# Knowledge Graph Recommendation Model Based on Feature Space Fusion
## 3 Our Model
### Model Introduction
#### 3.2.1 Behavioral Feature Module  
The model extracts the behavioral feature contained in the UIG through the behavioral feature module. Behavioral features are potential factors that influence users' decisions, as reflected by interaction history. For example, when a user watches a movie, in addition to being interested in the content of the movie, they may also be affected by factors such as conformity psychology (many users have watched the movie), publicity (attracted by the advertisements and posters of the film), friend recommendation, etc. The role of these factors is reflected through the user's interaction history, which depicts the user's behavior features. The behavior feature from the UIG and the content feature contained in the KG affect users' decisions from different aspects. As shown in Figure 1, User A interacts with \"Forrest Gump\" and \"The Shawshank Redemption\" because both belong to \"feature films\" in terms of content. User B's interactions \"Titanic\" and \"The Shawshank Redemption\" are not due to their similar content feature in the knowledge graph but because they are both highly acclaimed and highly interactive \"high score movies\" with a large number of users. Users who have watched one of the two movies tend to watch the other movie. Therefore, \"Titanic\" is probably the movie that User A will watch. In summary, the behavior feature module is an important aspect of modeling users and items.  
In the UIG, the interactive items of users express the features of users, and the interactive users of items express the features of items [3]. As a result, the behavioral feature module aggregates the neighborhood information of nodes in the UIG by a graph convolution operation to obtain the behavioral feature vector representation of user \(u_{\text{cf}}\) and item \(v_{\text{cf}}\):  
\[u^{(k+1)}=\sum_{v\in\mathbb{N}_{u}}\frac{1}{\sqrt{|N_{u}|}\sqrt{|N_{p}|}}v^{( k)} \tag{1}\]  
Figure 3: Illustration of the proposed KGRFSF model.  
\[v^{(k+1)}=\sum_{u\in N_{v}}\frac{1}{\sqrt{|N_{v}|}\sqrt{|N_{u}|}}u^{(k)} \tag{2}\]  
where \(u^{(k)}\) and \(v^{(k)}\) are the vector representations of user \(u\) and item \(v\) after convolution at layer \(k\), \(N_{u}\) is all items for user \(u\) interaction, and \(N_{v}\) is all users for item \(v\) interaction.  
After \(K\) layer convolution, \(K\) vectors are obtained, and each vector contains neighborhood information of different ranges. The larger \(K\) is, the wider the neighborhood range is, and the less information contained is associated with the central node. The model integrates the results of the \(k\)-layer convolution to obtain the behavioral feature vectors of user \(u_{\text{cf}}\) and item \(v_{\text{cf}}\):  
\[u_{\text{cf}}=\sum_{k=0}^{K}\alpha_{k}u^{(k)} \tag{3}\]  
\[v_{\text{cf}}=\sum_{k=0}^{K}\alpha_{k}v^{(k)} \tag{4}\]  
where \(\alpha_{k}\) is the weight of the \(k\) layer vector, with a value of \(1/(k+1)\). For the central node, the closer the node is, the more important its information and the higher its weight.  
To understand graph convolution from an overall perspective and facilitate batch processing, the matrix form of each layer of convolution is given here:  
\[E^{(k+1)}=\Big{(}D^{-\frac{1}{2}}AD^{-\frac{1}{2}}\Big{)}E^{(k)} \tag{5}\]  
where \(A\) is the adjacency matrix of nodes in the UIG, \(D\) is the degree matrix of nodes in the UIG, and \(E\) is the embedding vector matrix of nodes in the UIG. The calculation formula of the output feature matrix of the comprehensive \(K\)-layer convolution is as follows:  
\[E=\alpha_{0}E^{(0)}+\alpha_{1}E^{(1)}+\alpha_{2}E^{(2)}+\ldots+\alpha_{K}E^{( K)} \tag{6}\]  
where \(\alpha_{K}\) represents the weight of the \(K\) layer vector, and the value is \(1/(K+1)\).