# Social Recommendation Based on Quantified Trust and User's Primary Preference Space
## 3 Social Recommendation Framework Based on Trust and Preference
### Overview of Our Framework  
The TPSR framework consists of two parts: a user trust quantification method (TQ_RWR) and a social recommendation model (UPPS). Among them, TQ_RWR is a user trust quantification algorithm based on restart random walk, which measures the trust value between users by mining the user reputation hidden in the trust network topology. UPPS is a social recommendation model based on primary preference space, which is implemented by the probabilistic matrix factorization method.  
In TQ_RWR, we use the method of random walk with restart (RWR) to mine the implied credibility in the binary trust network and further quantify the trust between users.  
The existing social recommendation methods utilize binary data in the social trust network, where \(T_{ik}=1\) indicates that user \(u_{i}\) trusts user \(u_{k}\), and \(T_{ik}=0\) indicates that the user \(u_{i}\) does not trust user \(u_{k}\). In other words, the existing recommendation methods only utilize the local trust relationships, ignoring the overall structure and global information of the trust network [22]. Figure 1 shows an example of trust network; the nodes represent users and the edges indicate the trust relationships between users. As we all know, there is a serious \"star effect\" on today's various shopping or social platforms. For example, there is a \"Big V\" with a very low attention of hundreds of millions on microblogs. These \"Big V\" microblogs have a high credibility (also known as a reputation) in the hearts of many fans, and some of their behaviors will cause many fans to follow suit. It can be seen that credibility has a great influence on the generation of trust relationships in trusted social networks. Similar to the central user nodes of node 33 and node 34 in Figure 1, these central users have trust relationships with a large number of other users, and these central users have a higher credibility compared with others, while the binary trust network cannot express this global network information. TQ_RWR considers that the central users represented by nodes 33 and 34 are trusted by a large number of other users, and therefore have a high degree of credibility, which reflects the trustworthiness of users.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(R\) & user-item rating matrix \\ \(T\) & trust matrix \\ \(U\) & user feature matrix \\ \(\tilde{U}\) & primary user feature matrix \\ \(V\) & item feature matrix \\ \(M_{R}\) & interaction space projection matrix \\ \(M_{B}\) & truster space projection matrix \\ \(M_{W}\) & trustee space projection matrix \\ \(N,M\) & total number of user and item \\ \(N(x|\mu,\sigma^{2})\) & Gaussian distribution with mean \(\mu\) and variance \(\sigma^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Explanation of notations.  
In the UPPS model, we assume that users with social relationships exhibit a certain similarity based on interest preferences, and the higher trust between users, the more similar their interest preferences are. UPPS believes that the users consider different factors when building trust relationships or when rating items, as shown in Figure 2.  
Users may consider factors such as product quality, brand and appearance in the rating scene, and take into account factors such as the user's occupation, personality and background in the social scene. Each user has their own values, including acceptance, achievement, ambition, attractiveness, etc. Whether in the rating scene or in the social scene, the influencial factors come from users' own values. Therefore, the preference vectors of users in different scenarios are not always the same. That is, social networks and ratings exist in different preferences spaces. UPPS proposes the concept of a primary preference feature space, and treats the feature vector of users in different scenarios as the projections from primary preference vectors.