the model with delivery ratios that show impacts on selfish nodes in simulations under two typical incentive schemes. Then, network capacity is analyzed with this model and a sparse clustering regime that has similar mobility to opportunistic networks in order to show factors for future designs.  
opportunistic network; analytical model; incentive scheme; network capacity +
Footnote †: journal:  
0000-0001-4700-0000-0000-000  
complex interaction among nodes accurately in the decision-making process. A model with these features can evaluate most incentive schemes and reveal their common character. However, the above model uses an empirical utility formula and focuses on numerical analysis, which simply places datasets into the model without other comparisons. In other words, the model does not have enough factors to work as a guideline to design new incentive schemes.  
A well-defined formula should consider the characteristics of incentive schemes as well as other factors that affect cooperation. A node usually spends payoff, such as virtual money, reputation, exchanging messages, or something else, to stimulate others to forwarding, which can obtain the necessary currency. Cost and reward can represent an incentive scheme in a mathematical model with other properties of nodes. Several incentive schemes take energy usage and buffer usage into account. Therefore, this paper proposes an analytical model based on quantum game theory considering the energy usage and buffer usage, reward, cost, and cooperation degree, which shows the incentive degree and compares incentive degrees with the delivery ratio by simulations.  
Network capacity analysis using the proposed model shows the impact of incentive schemes on network performance. During analysis, a mobile model in opportunistic networks is needed as well. The sparse clustering regime is similar to the mobility of opportunistic networks with nodes that work in groups with high cooperation possibility and clusters that move randomly where communications in groups are stable and movement is otherwise needed for opportunities. Furthermore, the scheduling policy is similar to the scenario in [4] for nautical wireless ad hoc networks. Therefore, the analytical model for incentive schemes is applied to a sparse clustering regime to analyze network capacity and discuss various factors.  
As a summary, the contributions of this paper are as follows: (1) An analytical model based on quantum game theory considering the energy usage, buffer usage, reward, cost and cooperation degree is proposed. It is verified by a comparison between the incentive degree and delivery ratios according to simulations. (2) Network capacity is analyzed with the above model to show how incentive schemes affect network performance. The factors can be used as a guideline for designing efficient incentive schemes by considering these relations.  
The rest of this paper is organized as follows. The mathematical model is detailed in Section 2. Verification with simulations is described in Section 3. Network capacity is analyzed with the model and sparse clustering regime in Section 4. The factors are discussed in Section 5. Finally, we conclude the paper in Section 6.