## 5 Numerical Results
### Performance Analysis  
The numerical results presented in this section arise from the simulation parameters detailed below. We consider a circular area network with a radius of 200 m, a D2D community located far away from the SN \(\mathcal{S}\), within a circular area with radius of 15 m. The transmission power between the SN \(\mathcal{S}\) and one ED is set as \(P_{\mathcal{S}}=0.9\) W, while the D2D transmission power is \(P_{i}=0.6\) W. The reputation coefficient \(\rho_{i}\) assumes values within the interval \((0,1)\). Finally, the path loss exponent \(\alpha\) has been set to 2.5. The proximity graph \(\mathcal{G}_{d}\) is built establishing an edge between two EDs if their mutual distance is less than 10 m. The performance analysis is expressed in terms of delivery transmission delay, number of selected relays and, under the assumption of an equal power consumption for any ED acting as a relay, also in terms of overall power consumption.  
Figure 2 highlights the delivery delay normalized to the packet size achieved using the three different relay selection strategies. As can be seen, the proposed matching-based approach reaches lower values of delivery delay, ensuring a higher system responsiveness compared to the considered alternative approaches. As it is evident to note, by increasing the number of EDs, the delivery delay value decreases. This is due to the fact that the EDs are close to each other and the number of edges in \(\mathcal{G}_{d}\) grows with the size of the EDs set. Hence, with a greater ED density, the EDs are much closer to each other, and the delay is less than when the EDs density is low, i.e., when the EDs are more spread within the network area. Instead, in the same figure, the delay increases when the number of the EDs grows, since it is strictly related to the \(\rho\) coefficients. In fact, considering low values of EDs, the proposed approach is able to clearly select trusted (i.e., with high \(\rho_{i}\) values) devices, avoiding the direct links. Instead, by increasing the density of the EDs in the selected area, our algorithm more often selects direct links, in order to guarantee a higher level of trustworthiness.  
Therefore, it is evident to note the remarkable advantages achieved by applying the relay selection in comparison to the alternatives. Furthermore, the previous result is confirmed by Figure 3, in which the number of EDs is fixed and equal to 50, and the D2D area radius changes within the set \(\{5,10,15,20,25\}\) m. The relay selection reaches better results when the radius has low values, hence when the D2D community is concentrated in a small area. Additionally, in this case the proposed algorithm has better performance for all the radius values.  
Likewise, the validity of the proposed relay selection strategy is again evident in Figure 4, in which the number of the involved relays is lowered in comparison with the MDG and RA alternatives. Finally, Figure 5 illustrates the overall power consumption normalized to the individual relay power consumption (assumed equal for all the involved relays) as a function of the EDs number. This Figure highlights again a better behavior of the proposed algorithm in comparison with the considered alternatives. Figure 6 concludes our analysis by showing the impact of the reputation coefficient value on the relays selection. From this Figure, it is clearly evident that the proposed strategy favors the relays selection with high reputation levels.  
Figure 2: Delivery delay comparisons.  
Figure 4: Number of selected relays.  
Figure 5: Normalized overall system power consumption comparison.  
Figure 3: Normalized delivery delay comparisons for the case of 50 end devices (EDs).