# User Trust Inference in Online Social Networks: A Message Passing Perspective
## 3 The Proposed Model
### Probabilistic Model Inference and Interpretation
U\({}_{\text{A}}\), U\({}_{\text{B}}\), T\({}_{\text{AB}}\) are the random variables of the trustor node A, the trustee node B and the trustRelation node linked to them, respectively. Since the trustRelation node is linked to a trustor node and a trustee node by only two edges, with Equation (7), the belief at it is proportional to  
\[b_{\text{T}_{\text{AB}}}(\text{T}_{\text{AB}})\propto\varphi_{\text{T}_{ \text{AB}}}(\text{T}_{\text{AB}})\prod_{i\in N(\text{T}_{\text{AB}})}m_{i \rightarrow\text{T}_{\text{AB}}}(\text{T}_{\text{AB}}), \tag{10}\]  
and the two messages (\(M_{A}\) and \(M_{B}\) in the figure) sent to the trustRelation node are, respectively  
\[m_{\text{U}_{\text{A}}\rightarrow\text{T}_{\text{AB}}}(\text{T}_{\text{AB}} )\leftarrow\sum_{\text{U}_{\text{A}}}\varphi_{\text{U}_{\text{A}}}(\text{U}_ {\text{A}})\varphi_{\text{U}_{\text{A}}\text{T}_{\text{AB}}}(\text{U}_{\text {A}},\text{T}_{\text{AB}})\prod_{k\in N(\text{U}_{\text{A}})\setminus\text{T }_{\text{AB}}}m_{k\rightarrow\text{U}_{\text{A}}}(\text{U}_{\text{A}}), \tag{11}\]  
\[m_{\text{U}_{\text{B}}\rightarrow\text{T}_{\text{AB}}}(\text{T}_{\text{AB}} )\leftarrow\sum_{\text{U}_{\text{B}}}\varphi_{\text{U}_{\text{B}}}(\text{U}_ {\text{B}})\varphi_{\text{U}_{\text{B}}\text{T}_{\text{AB}}}(\text{U}_{\text {B}},\text{T}_{\text{AB}})\prod_{k\in N(\text{U}_{\text{B}})\setminus\text{T }_{\text{AB}}}m_{k\rightarrow\text{U}_{\text{B}}}(\text{U}_{\text{B}}). \tag{12}\]  
Let \(\Lambda_{x}(y)\) be the multiplication of the local evident at node \(x\) and edge \(\langle x,y\rangle\), and substitute Equations (11) and (12) and the messages demonstrated in the figure into Equation (10). The belief at the trustRelation node amounts to  
Figure 5: An illustration of passing messages through a trustor node U\({}_{\text{A}}\) and a trustee node U\({}_{\text{B}}\) to a trustRelation node T\({}_{\text{AB}}\). \[b_{\mathrm{T_{AB}}}(\mathrm{T_{AB}}) = \kappa\varphi_{\mathrm{T_{AB}}}(\mathrm{T_{AB}})(M_{A}M_{B}) \tag{13}\] \[= \kappa\varphi_{\mathrm{T_{AB}}}(\mathrm{T_{AB}})\left[\sum_{ \mathrm{U_{A}}}\Lambda_{\mathrm{U_{A}}}(\mathrm{T_{AB}})(M_{1}M_{2})\right] \left[\sum_{\mathrm{U_{B}}}\Lambda_{\mathrm{U_{B}}}(\mathrm{T_{AB}})M_{3} \right], \tag{14}\]  
where \(\kappa\) is a normalization constant. It follows that the trustRelation node's belief is proportional to all local evidence at this node and two messages sent by the trustor node and the trustee node. The local evidence is the calculated potential through the compatibility function (energy function, refer to Equations (5) and (6). Based on the trustor's attitude, experience, awareness of other users' expertise and trustworthiness, etc., collected by Equation (11), the message \(M_{A}\) sent by the trustor node tells the trustRelation node how much the trustor believes they will issue or not issue a trust relationship to the trustee. Likewise, through the trustee's expertise, experience and evidential suggestions from other users' trust, etc., constituted by Equation (12), the message \(M_{B}\) sent from the trustee node tells the trustRelation node how much the trustee thinks their behaviors should be recognized by the potential trustor so that the trustor will trust them or not. One can conclude that both the trustor and the trustee are involved in forming a trust relationship, obviously, in a real-world social network, and they have different contributions to the relationship formation, which is in accordance with Assumption 2, which we made previously.