## 4 Theoretical Analysis
### Impact of the Stretch Factor
#### 4.2.2 Load Balance  
The relationship between \(\epsilon\) and the network's load balance is not straightforward. In the following, we present a preliminary and intuitive theoretical analysis of this relationship. Let us consider a simple network topology consisting of only one hole and one source-destination pair as shown in Figure 3. Let us denote by \(\mathcal{C}\) the hole, by \(s\) the source node and by \(t\) the destination node. Denote by \(l_{1}\) and \(l_{2}\) the two Euclidean paths that bypass \(\mathcal{C}\) and that have all the vertices be the vertices of \(\mathcal{C}\) as shown in Figure 3. Suppose that \(l_{1}\) is shorter than \(l_{2}\). Then, if \(\epsilon\) is sufficiently large, the base path candidates are both \(l_{1}\) and \(l_{2}\). Otherwise, the base path candidates consist of only \(l_{1}\). Suppose \(l^{\prime}\) is a Euclidean path. Without loss of generality, we assume that \(l^{\prime}\) is the image of \(l_{1}\) through a homothetic transformation with the center \(I\) and the scale factor \(\xi\). Please note that \(\xi\) is defined by Formula (1), and it can be written as:  
\[\xi=\gamma_{1}\epsilon+\gamma_{2} \tag{2}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) are constants that the same for all packets from \(s\) to \(t\). Suppose \(V_{1},...,V_{n}\) are the vertices of \(\mathcal{C}\) that are vertices of \(l_{1}\). Let \(V_{i}^{\prime}\) be the image of \(V_{i}\) through the homothetic transformation (\(\forall i=1,...,n\)). Then, obviously \(V_{i}^{\prime}\) is the image of \(I\) through the homothetic center with the center \(V_{i}\) and the scale factor \(\xi-1\). As \(I\) locates inside \(\mathcal{C}\), \(V_{i}^{\prime}\) must stay inside the image of \(\mathcal{C}\) through the homothetic transformation with the center \(V_{i}\) and the scale factor \(\xi-1\) (denoted as \(\mathcal{C}_{i}\)) (see Figure 3). Let \(\mathcal{B}\) be the union of all \(\mathcal{C}_{i}\) (\(\forall i=1,...,n\)). Please note that when \(\epsilon\) is sufficiently small such that \(\mathcal{B}\) stays inside the network, then \(\mathcal{B}\) is the trajectory of all routing paths from \(s\) to \(t\). Accordingly, the trajectory of sensor nodes that participate in forwarding packets are polygons that are limited by the boundaries of \(\mathcal{C}_{i}\) and the base paths. In Figure 3, this trajectory are polygons whose boundaries are the green lines and the blue lines. Let \(\epsilon_{1}\) and \(\epsilon_{2}\) are two positive number where \(\epsilon_{1}>\epsilon_{2}\). Let \(\mathcal{B}_{1}\) and \(\mathcal{B}_{2}\) be the trajectories of sensor nodes participating in forwarding packets when \(\epsilon=\epsilon_{1}\), and \(\epsilon=\epsilon_{2}\), respectively. Figure 4 illustrates \(\mathcal{B}_{1}\) and \(\mathcal{B}_{2}\), in which the boundary of \(\mathcal{B}_{1}\) is colored orange and the boundary of \(\mathcal{B}_{1}\) is colored green. In the following we will prove the following statement: _\"If \(\mathcal{B}_{1}\) and \(\mathcal{B}_{2}\) stay inside the network, then the probability for a node to forward a packet in the case \(\epsilon=\epsilon_{1}\) does not exceed that in the case \(\epsilon=\epsilon_{2}\)\"_.  
Figure 4: The trajectories of Euclidean routing paths with different values of \(\epsilon\). The trajectory of Euclidean routing paths when \(\epsilon=\epsilon_{1}\), is bounded by the green line. The trajectory of Euclidean routing paths when \(\epsilon=\epsilon_{2}\), is bounded by the orange line [38]. Figure 3: The trajectory of Euclidean routing paths. The base paths \(l_{1}\) and \(l_{2}\) are colored blue, the Euclidean path \(l^{\prime}\) is colored red [38]. Denote \(\xi_{1}\) and \(\xi_{2}\) the scale factors in the homothetic transformations in the cases \(\epsilon=\epsilon_{1}\) and \(\epsilon=\epsilon_{2}\), respectively. According to Formula (2), \(\xi_{1}\) and \(\xi_{2}\) is proportional to \(\epsilon_{1}\) and \(\epsilon_{2}\), respectively. Let \(\mathcal{R}_{1}\) be a Euclidean routing path obtained in the case \(\epsilon=\epsilon_{1}\).