## 2 D2D Communication System Model
### Social Trust Model  
This section initially explains why relay devices do not always transmit at its maximum power. Then, it presents social trust model that uses Pareto distribution. This trust model is used to determine transmission power of a relay device.  
Generally, mobile devices in communication networks are power constrained. These devices tend to conserve their battery power whenever possible. Even in a cooperative D2D communication network, a relay device may not always transmit at the maximum power possible. This is because when the device transmits at a high power, it drains more of its battery power. A cooperative relay device, which is capable of transmitting at a variable power, prefers to transmit at a low power that is equal to or greater than a minimum threshold value of transmission power. This transmission power is lower than the maximum allowable transmission power.  
The transmission power of a relay is assumed to be linearly proportional to the social trust strength between the relay and the source [12]. The transmission power of a relay can be expressed as  
\[P_{r}=(P_{max}-P_{min})\times\beta_{r,s}+P_{min}, \tag{1}\]  
where \(P_{max}\) is maximum allowable transmission power, \(P_{min}\) is minimum allowable transmission power and \(\beta_{r,s}\) is social trust value that relay has for source.  
The rationale behind this assumption is that when people carrying mobile devices have strong social trust with each other, their willingness for mutual help is comparatively more than those with weak social trust [6; 7]. This behaviour of people can be utilized in designing a relay selection scheme of D2D communications. The willingness of helping a device with high social trust is reflected by transmitting at a higher power. The different ways to collect social information of users to estimate social trust values are described in our previous paper [10]. In this paper, we use Pareto distribution to model social trust among users in the network.  
Generally, social trust among majority of people in a community are weak. Only few people have high social trust between them [12]. The users having strong social relationship with each other usually have high social trust between them. A mathematical model of a social trust would therefore have a heavy tailed distribution. The authors of References [6; 12] have modeled social trust using Pareto distribution to represent a heavy-tailed distribution. Pareto distribution has a probability density function defined as  
\[f_{X}(x)=\frac{\alpha L^{\alpha}x^{-\alpha-1}}{1-\left(\frac{L}{H}\right)^{ \alpha}},L\leq x\leq H,\text{and }\alpha>0, \tag{2}\]  
where \(\alpha\), \(L\) and \(H\) denote the shape, scale and upper limit parameters, respectively [32]. The value of \(\alpha\) determines how fast the distribution tail decays and \(L\) represents the lower limit of the support. Social trust among devices in the network can be varied by varying \(\alpha\) and \(L\) values, while \(H\) is kept constant. The random variable X in Equation (2) is the same as the \(\beta_{r,s}\) used to represent social trust in Equation (1). Each node knows social trust value it has for other nodes.  
In a survey [26], at least 75% of the time, social trust among users in the network are found to be bidirectional. We assume that social trust among users are bidirectional in this paper. When a source can estimate social trust values of other devices, this information helps to determine which devices have the potential to transmit at a high power.