## 3 Key Technologies in Trust Management for WSNs
### Trust Evaluation Methods
#### Success-failure methods  
The two most common approaches in success-failure methods are Bayesian beta methods and entropy-based methods. [PERSON] et al. [5] analyzed, in detail, the applicability of Bayesian formulations and beta distributions to WSN trust systems. The calculation of trust values using the Bayesian beta method is shown in Equation (4):  
\[T_{ij}=E\big{(}R_{ij}\big{)}=E\big{(}Beta\big{\{}\alpha_{j}+1,\beta_{j}+1\big{\}} \big{)}=\frac{\alpha_{j}+1}{\alpha_{j}+\beta_{j}+2} \tag{4}\]  
where \(\alpha_{j}\) and \(\beta_{j}\) represent the cooperative and noncooperative interactions between nodes \(i\) and \(j\), respectively (from the perspective of node \(i\)). [PERSON] et al. [10] enhanced the Bayesian formulation by incorporating reward and penalty factors. These factors reward or penalize based on the number of successes or failures, as shown in Equation (5):  
Figure 6: Comparison of three threshold-limiting methods. \[T_{x,y}^{Metric}=\frac{N_{x,y}^{+}+1}{N_{x,y}^{total}+2}\cdot\frac{N_{x,y}^{+}}{N_{x,y}^{+}+1}\cdot\frac{1}{\sqrt{N_{x,y}^{-}+1}} \tag{5}\]  
where \(x\) is the evaluating node, which calculates the trust value and counts the number of success and failure hits, and \(y\) is the evaluated node by node \(x\). \(N_{x,y}^{+}\) is the successful count that node \(x\) recorded about node \(y\). \(N_{x,y}^{-}\) is the number of failure hits that node \(x\) experiences with node \(y\). \(T_{x,y}^{Metric}\) is the trust metric computed by \(x\) for \(y\). In Equation (2), the first term is the Bayesian factor, followed by the reward and penalty terms. The rewarding factor ensures a gradual increase in the trust value, whereas the penalty factor ensures a rapid decrease in the trust value after failure. [PERSON] and [PERSON] [3] used penalty and reward parameters that can be flexibly adjusted according to the network and the application, as shown in Equation (6):  
\[T_{x,y}(\Delta t)=\left[4\times\left(\frac{S_{x,y}(\Delta t)}{S_{x,y}(\Delta t )+U_{x,y}(\Delta t)}\right)*\frac{1}{\sqrt{\mu*U_{x,y}(\Delta t)}+1}*\varnothing ^{S_{x,y}(\Delta t)}\right] \tag{6}\]  
where \(T_{x,y}(\Delta t)\) represents the trust value evaluated by node \(x\) for node \(y\) within the time window \(\Delta t\). \(S_{x,y}(\Delta t)\) and \(U_{x,y}(\Delta t)\) represent the number of successful and unsuccessful interactions between node \(x\) and node \(y\), respectively. \(\mu\) and \(\varnothing^{S_{x,y}(\Delta t)}\) are the adjustable penalty and reward parameters, respectively. The value of \(\varnothing^{S_{x,y}(\Delta t)}\) increases with the number of successful interactions and can be adjusted to reflect the network and the application. [PERSON] et al. [32] similarly added a penalty factor in the model they proposed. The entropy trust model uses an entropy function to calculate the trust value based on the Bayesian beta method so that the trust value is not a linear function of probability, as shown in Equations (7) and (8) and Figure 7:  
\[T_{Entr}= \begin{cases}1-H(p),\;\;for\;0.5\;\leq\;p\;\leq 1;\\ H(p)-1,\;\;for\;0\;\leq\;p<0.5.\end{cases}  \tag{7}\]  
\[H(p)=-plog_{2}p-(1-p)log_{2}(1-p) \tag{8}\]  
where \(p\) is the trust value calculated using the Bayesian beta method and \(H(p)\) is the entropy function. Figure 7: Function plot of \(T_{Entr}\). [PERSON] and [PERSON] [37] applied source-level trust assessment based on the Bayesian beta method and the entropy-based method. Through mathematical analysis and simulation experiments, they verified that in selective forwarding attack scenarios, it is much more difficult to obtain a high trust value in trust models using the entropy-based method trust model compared to trust models using the Bayesian beta method. This is because, with the entropy-based method, nodes need to have a very low packet loss rate to obtain a high trust value. The entropy-based method trust model allows faster identification of attack victims as the increased number of lost packets causes the trust value to decrease further.