Graph Matrices
$$\mathbf{L} = \mathbf{D} - \mathbf{A}$$ $$\mathcal{L}_{\text{sym}} = \mathbf{I} - \mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}$$ $$\mathbf{f}^T \mathbf{L} \mathbf{f} = \frac{1}{2} \sum_{(i,j) \in E} (f_i - f_j)^2$$ $$\tilde{\mathbf{A}} = \mathbf{D}^{-1/2}(\mathbf{A}+\mathbf{I})\mathbf{D}^{-1/2}$$Message Passing (MPNN)
$$\mathbf{m}_{ij}^{(l+1)} = \phi^{(l)}\left(\mathbf{h}_i^{(l)}, \mathbf{h}_j^{(l)}, \mathbf{e}_{ij}\right)$$ $$\bar{\mathbf{m}}_i^{(l+1)} = \bigoplus_{j \in \mathcal{N}(i)} \mathbf{m}_{ij}^{(l+1)}$$ $$\mathbf{h}_i^{(l+1)} = \gamma^{(l)}\left(\mathbf{h}_i^{(l)}, \bar{\mathbf{m}}_i^{(l+1)}\right)$$ $$\mathbf{h}_G = \text{READOUT}\left(\{\mathbf{h}_i^{(L)}\}\right)$$GCN
$$\mathbf{H}^{(l+1)} = \sigma\left(\tilde{\mathbf{D}}^{-1/2}\tilde{\mathbf{A}}\tilde{\mathbf{D}}^{-1/2} \mathbf{H}^{(l)} \mathbf{W}^{(l)}\right)$$GraphSAGE
$$\mathbf{h}_i^{(l+1)} = \sigma\left(\mathbf{W}^{(l)} \cdot \text{CONCAT}\left(\mathbf{h}_i^{(l)}, \text{AGG}(\{\mathbf{h}_j^{(l)}\}_{j \in \mathcal{N}(i)})\right)\right)$$GAT
$$e_{ij} = \text{LeakyReLU}\left(\vec{a}^T [\mathbf{W}\mathbf{h}_i \| \mathbf{W}\mathbf{h}_j]\right)$$ $$\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k \in \mathcal{N}(i)} \exp(e_{ik})}$$ $$\mathbf{h}_i^{(l+1)} = \sigma\left(\sum_{j \in \mathcal{N}(i)} \alpha_{ij} \mathbf{W}\mathbf{h}_j^{(l)}\right)$$GIN
$$\mathbf{h}_i^{(l+1)} = \text{MLP}^{(l)}\left((1 + \epsilon^{(l)}) \mathbf{h}_i^{(l)} + \sum_{j \in \mathcal{N}(i)} \mathbf{h}_j^{(l)}\right)$$ChebNet
$$\mathbf{g} * \mathbf{f} \approx \sum_{k=0}^{K} \theta_k T_k(\tilde{\mathbf{L}})\mathbf{f}$$R-GCN
$$\mathbf{h}_i^{(l+1)} = \sigma\left(\sum_{r \in \mathcal{R}} \sum_{j \in \mathcal{N}_r(i)} \frac{1}{c_{i,r}} \mathbf{W}_r^{(l)} \mathbf{h}_j^{(l)} + \mathbf{W}_0^{(l)}\mathbf{h}_i^{(l)}\right)$$Knowledge Graph Embeddings
TransE: $\|\mathbf{h} + \mathbf{r} - \mathbf{t}\|$
DistMult: $\langle \mathbf{h}, \mathbf{r}, \mathbf{t} \rangle$