# Recharging Schedule for Mitigating Data Loss in Wireless Rechargeable Sensor Network
## 3 Network Model and Charging Scheme
### Node's Criticality Definition  
In this work, the criticality is regarded as a quantification of node's contribution in routing path construction, or connectivity maintenance. We introduce a metric called criticality index to measure node's criticality. For each node \(i\), we define its neighbor sets as \({N_{b}}^{i}=\{j\in N|d(i,j)\leq d_{r},i\
eq j\}\), where \(d(i,j)\) is the distance between node \(i\) and node \(j\). The node \(i\)'s criticality index \(r_{i}\) is given by:  
\[\psi_{ij}=\frac{\left|{N_{b}}^{j}\right|-\left|{N_{b}}^{j}\cap{N_{b}}^{i} \right|}{\left|{N_{b}}^{j}\right|} \tag{1}\]  
\[r_{i}=\sum_{j\in{N_{b}}^{i}}\psi_{ij} \tag{2}\]  
where \(\psi_{ij}\) is the dissimilarity ratio, which is a normalized indicator that takes a value between 0 and 1 to measure the difference between node \(i\) and node \(j\)'s neighbor sets. If node \(i\) and node \(j\) have a large number of common neighbors, the dissimilarity ratio \(\psi_{ij}\) is small and, on the contrary, \(\psi_{ij}\) is large if node \(i\) and node \(j\) have little common neighbors. The criticality index \(r_{i}\) is calculated by summing the dissimilarity ratios of node \(i\)'s neighbors, which can be treated as an indicator of the node's contribution in network connectivity. As such, the criticality index is determined by the node's degree and the diversity of its neighbors. Firstly, if a node has higher node degree, it would have more neighbors and own greater possibility to become  
Figure 2: Illustration of the network architecture. a relay node in numerous shortest routing paths than other nodes with a lower degree, which implies that the node owns higher criticality in connectivity maintenance. Besides, if a node is an isolated node and does not have any neighbors from the beginning of the network, its criticality index is 0. Secondly, the greater dissimilarity between the node's and its neighbors' neighbor sets, the higher criticality the node owns. We use the example network of Figure 3a to explain this part. In this network, nodes B and E have the same node degree, however, they own different criticality. Node B has two neighbor nodes, A and C. Node A's neighbor is node B, while node C's neighbors are node B, D, and E, thus node B doesn't have the same neighbor nodes with node A or C. Like node B, node E also has two neighbor nodes, C and D. According to Equation (2), the criticality index of node E is smaller than that of node B due to the fact that C and D have the same neighbor E. The link between node C and D enhances the redundancy of connections, thus if node E is removed from the network, shown in Figure 3b, node C and D will be still connected. On the contrary, shown in Figure 3c, if node B is removed, the network will be split into two parts, and then node A will become a disjointed node. It demonstrates that node with a higher diversity of neighbors has a higher influence in network connectivity, and it should be treated as a higher criticality node. However, only considering the node's connectivity contribution is not enough to make a proper charging schedule for the mobile charger, where the energy criticality is also an important indicator that should be considered. If we ignore the nodes' charging demand and only care about the connectivity contribution, the node with great contribution in connectivity will always be charged even if their energy is full capacity, on the other side, the node with less contribution like the edge nodes will never get the chance to be charged, causing a coverage hole in the network. To avoid that, the energy criticality, which represents a normalized ratio of the node's consumed energy to the full energy capacity, is adopted to weight the criticality index and is defined as below:  
\[\rho_{i}=\frac{E_{max}-E_{i}}{E_{max}-E_{min}} \tag{3}\]  
\[\hat{r}_{i}=\rho_{i}r_{i} \tag{4}\]  
where \(E_{i}\) denotes the residual energy level of node \(i\) when the mobile charger starts a new trip; \(\rho_{i}\) denotes the energy criticality and \(\hat{r}_{i}\) denotes the weighted criticality index.