## 3 Materials and Methods
Specifically, the output of the FM component is the sum of addition and inner product units:  
\[y_{FM}=\;<w,x>+\sum\
olimits_{j_{1}=1}^{d}\sum\
olimits_{j_{2}=j_{1}+1}^{d}< V_{i},V_{j}>x_{j_{1}}\cdot x_{j_{2}}, \tag{6}\]  
where \(w\in R^{d}\) and \(V_{i}\in R^{k}\). The addition unit focuses on first-order features, whereas the inner product unit focuses on second-order interactions. (4) AFM  
The AFM model introduces an attention mechanism by adding an attention network between the feature crossover layer and final output layer. The role of the attention network is to provide weights for each cross-feature. The pooling process of the AFM after adding the attention score is as follows:  
\[f_{Att}\big{(}f_{PI}(\varepsilon)\big{)}=\sum\
olimits_{(i,j)\in R_{x}}a_{ij} \big{(}v_{i}\odot v_{j}\big{)}x_{i}x_{j}, \tag{7}\]  
where \(f_{PI}(\varepsilon)=\big{\{}\big{(}v_{i}\odot v_{j}\big{)}x_{i}x_{j}\big{)} \;(i,j)\in R_{x}\)(where \(\odot\) means the dot product and \(R_{x}=\big{\{}(i,j)_{i\in x,j\in x,j>i}\big{\}}\)), \(v_{i}\) and \(v_{j}\) respectively represent the corresponding hidden layer weights vectors of \(x_{i}\) and \(x_{j}\), and \(a_{ij}\) is the attention score. B. Details of ARFM model As mentioned above, compared with the traditional machine-learning model, the largest improvement in ARFM lies in the innovative proposal of the aging mechanism and the integration of multiple residual layers into the model. The following section explains in detail the structure and algorithm principles of these two parts. The starting point for the study of the aging mechanism is our understanding of the long-tail distribution of the datasets. The so-called long-tail distribution refers to the fact that, in the recommendation system, the most popular and viewed items are the least popular, and most items cannot be exposed to users. To illustrate the long-tail distribution of user behavior, we selected raw data from the Delicious and CiteULike datasets for analysis. Figure 2 shows the distribution curves for the popularity of the items in the Delicious and CiteULike datasets. The horizontal axis represents the popularity \(K\) of the item and the vertical axis represents the total number of items with popularity \(K\). Figure 3 shows the distribution curve of user activity in the Delicious and CiteULike datasets. The horizontal axis represents user activity \(K\), and the vertical axis represents the total number of users with activity \(K\). Therefore, under the influence of a long-tail distribution, new items cannot be well exposed; only popular items receive attention, and the efficiency of the recommendation system is affected. Therefore, it can be seen that the starting point of the aging mechanism is very necessary. In addition, while studying the distribution of the long tail, a new question arises: Will a user's time using a certain platform affect the user's choice of items with different popularity? New users tend to browse popular items because they are unfamiliar with the site and can only click on popular items on the front page, whereas older users gradually  
Figure 3: The long-tail distribution of user activity. Figure 2: The long-tail distribution of item popularity. begin browsing unpopular items. Figure 4 shows the relationship between user activity and item popularity in the MovieLens dataset, where the horizontal axis is user activity and the vertical axis is the average popularity of items rated highly by all users at a certain activity level. The problem of long-tail distribution also appears in the problem of cold starts [35]. Cold start can be divided into user cold start and item cold start, which correspond to situations in which there is no historical behavior data after new users and items are added to the recommendation system. From a new perspective, solving cold-start and long-tail distribution problems involves killing two birds with one stone.