## 4 Proposed Model
### Observation from Rating Distribution  
(1) Global observation  
As argued in many related works [16; 31; 32], the overall ratings follow the normal distribution in recommender systems when there are enough rating behaviors. Theoretically, the rating densities follow \(\varphi(y)=N\big{(}\mu,\sigma^{2}\big{)}\), with the probability density as:  
\[\varphi(y)=N\big{(}\mu,\sigma^{2}\big{)}=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{ (y-y)^{2}}{2\sigma^{2}}}, \tag{3}\]  
where \(y\in[\mu-\frac{Len_{\text{sys}}}{2},\mu+\frac{Len_{\text{sys}}}{2}]\), \(\mu\) is the mean value of whole ratings in the system, \(\sigma^{2}\) is the global variance in all ratings, and \(Len_{\text{sys}}\) is the length of the system's rating scale. Additionally:  
\[\sigma=\frac{1}{N_{\text{sys}}}\sum\
olimits_{n=1}^{N_{\text{sys}}}(r_{n}-\mu), \tag{4}\]  
represents the global standard deviation of ratings, where \(N_{\text{sys}}\) is the number of all ratings described. We selected five real datasets for the global observation, including large-scale datasets Flixster and Movielens-10 M, middle-scale dataset FilmTrust, small-scale dataset ml-latest-small, and mini-scale dataset MiniFilm. The related properties of these datasets are described in Section 6. Figure 2 shows the rating densities' distribution with different rating segments. We observed that these ratings follow the general normal distribution trend, especially when there are more ratings. This observation proves that the normal distribution probability can adapt to the global distribution for overall ratings in all recommender systems. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Description** \\ \hline \(u,v\) & user ID \\ \(i,j\) & item ID \\ \(\mu\) & overall average rating \\ \(\mu_{u}\) & average rating given from \(u\) \\ \(\tau_{i}\) & average rating of \(i\) \\ \(r_{ni}\) & observed rating of \((u,i)\) \\ \(\tau_{ni}\) & predicted rating of \((u,i)\) \\ \(b_{u}\) & bias of \(u\) \\ \(b_{i}\) & bias of \(i\) \\ \(\alpha_{u}\) & Local standard deviation of \(u\) \\ \(\sigma\) & Global standard deviation \\ \(N_{sys}\) & The number of all ratings in system \\ \(N_{u}\) & The number of ratings given by user \(u\) \\ \(Q\in\mathbb{R}^{d\times m}\) & factor-item matrix \\ \(p_{u}\in\mathbb{R}^{1\times d}\) & latent feature vector of user \(u\) \\ \(q_{i}\in\mathbb{R}^{1\times d}\) & latent feature vector of item \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some related symbols. Figure 2: Global distributions on different datasets: (**a**) Flixster, (**b**) MiniFilm, (**c**) FilmTrust, (**d**) MovieLens\({}_{10}\) M, and (**e**) MovieLens\({}_{\ast}\)Latest\({}_{\ast}\)Small. (2) Local observation  
To investigate the personalized rating features, we further observed the local distribution of ratings given from different kinds of personalized users. We selected Flixster for the example demonstration to make the description more convenient. Other datasets with enough ratings also follow a similar regularity. As described in Section 2, we found that four types of personalized users exist in recommender systems, including normal users, strict users, middle users, and lenient users. We calculated the statistics on all 8,196,077 ratings of Flixster according to different users' rating segments. Table 2 shows the statistical results. The rating range of normal users is from 0 to 5, that of strict users is from 0 to 4, that of middle users is from 1 to 4, and that of lenient users is from 1 to 5. We also investigated related segments' rating densities and calculated their mean values \(\mu_{u}\) and standard deviations \(\sigma_{u}\). Meanwhile, we calculated the fitted mean value \(\mu_{f}\) and fitted standard deviation \(\sigma_{f}\) for different kinds of users' ratings according to a normal distribution.