## 2 D2D Communication System Model  
### Network Model  
Consider a cooperative network which consists of \(n_{total}\) mobile devices under the coverage of a BS. When a source (\(s\)) mobile device wants to have D2D communications with a destination (\(d\)) mobile device using a cellular channel, bypassing the BS, it can be done in two ways. The first way is to send data packets directly from the source to the destination. This is referred to as direct D2D communication. Another way is to initially send the data packets from the source to a relay (\(r\)) mobile device in proximity. The relay receives the packets and forwards them to the destination. This is referred to as D2D communication via a relay. Decode-and-forward (DF) relaying [7; 8; 12; 25] is used for D2D communications to forward data. All mobile devices which are not engaged in communication are considered to be interested to act as a relay. The relay devices do not necessarily transmit at the maximum power while relaying data for other devices [6; 7; 12].  
Figure 1 illustrates the search location of a socially trusted relay device that improves the D2D communication between mobile devices. The circles with \(n\) inside them represent devices in the network, where \(n\) represents the device number. An arrow in a device represents the direction of mobility and \(V_{n}\) underneath represents its velocity. The double-lined circles are source device \(s\) and destination device \(d\). The dashed circle with radius \(R\) represents a relay selection region which is centered at the midpoint \(M\) of the distance between \(s\) and \(d\). The circles (with white background) are devices outside relay selection region. The circles (with green background) are friends of source with high battery power levels within the circular region. Similarly, the circles (with red background) are friends of source with low battery power levels. The circles (with yellow background) are devices which are not friends of source. The social trust value that a device \(n\) has for the source \(s\) is represented by \(\beta_{n,s}\).  
We assume that the BS knows location information of devices. The devices in the network have a Global Positioning System (GPS) or a location tracker that periodically updates the BS about their location [3; 21]. This helps the BS to locate devices within its coverage area and recommend a list of devices that are located within a circular region for relay selection. A socially trusted relay device is selected among the devices that are located within the circle.  
Operator controlled link establishment is used for both direct D2D communication or D2D communication via a relay as envisioned in Reference [26]. When a relay is selected for D2D communications, the same frequency is used for communication between the source to the relay  
Figure 1: A system model for device-to-device (D2D) communications.  
and that from the relay to the destination. The transmission power of D2D communication between mobile devices is determined by the BS. The maximum allowable transmission power, \(P_{max}\), is based on network parameters and to avoid interference to neighbouring cellular devices [7; 8]. The mobile devices having D2D communications transmit at a power \(P_{i,j}\leq P_{max}\). A source always transmits at the maximum allowable transmission power i.e. \(P_{s,j}=P_{max}\). However, a relay may not transmit at \(P_{max}\). We assume the background noise is constant and there is no interference in the system due to high frequency reuse factor [27].  
We explain our proposed relay selections schemes in detail, supported by an example scenario, in Section 3.1.