# A Bidirectional Trust Model for Service Delegation in Social Internet of Things
## 3 Trust and Service Delegation Model
### Trust Model
#### 3.1.3 Indirect Trust  
In addition to direct trust evaluation, A will also ask C and D for relevant opinions about B. This paragraph solves the fusion problem of recommendation opinions by designing the discounting and consensus operators. The recommendation opinions from objects C and D are expressed as follows, respectively. \[\overrightarrow{T}_{B\gets C}^{rec}(\varphi)=(b_{B\gets C}(\varphi), d_{B\gets C}(\varphi),u_{B\gets C}(\varphi),a_{B\gets C}(\varphi)) \tag{18}\]  
\[\overrightarrow{T}_{B\gets D}^{rec}(\varphi)=(b_{B\gets D}(\varphi), d_{B\gets D}(\varphi),u_{B\gets D}(\varphi),a_{B\gets D}( \varphi)) \tag{19}\]  
The objective in this part is to construct the suitable function \(f_{ind}(\cdot)\) to integrate the \(\overrightarrow{T}_{B\gets D}^{rec}(\varphi)\) and \(\overrightarrow{T}_{B\gets C}^{rec}(\varphi)\), which is formulated as follows:  
\[\overrightarrow{T}_{B\gets A}^{ind}(\varphi)=f_{ind}(\overrightarrow{T}_ {B\gets C}^{rec}(\varphi),\overrightarrow{T}_{B\gets D}^{rec}( \varphi)), \tag{20}\]  
and \(\overrightarrow{T}_{B\gets A}^{ind}(\varphi)=(b_{B\gets A}^{ind},a_{B \gets A}^{ind},u_{B\gets A}^{ind},a_{B\gets A}^{ind})\). * **Discounting and Consensus Operator** In the subjective logic framework, the discounting rule does not have a natural interpretation of evidence handling [36]. To this end, we use the trust in opinion space to discount the trust in evidence space. The ideas and principles are shown in Figure 4. We use the symbol \(\otimes\) to represent the discounting operator. Thus we have \(\overrightarrow{T}_{B\gets C\gets A}^{ind}(\varphi)= \overrightarrow{T}_{B\gets C}^{rec}(\varphi)\otimes\overrightarrow{T}_{C \gets A}^{d}(\varphi)\). The specific discounting rule \(\otimes\) in evidence space is shown as follows. \[\left\{ \begin{array}{l}\gamma_{B\gets C\gets A}^{ind}(\varphi)= \overrightarrow{T}_{G\gets A}^{d}(\varphi)\gamma_{B\gets C}^{rec}( \varphi)\\ \overrightarrow{T}_{B\gets C\gets A}^{ind}(\varphi)= \overrightarrow{T}_{C\gets A}^{rec}(\varphi)\overrightarrow{T}_{B \gets C}^{rec}(\varphi)\end{array} \right. \tag{21}\]  
Based on Equations (12) and (21), we can calculate the indirect trust vector \(\overrightarrow{T}_{B\gets C\gets A}^{ind}(\varphi)\), which is shown as follows. \[\left\{ \begin{array}{l}b_{B\gets C\gets A}^{ind}(\varphi)= \frac{\hat{T}_{C\gets A}^{ind}(\varphi)b_{B\gets C}^{rec}(\varphi)}{ \hat{T}_{C\gets A}^{dec}(\varphi)+\hat{T}_{C\gets A}^{dec}(\varphi)+ \hat{T}_{B\gets C}^{rec}(\varphi)}\\ d_{B\gets C\gets A}^{ind}(\varphi)=\frac{\hat{T}_{C\gets A}^{dec}( \varphi)}{\hat{T}_{C\gets A}^{dec}(\varphi)b_{B\gets C}^{rec}(\varphi)+ \hat{T}_{C\gets A}^{dec}(\varphi)+\hat{T}_{B\gets C}^{rec}(\varphi)}\\ u_{B\gets C\gets C\gets A}^{ind}(\varphi)=\frac{u_{B\gets C}^{ rec}(\varphi)}{\hat{T}_{C\gets A}^{dec}(\varphi)b_{B\gets C}^{rec}(\varphi)+ \hat{T}_{C\gets A}^{dec}(\varphi)+\hat{T}_{B\gets C}^{rec}(\varphi)}\\ a_{B\gets C\gets A}^{ind}(\varphi)=\hat{T}_{C\gets A}^{d}(\varphi)a_{B \gets A}^{rec}(\varphi)\end{array} \right. \tag{22}\]  
The consensus operator is designed for integrating the recommendation opinions from different sources. We use the weighted sum method to design the consensus operator. Similar to the design idea of discounting operator, we use the trust in opinion space as weight parameters. The symbol \(\oplus\) represents the consensus operator, and thus we have \(\overrightarrow{T}_{B\gets A}^{ind}(\varphi)=\overrightarrow{T}_{B \gets C\gets A}^{ind}(\varphi)=\overrightarrow{T}_{B\gets C \gets A}^{ind}(\varphi)\oplus\overrightarrow{T}_{B\gets D\gets A }^{ind}(\varphi)\). The specific consensus operator in evidence space is shown as follows. \[\left\{