# A Bidirectional Trust Model for Service Delegation in Social Internet of Things
## 3 Trust and Service Delegation Model
### Trust Model
#### 3.1.1 Subjective Logic  
Subjective logic is an uncertain probabilistic logic that was initially introduced by [PERSON] to address formal representations of trust [35]. The subjective logic constructs a bijective mapping between opinion space and evidence space, which can help SR to form its own opinion based on the existing direct evidence, and to integrate the recommendation opinions from others to form a comprehensive opinion. **Definition 1** (Opinion Space). : _[PERSON]'s direct opinion about object \(B\) for the task \(\varphi\) is a vector:_  
\[\overrightarrow{T}_{B\gets A}(\varphi)=(b_{B\gets A}(\varphi),d_{B \gets A}(\varphi),u_{B\gets A}(\varphi),a_{B\gets A}(\varphi)), \tag{5}\]  
_where \(b_{B\gets A}(\varphi)\) represents the degree to which \(A\) believes \(B\) will successfully perform the task \(\varphi\), and \(d_{B\gets A}(\varphi)\) represents the degree to which \(A\) disbelieves \(B\) will successfully perform the task \(\varphi\). \(u_{B\gets A}(\varphi)\) represents the degree to which \(A\) is uncertain about whether \(B\) will successfully perform the task \(\varphi\), and \(a_{B\gets A}(\varphi)\) is the base rate. The opinion satisfies the additivity requirement as follows:_  
\[b_{B\gets A}(\varphi)+d_{B\gets A}(\varphi)+u_{B\gets A}(\varphi )=1, \tag{6}\]  
_and the projected probability of the opinion \(\overrightarrow{T}_{B\gets A}(\varphi)\) is defined as_  
\[\hat{T}_{B\gets A}(\varphi)=b_{B\gets A}(\varphi)+a_{B\gets A}( \varphi)u_{B\gets A}(\varphi). \tag{7}\]  
Figure 2: The design framework of the trust model and service delegation. In our trust model, we use \(\overrightarrow{T}_{B\gets A}^{d}(\varphi)\) to represent the direct trust vector of object B from the viewpoint of object A for the task \(\varphi\). \(\hat{T}_{B\gets A}^{d}(\varphi)\) is used for representing the projected trustworthiness of object B.  
Evidences are fundamental for forming opinions, which can be presented as a series of the binary comments such as \"satisfaction\" and \"dissatisfaction\". The amount of evidence will affect the certainty of the opinion. In subjective logic, the Beta function is used for constructing the evidence space. The probability density function is as follows:  
\[Beta(p_{x},\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma( \beta)}p_{x}^{\alpha-1}(1-p_{x})^{\beta-1}, \tag{8}\]  
where \(\Gamma()\) is the gamma function. The beta function can be used to represent the probability distribution of binary events. Therefore, the evidence space can be defined as follows:  
**Definition 2** (Evidence Space). : _An evidence space can be depicted by a beta probability distribution:_  
\[Beta(\overrightarrow{T}_{B\gets A}^{d}(\varphi),\alpha_{B\gets A} (\varphi),\beta_{B\gets A}(\varphi)), \tag{9}\]  
_where \(\overrightarrow{T}_{B\gets A}^{d}(\varphi)\) represents the direct trust vector of B from the viewpoint of A in evidence space. \(\alpha_{B\gets A}(\varphi)\) and \(\beta_{B\gets A}(\varphi)\) are defined as:_  
\[\left\{ \begin{array}{l}\alpha_{B\gets A}(\varphi)=\gamma_{B\gets A }(\varphi)+2a_{B\gets A}(\varphi)\\ \beta_{B\gets A}(\varphi)=\overline{\gamma}_{B\gets A}(\varphi)+2(1- a_{B\gets A}(\varphi))\end{array} \right. \tag{10}\]  
_where \(\gamma_{B\gets A}(\varphi)\) and \(\overline{\gamma}_{B\gets A}(\varphi)\) are the evidence strength which is based on the historical interactions between objects A and B. \(\gamma_{B\gets A}(\varphi)\) denotes the positive evidence strength, which indicates that the B is trustworthy. \(\overline{\gamma}_{B\gets A}(\varphi)\) denotes the negative evidence strength, which indicates that B is untrustworthy._  
_The expected probability \(E(\overrightarrow{T}_{B\gets A}^{d}(\varphi))\) is defined as the projected trustworthiness in evidence space, which is expressed as follows:_  
\[