## 3 Link Persistence Analysis  
The failure of any link within a path results in the failure of the entire path. Hence, the stability of a path is contingent upon the stability of all the links that constitute it [25]. In periods of non-breakage, the optimal path is characterized by maximal capacity and minimal interference. Therefore, an analysis of link persistence and interference is essential for enhancing network reliability, and an objective function is necessary to quantify topology performance. In this model, link persistence is defined as the probability that an active link exists between two mobile nodes at the time \(\mathrm{t}_{0}+\mathrm{t}\), provided that this link was available at the time \(\mathrm{t}_{0}\)[13]. To illustrate the link connectivity between two nodes over a given time span, it is essential to analyze the motion distribution of individual initially. Subsequently, the connection probability of a node is calculated by considering the magnitude and direction of one moving node relative to another, which represents the impact of motion on link persistence. At time \(\mathrm{t}\), the relationship between the distance separating two nodes and link persistence is established. The distance \(\overset{\rightarrow}{\mathrm{D}_{M,\ N}}(\mathrm{t})\) is the result of the sum of the random mobility vector \(\overset{\rightarrow}{\mathrm{s}}(\mathrm{t})\) and the constant vector \(\overset{\rightarrow}{\mathrm{c}}\), as illustrated in Figure 4. The solution corresponds exactly to the general problem of determining the sum of a Rayleigh-distributed vector and a constant. By decomposing \(\overset{\rightarrow}{\mathrm{s}}(\mathrm{t})\) into its X and Y components from the Gaussian distribution and adding these to the \(\mathrm{x}_{\mathrm{c}}\) and \(\mathrm{y}_{\mathrm{c}}\) components of the initial distance, the link persistence distance probability distribution function is obtained:  
\[ \begin{array}{ll}\mathrm{f}_{\mathrm{D}_{M,\ N},\mathrm{\theta}_{D}}( \mathrm{d},\theta)&=\mathrm{f}_{\mathrm{x},\mathrm{y}}(\mathrm{x},\mathrm{y}) \begin{pmatrix}\cos\theta&-\mathrm{d}\mathrm{s}\mathrm{i}\,\theta\\ \sin\theta&\mathrm{d}\mathrm{d}\mathrm{s}\,\theta\end{pmatrix}\\ &=\frac{\mathrm{d}}{\pi\alpha_{\mathrm{M,\ N}}}\exp\left[-\frac{(\mathrm{d} \cos\theta-\mathrm{x}_{\mathrm{c}})^{2}+\left(\mathrm{d}\mathrm{s}\mathrm{i} \,\theta-\mathrm{y}_{\mathrm{c}}\right)^{2}}{\alpha_{\mathrm{M,\ N}}}\right]\\ &=\frac{\mathrm{d}}{\pi\alpha_{\mathrm{M,\ N}}}\exp\left[-\frac{\left(\mathrm{d} ^{2}+\mathrm{C}^{2}\right)-2\mathrm{d}\mathrm{x}_{\mathrm{c}}\mathrm{cos} \,\theta+\mathrm{y}_{\mathrm{c}}\mathrm{s}\mathrm{i}\,\theta\right)}{\alpha_{ \mathrm{M,\ N}}}\end{pmatrix},\end{array}  \tag{3}\]  
where \(\mathrm{C}^{2}=\mathrm{x}_{\mathrm{c}}^{2}+\mathrm{y}_{\mathrm{c}}^{2}\). When node N reaches the edge of node M's communication range, the angle between line M'N' and the X-axis can be determined using the law of cosines, resulting in an angle of  
\[\theta_{D}\ =\ \theta_{Z}\ +\ \theta_{Z,\ D}\ =\ \theta_{Z}+\arcsin\left\{ \frac{\mathrm{C}}{\mathrm{R}}\mathrm{sin}[\theta_{Z}\ +\ (180^{\circ}-\ \theta_{\mathrm{c}})]\right\}. \tag{4}\]  
Figure 3: Single-node joint mobility transformation. The link persistence \(\rm{A_{M,\ N}}\) corresponds to the probability \(\rm{P_{M,\ N}}\) that \(\rm{D_{M,\ N}(t_{0})}\ \leq\ R\), and remains \(\rm{D_{M,\ N}(t_{0}+t)}\ \leq\ R\), which can be calculated by integrating the probability distribution function of distance. The probability distribution function of distance is shown in Equation (3). Hence, the link persistence is expressed as follows:  
\[ \begin{array}{ll}\rm{A_{M,\ N}}&=\ \rm{P_{M,\ N}}[\ D_{M,\ N}(t_{0}+t)\ \leq\ R\ |\ D_{M,\ N}(t_{0})\ \leq\ R]\\ &=\int_{0}^{R}\int_{0}^{2\pi}\rm{f_{D_{M,\ N}\Theta_{D}}}(d,\theta)d\theta d \theta d\theta d\\ &=\frac{d}{\pi\alpha_{M,\ N}}\int_{0}^{R}\int_{0}^{2\pi}\exp\left[-\frac{\left( d^{2}+C^{2}\right)+2d(Cos\theta_{C}+cos\theta+Csin\theta_{C}sin\theta)}{\alpha_{M,\ N}}\right]d\theta d \theta d,\end{array}