## 3 Model and Assumptions
### An Example  
To illustrate this deployment paradigm, consider a square sensor field of side length \(1000\) units as used in the simulation experiments we conduct and by [PERSON] _et al._[18]. Suppose the sensor network architect decides to partition this field into territories that are \"grids\" with four rows and five columns, as shown in Figure 1. In this particular example, there are \(1000\) nodes; the communication range for each node and the standard deviation, \(\sigma\), were both set to \(50\) distance units. These \(1000\) sensor nodes are now evenly distributed into these \(20\) groups with \(50\) nodes in each group. The group deployment points for each group are the center of the grid territory that the nodes were being deployed in, as shown in the figure. We use this particular network topology in our simulation experiments and for further discussions in this paper for ease of exposition, but we stress that the functionality of **Disccard** is not limited to this particular grid topology.  
In our protocol, we distinguish between nodes that are deployed close to their intended deployment point from nodes that are located further away from their deployment point. To this end, we define the home zone of a group as the region of its territory that is no more than distance \(r\) units from its deployment point. In other words, the home zone of a group is defined as a circle of radius \(r\), which is a parameter of the system, centered at the deployment point as shown in Figure 1. Nodes of a group \(G\) that are deployed within distance \(r\), _i.e._, within the home zone, are called safe nodes, and nodes that are located outside the home zone are called unsafe nodes. The terms safe and unsafe are meant to reflect the roles that these nodes play in our algorithm, as we describe in Section 5.  
Intuitively, safe nodes are tasked with the responsibility of detecting and disseminating the identities of the duplicate nodes and are implicitly trusted as being genuine (not duplicates) until proven otherwise, while unsafe nodes are burdened with the task of having to prove themselves as genuine and are treated as duplicates until they have proven otherwise.  
Population Density  
Since we chose to model our deployment process as a two-dimensional Gaussian distribution, the probability of a node to reside in its home zone, and consequently be a safe node, can be expressed as \(1-e^{-\frac{r^{2}}{2\sigma^{2}}}\), where \(r\) is the home zone radius and \(\sigma\) is the standard deviation of the Gaussian distribution. Suppose, we choose the home zone radius \(r\) to be \(100\), as shown in Figure 1. Further, set the communication range of each node and the standard deviation \(\sigma\) to each be \(50\) distance units.  
Figure 1: Group-deployed sensor network topology.  
The probability of a node to be safe is about \(0.86\). Therefore, for a group consisting of \(50\) nodes, we expect to see about six unsafe nodes and \(44\) safe nodes in a particular group territory. In our empirical experiments, we observed that the number of unsafe nodes is about five per group on average, which implies that 45 nodes reside in their home zone. Indeed, if the deployment accuracy were poor, _i.e._, a higher standard deviation for the distribution, we would find a greater number of unsafe nodes. Since our algorithm differentiates between the safe and unsafe nodes, the costs associated with our algorithm are quite dependent on the deployment accuracy. In this paper, we conduct our simulation experiments and present the costs associated with \(\mathbf{Discard}\) for \(\sigma=50\), \(r=100\), and the communication range of each node at full power equals 50, as done in prior work.