## 3 System Framework  
According to its definition, a legacy quorum system ensures that all the nodes in a network can successfully and directly discover and rendezvous with other nodes, and the system focuses on only pairwise and direct discovery and rendezvous. Although implementing it is extremely easy, the energy consumption for each node is high. However, through the proposed extended quorum system, the concept of indirect reachability can be used to discover and rendezvous through already discovered neighbors. For the example in Figure 4b, the information of node B can be indirectly propagated to node D through the intermediate node A. Thus, the redundant active time slots can be eliminated. However, the extended quorum graph does not completely support the mobile environment, especially in the flexible network design and device energy consideration. Mobile devices are represented as mobile nodes in a mobile environment, and they constantly reconfigure. The schedule of active time slots should be modified for the nodes that join or leave, when the device owners enter or leave the area. However, according to the extended quorum graph, the node system takes time to rearrange the active time slots when the node structure is changed. In addition, the battery life for each node is different. To extend the network life and strengthen the network  
Figure 3: Example of (**a**) legacy quorum graph; and (**b**) extended quorum graph. performance, the proposed method should consider the battery usage. Thus, we designed and proposed our Weighted Quorum method (WQS). The main concept of the proposed WQS method is simple and based on the following observations: (1) flexibility: the optimizing arrangement of active time slots; (2) mobility: in a mobile environment, configuring schedules for the new nodes in the network is easy; and (3) energy efficiency: the consideration of the nodes' energy state in the active time slot arrangement. In our developing idea, we try to maintain the most connections per row in the quorum graph, and the redundant connections would be deleted to increase the energy efficiency. For maximal contribution to pairwise connections in our study, we defined the decision factor in Equation (1) for a vertex \(x_{i}\) in each row, \(f_{x_{i}}\), is used as the idea to select the needed pairwise connection. Thus,  
\[ \begin{array}{c}f(x_{i})=(1-\alpha)b_{x_{i}}+\alpha a_{x_{i}}\\ f_{y}=\ (1\ -\ \alpha)b_{y}+\ \alpha a_{y}\end{array}  \tag{1}\]  
where \(b_{x_{i}}\) is the node energy status, and the accumulative wakeup factor is \(a_{x_{i}}\), which is the reciprocal of the connecting numbers. The energy status is the ratio of the remaining power for each node. The decision factor \(f(x_{i})\) is used to decide the candidates among the selected vertices. The \(\alpha\) depends on the system requirement. Because the proposed system is used in a mobile environment, the energy status could be distinct for each node in the environment. The accumulative wakeup factor is the reference status for each node. The more the node was referenced, the lower the reference status is than those of the other nodes. In our experiment, \(\alpha\) is defined as 0.5 in our system evaluation. For the purpose to illustrate the importance of reference rate and energy status, we compare the distinct results of \(\alpha\) value in the experiment 4.3. Accordingly, we divide the quorum graph into two parts by the row with the most vertex numbers in one row. In other words, assume that the network has \(M\) nodes and \(N\) active slots. Thus, we have \(X=[x_{1},x_{2},...,x_{M}]\) and \(A=[A_{1},A_{2},...\,A_{N}]\), a total of \(N\times(N\ -1)\) connections is retained with a minimal number of active slots in the legacy quorum graph. For the active slot \(A_{i}\), we select the slot with many nodes as \(A_{q}\), \(1\leq i,q\leq N\), Then, then the quorum has been divided, \(A=[A_{1},A_{2},...\,A_{N}]=[\overline{A},A_{q},\underline{A}]\), where \(\overline{A}=[A_{1},A_{2},...\,,A_{q-1}]\) and \(\underline{A}=[A_{q+1},A_{q+2},...\,A_{N}]\).