## 5 Performance Analysis
One of them would send or receive \(k\) detection information to another, so we assume \(E(k)\) is the average number of detection requests:  
\[E(k)=\sum_{k=0}^{\sqrt{N}}k\times\mathrm{P}_{e}(k) \tag{52}\]  
Substituting Equation (51) into (52) to obtain \(Ce\):  
\[E(k)=1 \tag{53}\]  
Since one sensor node encounters an average of \(pN\) nodes, so the average number of sending or receiving packets including detection request and detection information is \(Ce\):  
\[Ce=p\times N\times E(k)=p\times N \tag{54}\]  
Substituting Equations (48), (50) and (54) into (36) to obtain \(C_{com}\):  
\[C_{com}=p\times(N+4\sqrt{N}) \tag{55}\]  
so the communication cost of ESHT scheme is \(O(N)\). In our protocol, each node needs to use a digital signature when it sends a packet. Conversely, as soon as a node retrieves a claim, it is needed to verify up to the signature. So the computation cost is in proportion to the communication cost, the computation cost is \(O(N)\). In the ESHT scheme, in order to detect replica attacks, each witness node need to obtain sample, a sample \(v_{L}^{i}\) is computed from two consecutive time-location claims of node \(u_{L}\), according to Equation (3), when a sample \(v_{L}^{i}\) is retrieved, the former time-location claim \(C_{L}^{i-1}\) is abandoned, and only present time-location claim \(C_{L}^{i}\) is stored to wait for the next time-location claim \(C_{L}^{i+1}\). At the same time, to detect replica attacks in the third stage, a queue in which the detection information is stored is maintained. So the fixed length of storage space including a time-location and a queue is required for each node. Every node has \(\sqrt{N}\) tracked nodes. Conversely, every node has \(\sqrt{N}\) trace nodes, so the storage cost of every node is \(O(\sqrt{N})\). The comparison of system overhead between [15; 24; 25] and our proposed scheme is summarized in Table 3.  
An analogue test is carried out to test the feasibility and accuracy of the scheme by using OMNeT++ platform. OMNeT++ is a scalable, modular simulink and framework, mainly for constructing network emulators. In the testing, \(N\) sensor nodes distribution are relatively uniform within a square area of size 1000 m \(\times\) 1000 m, where \(N\) varies from 100 to 1000. The communication range of each node varies from 50 m to 100 m. The Random Waypoint Mobility Model (RWP) is adopted as the movement model. We use the code of [PERSON] et al. [41] to construct the movement model based on Random Waypoint Mobility mode with steady-state distribution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **[PERSON] et al. [15]** & **[PERSON] et al. [24]** & **[PERSON] et al. [25]** & **Proposed Scheme** \\ \hline
**Method** & **Distributed** & **Centralized** & **Distributed** & **Distributed** \\ \hline Communication overhead & \(O(N^{2})\) & \(O(N\sqrt{N})\) & \(O(N)\) & \(O(N)\) \\ Computation overhead & / & \(O(N)\) & \(O(\sqrt{N})\) & \(O(N)\) \\ Storage overhead & \(O(N)\) & \(O(1)\) & \(O(N)\) & \(O(\sqrt{N})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison with [15; 24; 25]. In the movement model, every node randomly selects a speed which is in the interval of 2-8 m/s to move toward a specific waypoint. At the close of each speech, a node stops for a random duration \(T_{stop}\) after moving for \(T\) unit time, where \(T_{stop}\) varies from 0 s to 20 s.  
In the simulation experiment, only one comprised node and its one replica are deployed in the network. The communication between different nodes applies the standard unit-disc two-way communication pattern. At the same time, the IEEE 802.11 protocol is adopted as the medium access control protocol for each node. Assume the user-configured false negative rate \(\gamma=0.01\) and the user-configured false positive rate \(\eta=0.01\). Every experiment is carried out for 1000 simulation seconds, and the average value of 10 experimental results is discussed.