## 5 Performance Analysis
\begin{array}{c}{P_{d3}}^{\prime}(n)\geq{P_{3}(n)\times(\frac{1-\gamma-\eta }{1-\eta})}\\ \geq\sum\limits_{k=1}^{n}{p\times(1-p)^{k-1}\times P({N_{m}}^{\prime}=k)\times( \frac{1-\gamma-\eta}{1-\eta})}\end{array}  \tag{43}\]  
Considering the balance between storage cost and detection efficiency, every node has \(\sqrt{N}\) witness nodes, and thus the detection probability after n epochs in step 3 is:  
\[ \begin{array}{c}{P_{d3}(n)\geq 1-(1-{P_{d3}}^{\prime}(n))\overset{(\sqrt{N} }{2})}\\ \end{array}  \tag{44}\]  
The metrics used to evaluate efficiency of the proposed protocol are:  
* Communication overhead: the average amount of packets which are transmitted and received by each node when the protocol for detecting node clone attacks works in networks, which is expressed as \(C_{com}\). * Storage overhead: the average amount of copies of the time-location claims or detection information that need to be stored in a sensor when the protocol for detecting node clone attacks works in networks, which is expressed as \(C_{mem}\). * Computation overhead: the average amount of public key signing and verification operation for each node, which is denoted as \(C_{cp}\). In the ESHT scheme, the communication overhead \(C_{com}\) is calculated as: \[C_{com}=C_{t}+C_{f}+C_{e}\] (45) where \(C_{t}\) is the communication overhead of receiving time-location claim requests from the encountered track nodes, and \(C_{f}\) is the communication cost of replying time-location claims to the track nodes, \(C_{e}\) is the communication cost of forwarding the detection information between the trace node. The probability that each node encounters \(i\) trace nodes in one epoch is \(P_{t}(i)\): \[P_{t}(i)=\frac{\left( \begin{array}{c}\sqrt{N}\\ i\end{array} \right)\left( \begin{array}{c}N-\sqrt{N}\\ pN-i\end{array} \right)}{\left( \begin{array}{c}N\\ pN\end{array} \right)}\] (46) where \(p\) is the chance that any two nodes encounter in one epoch, \(N\) denotes the amount of sensor nodes in the mobile wireless sensor networks, \(pN\) denotes the average number of nodes which one sensor node encounters in one epoch. We assume a sensor node encounters \(i\) witnesses in one epoch, and then this sensor node would receive \(i\) time-location claim requests and reply \(i\) time-location claims to those witnesses. Assume \(E(i)\) is the average amount of packets which are transmitted and received by each node when a sensor encounters its witnesses: \[E(i)=\sum_{i=0}^{\sqrt{N}}2i\times P_{t}(i)\] (47) Substituting Equation (46) into (47) to obtain \(C_{t}\): \[C_{t}=E(i)=2p\times\sqrt{N}\] (48) The probability that each node encounters \(j\) traced nodes in one epoch is \(P_{f}(j)\): \[P_{f}(j)=\frac{\left( \begin{array}{c}\sqrt{N}\\ j\end{array} \right)\left( \begin{array}{c}N-\sqrt{N}\\ pN-j\end{array} \right)}{\left( \begin{array}{c}N\\ pN\end{array} \right)}\] (49)  
In the same way, the communication cost of replying time-location claims to the track nodes is obtained as:  
\[C_{f}=E(j)=2p\times\sqrt{N} \tag{50}\]  
The probability that any two sensor nodes have k same tracked nodes in one epoch is \(P_{e}(k)\):  
\[Pe(k)=\frac{\left( \begin{array}{c}\sqrt{N}\\ k\end{array} \right)\left( \begin{array}{c}N-k\\ \sqrt{N}-k\end{array} \right)\left( \begin{array}{c}N-\sqrt{N}\\ \sqrt{N}-k\end{array} \right)}{\left( \begin{array}{c}N\\ \sqrt{N}\end{array} \right)^{2}} \tag{51}\]We assume two sensor nodes have \(k\) same tracked nodes.