## 5 Theoretical Analysis
### Trail Connectivity  
In TS, the connectivity of the trail determines the reliability of data transmission. Therefore, in this section, we derive the condition to ensure the connectivity of the trail, which can be used as the guideline of setting trail update interval. At first, we provide the formal definition of the trail connectivity. Let \(\mathbf{N}(A_{\theta})\) denote the set of sensor nodes located in the transmission range of the actor when it updates footprint \(\theta\) at time slot \(t_{\theta}\). **Definition 2** (Trail Connectivity). : _For any given footprint \(\theta\), the trail is connected when there exists at least one pair of sensor nodes \((i,j)\) that satisfies \(d_{i,j}\leq R\), where \(i\in\mathbf{N}(A_{\theta})\) and \(j\in\mathbf{N}(A_{\theta+1})\)._  
**Theorem 1**. : _With given grid length \(L\) and the speed of the actor \(v\), the trail is ensured to be connected when the update interval \(T_{u}\) satisfy,_  
\[T_{u}\leq\sqrt{2}\cdot\frac{L}{v} \tag{1}\]  
Proof. : Let \(A_{1}(x_{1},y_{1})\) denote the coordinate of the actor at time \(t_{\theta}\) and \(P(x_{p},y_{p})\) denote a host on the trail that selects the next hop in its neighbor table \(\mathbf{N}(P)\). The transmission range of sensor nodes and actors are constrained by \(R\in(\sqrt{2}L,2L)\). In TS, node \(P\) is closest to \((x_{1},y_{1})\) by comparing with its neighbors, such that, \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(N_{s}\) & The number of sensor nodes in the network area \\ \((x_{i},y_{i})\) & The coordinate of node \(i\) \\ \(H\) & The side length of the network area \\ \(L\) & The side length of a grid \\ \(R\) & The transmission range of nodes \\ \(T_{u}\) & The update interval of the actor \\ \(e\) & The threshold value of trail age \\ \(\theta_{i}\) & The footprint left at node \(i\) \\ \(a_{i}\) & The age of footprint \(\theta_{i}\) \\ \(d_{ij}\) & The distance between sensor node \(i\) and \(j\) \\ \(\mathbf{N}(i)\) & The neighbors set of sensor node \(i\) \\ \(\mathbf{N}(A_{\theta})\) & The neighbors set of the actor when it updates footprint \(\theta\) \\ \(\tau_{A}\) & The waiting time of the actor \\ \(\tau_{P}\) & The waiting time of packet \\ \(S\) & The initial TTL value \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables List. Figure 5: Network Model in Theoretical Analysis. \[ \begin{cases}x_{1}=x_{p}+a_{1}&-\frac{L}{2}\leq a_{1}\leq\frac{L}{2}\\ y_{1}=y_{p}+b_{1}&-\frac{L}{2}\leq b_{1}\leq\frac{L}{2}\end{cases}  \tag{2}\]  
Assume that the actor arrives at \(A_{2}(x_{2},y_{2})\) when it broadcasts the footprint information at time \(t_{\theta+1}\). Without loss of generality, consider a worst case that the actor moves consistently in the diagonal direction, then we have,  
\[ \begin{cases}x_{2}=x_{1}+D_{x}&-A_{x}\leq D_{x}\leq A_{x}\\ y_{2}=y_{1}+D_{y}&-B_{y}\leq D_{y}\leq B_{y}\end{cases}  \tag{3}\]  
In order to guarantee the connectivity of the trail, \(A_{x}\) and \(B_{y}\) need to satisfy the following condition: For arbitrarily chosen \(D_{x}\) and \(D_{y}\), there always exists a node in \(\mathbf{N}(P)\), denoted by \(H(x_{h},y_{h})\) that meets,  
\[ \begin{cases}x_{h}=x_{2}+a_{2}&-\frac{L}{2}\leq a_{2}\leq\frac{L}{2}\\ y_{h}=y_{2}+b_{2}&-\frac{L}{2}\leq b_{2}\leq\frac{L}{2}\end{cases}  \tag{4}\]  
To ensure that the packet can be transmitted successfully from \(P\) to \(H\), node \(P\) and \(H\) have to locate within the transmission range of each other. Then we can derive that,  
\[ \begin{cases}x_{h}=x_{p}\pm L\\ y_{h}=y_{p}\pm L\end{cases}  \tag{5}\]  
It is important to note that the aforementioned conditions can guarantee the connectivity in just one hop. However, the connectivity should be guaranteed for the whole trail. According to Equation (3), we can derive the location of the actor \(A_{n}\) when it broadcasts the footprint information at time \(t_{\theta+n}\),  
\[ \begin{cases}x_{n}=x_{1}+n\cdot D_{x}&-A_{x}\leq D_{x}\leq A_{x}\\ y_{n}=y_{1}+n\cdot D_{y}&-B_{y}\leq D_{y}\leq B_{y}\end{cases}