## 4 Analysis  
In this section, we evaluate the performance of our mechanism. We build models and analyze the performance of the proposed scheme based on a two-dimensional mobile sensor network where sensor nodes freely move throughout the deployed area. The unit disk graph (UDG) communication model is adopted as the analysis component of most of the literature about sensor networks [6, 15, 16]. All direct communication links between sensor nodes are assumed to be bidirectional. This communication model is common in the current generation of sensor networks [17]. We mainly focus on how to mitigate malicious attacks, especially how to improve detection performance for replication attacks in the commonly used communication models.  
The communication range of each node is denoted by \(R\). Since a perfect circular communication cutoff cannot be expected in a real environment, the value \(R\) actually defines the circular area where all the nodes can be reached properly. Although two nodes may still communicate over a distance greater than \(R\), we ignore these cases to make the value of \(R\) strict enough to ensure the reliability of neighborhood communication.  
We assume that the sink node transmits with higher power than sensor nodes and hence has a sink-to-sensor communication range of \(\mathtt{L}>\mathtt{R}\). Note that to achieve a communication range ratio \(\mathtt{L}/\mathtt{R}\), the sink needs to transmit with power \(P_{S}=\left(L\,/\,R\right)^{\gamma}(P_{n}\,/\,G)\), where \(P_{n}\) is the sensor transmission power, \(\gamma\) is the signal attenuation factor and \(\mathtt{G}\) is the directivity gain. Given that sensors are very low power devices, the higher transmission power assumption on the sink side is reasonable. A typical sensor has a communication range from 3 m to 30 m with a transmission power of \(P_{n}=0.75\) mW. Hence, the sink needs to transmit with a power \(P_{S}=75\) mW to achieve a communication range ratio \(L\,/\,R=10\) when \(\gamma=2\).  
We assume that GPS systems or localization schemes have been applied to the network such that all nodes know their own positions. The core idea of the proposed scheme is to identify the authenticity of the location claimed by a node with the aid of the benign witness nodes. Especially when a replication attack is locally detected, it is a crucial prerequisite that there are witness nodes in a specific region suitable for observation.  
Recall the statistics theory for spatial node distribution. The random deployment of the network nodes can be modeled as a spatially homogeneous Poisson point process [14]. The random placement of the nodes with a density \(\rho_{{}_{N}}\rightleftharpoons|N\,|\,/\mathcal{A}\) is equivalent to a sequence of events following a homogeneous Poisson point process of rate \(\rho_{{}_{N}}\rightleftharpoons|N\,|\,/\mathcal{A}\). Given that N events occur in area \(\rho_{{}_{N}}\rightleftharpoons|N\,|\,/\mathcal{A}\), these events are uniformly distributed within \(\rho_{{}_{N}}\rightleftharpoons|N\,|\,/\mathcal{A}\). Let S denote the set of nodes located in area \(\alpha\), giving us:  
\[P(|\,S\leftleftharpoons k)=\frac{(\rho_{{}_{N}}\alpha)^{k}}{k!}e^{-\rho_{{}_{N }}\alpha} \tag{4}\]  
Using Equation (2), we can deduce the probability that there are at least k nodes within area \(\alpha\) :  
\[P(|\,S\leftrightharpoons k)=1-\sum_{i=0}^{k-1}\frac{(\rho_{{}_{N}}\alpha)^{i}}{ i!}e^{-\rho_{{}_{N}}\alpha} \tag{5}\]