# CoDA: Collaborative Data Aggregation in Emerging Sensor Networks Using Bio-Level Voronoi Diagrams
## 3 System Model
### Structure of Voronoi-Cluster  
From the point of view of the Voronoi structure corresponding to the computational geometry, the plane is divided into \(N\) zones by the Voronoi diagram, which depends on the number of discrete points \(N\). The basic Voronoi diagram describes the areas that are nearest to a set of given points, and these can be viewed as zones of control [41]. Therefore, dividing the space plane into several regions and designating the sensor nodes into corresponding Voronoi-clusters, can ensure that all nodes in a single cluster are evenly distributed and make the distance between the member nodes to its cluster head as short as possible. In ESNs, the selection of cluster heads plays a very important role in ensuring energy efficiency, and the main objective of cluster head selection is to make the cluster distribution reasonable. On the one hand, it is necessary to make the member nodes evenly distributed, so the communication overhead can be maintained at a low level. On the other hand, considering the inter-cluster communication, the distance between cluster head nodes should be within a reasonable range so as to keep the multi-hop transmission losses small. Therefore, to reduce the transmission losses and ensure insignificant difference between the member nodes in the aspect of cluster-in energy consumption, the topological structure of the network should be optimized. Meanwhile, by reducing the energy consumption in the multi-hop transmission as much as possible, ultimately it can achieve the balance of energy consumption in the whole network, and improve the life cycle of the network. Due to the vulnerability of nodes and the variability of network topology, once some sensors with poor performance are selected as cluster heads it will seriously affect the quality of service of the network. The reliability of cluster heads and the distribution of clusters become the keys to determine whether the network can run efficiently. In order to analyze the influence of the number of cluster heads on the overall energy consumption of the network, the first order radio model can be used for measuring energy consumption between sensor nodes. Equation (2) represents the amount of energy consumption in transmitting a packet with \(l\) bits over \(d\) distance according to the first order radio model [42]:  
\[E_{Tx}(l,d)=\left\{ \begin{array}{l}lE_{elec}+l\varepsilon_{fs}d^{2},d<d_{0}\\ l\varepsilon_{elc}+l\varepsilon_{mp}d^{4},d\geqslant d_{0}\end{array} \right. \tag{2}\]  
Figure 1: The model of a bio-level Voronoi diagram. In which, \(E_{elec}\) is the electronics energy, \(\epsilon_{mp}d^{4}\) is the amplifier energy of the next CH, and \(d_{0}\) is a threshold. The energy consumption of \(b\) bits received by the receiver is \(E_{RX}=lE_{elec}\). The optimal number of clusters can be solved by calculating the energy consumed per round. It is assumed that the \(N\) sensor nodes are distributed evenly in the area of \(M\times M\), and the \(k\) cluster heads are generated by the election network. During the initial stage, the energy consumption depleted for message exchange between cluster head and its members can be calculated as:  
\[ \begin{split} E_{init}&=kE_{CH-broad}+(N-k)E_{ member-broad}\\ &=k(l_{0}E_{elec}+l_{0}\epsilon_{mp}R^{4})+(N-k)l_{0}E_{elec} \end{split}  \tag{3}\]  
In order to ensure that all nodes in the network receive the broadcast message, the radius \(R\) should be set as the maximum distance between any two nodes in the area. Under normal circumstances, \(R>d_{0}\) and power amplifier consumption uses multipath fading channel model. During the stage of the cluster formation, the \(k\) cluster heads receive the JOIN message sent by the respective member nodes and broadcasts the TDMA time slot table. Then, \(N-k\) member nodes send JOIN messages and receive the TDMA slot table. Because the distance between the cluster heads is relatively small, the power amplifier power consumption is free space channel model.