# Graph-Augmentation-Free Self-Supervised Learning for Social Recommendation
## 3 Proposed Model
### Graph Collaborative Filtering BackBone  
In this section, we present our model GAFSRec. In Figure 2, the schematic overview of our model is illustrated. Due to its strong ability to capture node dependencies, we adopted LightGCN [20] to aggregate neighborhood information. Formally, a general GCN is constructed using the following hierarchical propagation:  
\[\mathbf{Z}^{(l)}=\mathbf{D}^{-1}\mathbf{A}\mathbf{Z}^{(l-1)} \tag{3}\]  
where \(Z^{(l)}\) is the \(l\)-th layer embedding of the node, and \(G_{q}\in\{G_{r},G_{e},G_{i},G_{j}\}\) are the four views, with \(A\) being the adjacency matrix and \(D\) the degree diagonal matrix of \(A\). We encoded the original node vectors into the embedding vectors required by each view through gating functions. \[\mathbf{Z}^{(0)}_{q\in\{r,e,i,j\}}=\mathbf{Z}^{(0)}\odot\sigma(\mathbf{Z}^{(0)}\mathbf{W}^{q}_{ g}+\mathbf{b}^{q}_{g}) \tag{4}\]  
where \(\sigma\), \(\mathbf{W}^{q}_{g}\in\mathbb{R}^{d\times d}\), and \(\mathbf{b}^{q}_{g}\in\mathbb{R}^{d}\) are the activation function, weight matrix, and bias vector. \(q\in\{r,e,i,j\}\) represents four views. \(\mathbf{Z}^{(0)}\) denotes the initial embedding vector and \(\odot\) represents the dot product. We used the generalization ability of hypergraph modeling to capture more effective high-level user information. Therefore, in the encoder, a hypergraph convolutional neural network [21] was used. \[\mathbf{Z}^{(l)}=\mathbf{D}^{-1}\mathbf{A}\mathbf{Z}^{(l-1)}=\mathbf{D}^{-1}_{\sigma}\mathbf{H}\mathbf{W} \mathbf{D}^{-1}_{e}\mathbf{H}^{T}\mathbf{Z}^{(l-1)} \tag{5}\]  
where \(\mathbf{A}=\mathbf{H}\mathbf{W}\mathbf{D}^{-1}_{e}\mathbf{H}^{T}\) is the Laplacian matrix of the hypergraph convolutional network; H is the incidence matrix of each hypergraph; \(\mathbf{D}_{\sigma}\) and \(\mathbf{D}_{e}\) are the degree matrix of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Motif** & **Matrix Computation** & \(\mathbf{A}_{M_{k}}\) \\ \hline \(M_{1}\) & \(C=(U\cdot U)\odot U^{T}\) & \(C+C^{T}\) \\ \(M_{2}\) & \(C=(B\cdot U)\odot U^{T}+(U\cdot B)\odot\) & \(C+C^{T}\) \\ \(U^{T}+(U\cdot U)\odot B\) & & \\ \(M_{3}\) & \(C=(B\cdot B)\odot U+(U\cdot U)\odot\) & \(C+C^{T}\) \\ \(M_{4}\) & \(C=(B\cdot B)\odot B\) & \(C\) \\ \(M_{5}\) & \(C=(U\cdot U)\odot U^{T}+(U\cdot\cdot\) & \(C+C^{T}\) \\ \(U^{T})\odot U+(U^{T}\cdot U)\odot U\) & \(C=(U\cdot B)\odot U+(B\cdot U)\odot\) & \(C\) \\ \(M_{6}\) & \(C=(U^{T}\cdot B)\odot U+(U\cdot B)\odot\) & \(C\) \\ \(U^{T}+(U^{T}\cdot U)\odot B\) & & \\ \(M_{7}\) & \(C=(U^{T}\cdot B)\odot U^{T}+(B\cdot\) & \(C\) \\ \(U)\odot U+(U\cdot U^{T})\odot B\) & & \\ \(M_{8}\) & \(C=R\cdot R^{T}\) & \(C\) \\ \(M_{9}\) & \(C=(R\cdot R^{T})\odot B\) & \(C\) \\ \(M_{10}\) & \(C=(R\cdot R^{T})\odot U\) & \(C+C^{T}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computation of motif-based adjacency matrices. nodes and the degree matrix of the hyperedges; and \(\sigma\), \(\Theta\), and \(\mathbf{W}\) are activation functions, learnable filter matrices, and the parameters of the diagonal matrix. Considering node influence and credibility, we adopted motifs [18] to construct a hypergraph. Given the complexity of the actual construction of the Laplacian matrix for hypergraph convolution (which includes a large number of graph-induced hyperedges), matrix multiplication can be considered for simplification. Finally, the hypergraph convolutional neural network can be expressed as: \(\mathbf{Z}_{q\in\{r,\varepsilon,i,j\}}^{(l)}=\widetilde{D}_{q}^{-1}\mathbf{A}_{q}\mathbf{Z }_{q}^{(l-1)}\), where \(\widetilde{D}_{q}\in\mathbb{R}^{m\times m}\) is the degree matrix of \(\mathbf{A}_{q}\). It can be seen from this that the graph convolutional neural network is a special case of the hypergraph neural network.