# Light Graph Convolutional Recommendation Algorithm Based on Hybrid Spreading
## 3 Model Design
### Incorporating the Hybrid Spreading of the Embedding Matrix  
The recommendation system is represented as a bipartite network \(G=(U,O,E)\), where \(U=\{u_{1},u_{2},\cdots,u_{m}\}\), \(O=\{o_{1},o_{2},\cdots,o_{n}\}\), and \(E=\big{\{}\mathbf{e}_{1},\mathbf{e}_{2},\cdots,\mathbf{e}_{q}\big{\}}\) denote the \(m\) users, the \(n\) items, and the \(q\) interactions between pairs of \(m\) users and \(n\) items, respectively, it can be described using the adjacency matrix \(\mathbf{A}=\big{[}a_{ij}\big{]}_{m\times n}\), if there is an interaction relationship between user \(u_{i}\) and item \(o_{j}\), then \(a_{ij}=1\), otherwise \(a_{ij}=0\). Assuming that a user interacts with an item because the user is interested in the item, the basic task of a recommendation system becomes generating an interest list of items that the target user has not interacted with. All spreading-based recommendation algorithms assign a certain amount of initial resource \(\mathbf{f}=\left[f_{1},f_{2},\cdots,f_{n}\right]^{T}\) to the items for the target user \(u_{i}\), and then reallocate the resources by spreading \(\mathbf{f}=\mathbf{W}\mathbf{f}\). Where \(\mathbf{W}\) is referred to as the resource transfer matrix, and the transfer matrix elements \(w_{\alpha\beta}\) for ProbS and HeatS are denoted as:  
\[ \begin{split} w_{\alpha\beta}^{ProbS}&=\frac{1}{k( \theta_{\beta})}\sum\limits_{i=1}^{m}\frac{a_{i\alpha}a_{i\beta}}{k(u_{i})}\\ w_{\alpha\beta}^{HeatS}&=\frac{1}{k(\theta_{\alpha })}\sum\limits_{i=1}^{m}\frac{a_{i\alpha}a_{i\beta}}{k(u_{i})}\end{split}  \tag{6}\]  
where \(w_{\alpha\beta}^{ProbS}\) are the transfer matrix elements for ProbS, \(w_{\alpha\beta}^{HeatS}\) are the transfer matrix elements for HeatS, \(k(\theta_{\alpha})=\sum_{l=1}^{m}a_{l\alpha}\) is the degree of item \(o_{\alpha}\), \(k(o_{\beta})=\sum_{l=1}^{m}a_{l\beta}\) is the degree of item \(o_{\beta}\), \(k(u_{i})=\sum_{r=1}^{m}a_{ir}\) it the degree of user \(u_{i}\). ProbS has high accuracy but low diversity, and HeatS is designed to meet the challenge of diversity, but its accuracy is low. The spreading process for both is illustrated in Figure 3. It describes the spreading process of ProbS and HeatS, with weights calculated according to Equation (6). The changes in weights during the spreading process will alter the relative influence of items and users. If a certain user's weight is large, then the weights of other items will be greatly affected by that user, and the recommendation system will tend to recommend items that have a relationship with that user; if a certain item's weight is large, it indicates that the item has a strong influence on multiple users, likely categorizing it as a popular item, and the recommendation system will make this item more likely to be recommended. According to the spreading process of ProbS and HeatS, mark the user resource vector obtained after the first round of probabilistic spreading and heat spreading as \(\mathbf{c},\mathbf{c}^{\prime}\in\mathbb{R}^{m}\), and allocate a certain amount of initial resources \(\mathbf{f}\in\mathbb{R}^{n}\) for the item against the target user \(u_{i},i\in\{1,2,\cdots,m\}\), then:  
\[ \begin{array}{l}\mathbf{c}_{i}=\sum\limits_{\beta=1}^{n}a_{i\beta}\frac{\mathbf{f}_ {\beta}}{k(o_{\beta})}\\ \mathbf{c}^{\prime}_{i}=\sum\limits_{\beta=1}^{n}a_{i\beta}\frac{\mathbf{f}_{\beta}}{ k(u_{i})}\end{array}  \tag{7}\]  
where \(k(o_{\beta})\) it the degree of item \(o_{\beta}\), \(k(u_{i})\) is the degree of user \(u_{i}\). According to Equations (6) and (7), it can be obtained:  
\[ \begin{array}{l}\mathbf{t}_{\alpha}=\sum\limits_{\beta=1}^{n}\mathbf{f}_{\beta}\frac {\mathbf{g}_{\alpha}}{k(o_{\beta})}\sum\limits_{i=1}^{m}\frac{a_{i\alpha}a_{i \beta}}{k(u_{i})}\\ \mathbf{t}^{\prime}_{\alpha}=\sum\limits_{\beta=1}^{n}\mathbf{f}_{\beta}\frac{\mathbf{g}_ {\alpha}}{k(o_{\alpha})}\sum\limits_{i=1}^{m}\frac{a_{i\alpha}a_{i\beta}}{k(u_ {i})}\end{array}  \tag{8}\]  
Figure 3: Diagram of the spreading process in the user–item bipartite network. * a is the target user.