## 4 Objective Function Under Different Transmission Modes
### Direct Transmission  
According to the initial positions depicted in Figure 4, the initial positions of the nodes determine whether the source S and destination D are adjacent. Specifically, when the distance between S and D is less than the effective communication radius R, the nodes are deemed adjacent. Here, the source S is centrally located within the circle. For normal point-to-point direct transmission, the destination D must lie within the effective communication radius of the source S. In wireless communication, signal attenuation occurs during transmission. The received signal power \(\rm{P_{r}}\) can be denoted as \(\rm{P_{r}}=\ P_{t}\ \times\ d^{-\lambda}\), where \(\rm{P_{t}}\) represents the transmitted power from the antenna, d represents the distance between nodes, and \(\lambda\) is the path loss exponent. The Signal-to-Noise Ratio (SNR) of the received signal is calculated as  
\[\rm{\gamma_{S,\ D}(t)}=\frac{P_{t}\ \times\ d_{S,\ D}^{-\alpha}(t)}{N_{0}}. \tag{6}\]  
Figure 4: Relationships among variables.  
The outage capacity is employed to characterize the link rate, representing the achievable link capacity under a specified low outage probability \(\varepsilon\). For direct transmission, the outage capacity is given by  
\[\mathrm{C}_{\mathrm{DT}}^{\varepsilon}(\mathrm{t})=\mathrm{Blog}_{2}\left[1+ \frac{1.5\varepsilon\,\times\,\gamma_{\mathrm{S,\,D}}(\mathrm{t})}{0.2- \varepsilon}\right]. \tag{7}\]  
To guarantee the transmission quality, the interference along the transmission path should account for the cumulative effect of all nodes involved in the transmission process. Specifically, in the case of direct transmission between two nodes, the interfering nodes comprise those encompassed by both the source and destination nodes, so that  
\[\mathrm{I}_{\mathrm{DT}}\,=\mathrm{I}(\mathrm{S})\,\cup\,\mathrm{I}(\mathrm{ D}). \tag{8}\]  
Node mobility introduces variability into their motion states, impacting link connectivity and system stability, and thereby influencing network capacity. The relative motion speed between the source and terminal nodes, denoted as \(\mathrm{V}_{\mathrm{S,\,D}}\), can be represented as the effective value resulting from the sum of the randomly moving velocity vectors \(\mathrm{V}_{\mathrm{S}}\) and \(\mathrm{V}_{\mathrm{D}}\). As the destination node \(\mathrm{D}\) approaches the communication range boundary of the source node \(\mathrm{S}\), the relative motion distance between nodes, governed by the law of cosines, becomes  
\[Z=\mathrm{R}\,\times\,\frac{\sin[180^{\circ}-\,\theta_{\mathrm{Z}}-(180^{\circ }-\,\theta_{\mathrm{C}})-\theta_{\mathrm{Z,\,D}}]}{\sin[\theta_{\mathrm{Z}}+(1 80^{\circ}-\,\theta_{\mathrm{C}})]}, \tag{9}\]  
and the theoretical communication time is \(\mathrm{T}_{\mathrm{S,\,D}}=\frac{Z_{\mathrm{S,\,D}}}{|\mathrm{V}_{\mathrm{S, \,D}}|}\). The link persistence can then be extrapolated along the path. Given the nonlinear nature of link connectivity in MASS communication scenarios, individual links operate independently. Therefore, when a route is on the verge of failure, alternative routes should be preemptively selected based on link persistence. The estimated communication time that takes into account the possibility of link disruption can then be denoted as follows:  
\[\mathrm{T}_{\mathrm{P}}=\mathrm{T}_{\mathrm{S,\,D}}\,\times\,A_{\mathrm{S,\,D }}(\mathrm{t}). \tag{10}\]  
As described above, link persistence, rate, and interference constitute the three primary factors determining network capacity. To optimize network capacity during communication, it is essential to enhance link persistence and rate while minimizing interference. The traffic exchanged between nodes during communication time can be derived by integrating the outage capacity. Consequently, the derived objective function is  
\[\mathrm{g}_{\mathrm{DT}}(\mathrm{t})=\int_{0}^{\mathrm{T}_{\mathrm{S,\,D}} \times A_{\mathrm{DT}}(\mathrm{t})}\frac{\mathrm{C}_{\mathrm{DT}}^{\varepsilon }(\mathrm{t})}{\mathrm{I}_{\mathrm{DT}}}\mathrm{dt}. \tag{11}\]