# Deep Cross-Network Alignment with Anchor Node Pair Diverse Local Structure
## 3 Method
### Network Embedding with the Diverse Local Structure of Anchor Node
#### 3.2.1 Masked Network Encoder Based on Variational Graph Convolution  
Due to their similar characteristics and connectedness, nearby nodes generally have more influence than distant ones when comparing nodes across networks [39; 40]. To better use the network's local structure while preserving the whole network structure in a low-dimension embedding space and learning the distribution of each node simultaneously, we perform convolution operations between Gaussian distributions parameterized by a two-layer convolutional network as an encoder. Without loss of generality, we denote the matrix with capital letters, and use \(\mathbf{A}(i,:)\) to denote the \(i\)-th row of the matrix \(\mathbf{A}\). \(\odot\) is the Hadamard product. \(d\) is the dimension of each node embedding. Notice that degree distributions are power-law in most cases, and nodes with lower degrees are less likely than nodes with higher degrees to be connected to neighbors who are classified as anchors. As a result, these low-degree nodes receive less information from labeled anchor nodes, which leads to unsatisfactory or even subpar alignment performance. Therefore, we randomly mask some edges of nodes whose degree is larger than the average degree of all nodes in the network. For the network without attributes, we use the identity matrix \(\mathbf{I}\) and the adjacency matrix \(\mathbf{A}\) as the model input. Furthermore, we can also use the node-feature matrix as input if the network has the node attributes, which means our method can be extended easily to attribute networks. For each network, take \(G^{s}\) as an example, we define a masked adjacency matrix \(\mathbf{\tilde{A}}^{s}\). Specifically, we first collect the nodes whose degree is larger than the average degree of all nodes in \(G^{s}\), and for node \(v^{s}_{i}\) in the set  
\[\mathbf{\tilde{A}}^{s}(i,:)=\mathbf{A}^{s}(i,:)\odot\mathbf{m}, \tag{1}\]  
where \(\mathbf{m}\in\left\{0,1\right\}^{1\times n_{s}}\) is the masked vector for \(v^{s}_{i}\) and the sum of its elements is \((d(v^{s}_{i})-d_{avg})\). \(d(v^{s}_{i})\) and \(d_{avg}\) are the \(v^{s}_{i}\)'s degree and the average degree of all nodes in \(G^{s}\), respectively; similarly for \(\mathbf{\tilde{A}}^{t}\). Instead of embedding each node as a point vector, we represent a node by Gaussian distributions, i.e., the mean and the variance, to incorporate the uncertainty [41]. We use \(\mathbf{M}^{s}\in R^{n_{s}\times d}\), \(\mathbf{M}^{t}\in R^{n_{t}\times d}\), \(\mathbf{\Sigma}^{s}\in R^{n_{s}\times d}\), and \(\mathbf{\Sigma}^{t}\in R^{n_{t}\times d}\) to denote the matrix of means and variances for all nodes in the \(G^{s}\) and \(G^{t}\), respectively, where each row represents the mean/variance vector for a node  
\[\mathbf{M}^{s}=\mathbf{\tilde{A}}^{s}(\rho(\mathbf{\tilde{A}}^{s}\mathbf{I}\mathbf{W }^{s}_{0}))\mathbf{W}^{s}_{\mu} \mathbf{M}^{t}=\mathbf{\tilde{A}}^{t}(\rho(\mathbf{\tilde{A}}^{t}\mathbf{I}\mathbf{W }^{t}_{0}))\mathbf{W}^{t}_{\mu}, \tag{2}\] \[\mathbf{\Sigma}^{s}=\mathbf{\tilde{A}}^{s}(\rho(\mathbf{\tilde{A}}^{s}\mathbf{I} \mathbf{W}^{s}_{0}))\mathbf{W}^{s}_{\sigma} \mathbf{\Sigma}^{t}=\mathbf{\tilde{A}}^{t}(\rho(\mathbf{\tilde{A}}^{t}\mathbf{I} \mathbf{W}^{t}_{0}))\mathbf{W}^{t}_{\sigma^{t}}, \tag{3}\]  
where \(\mathbf{W}^{s}_{0}\in R^{n_{s}\times F}\) is parameter matrix shared by \(\mathbf{M}^{s}\) and \(\mathbf{\Sigma}^{s}\), \(\mathbf{W}^{s}_{\mu}\in R^{F\times d}\) and \(\mathbf{W}^{s}_{\sigma}\in R^{F\times d}\) are parameters for the \(\mathbf{M}^{s}\) and \(\mathbf{\Sigma}^{s}\), respectively, and \(F\) is the number of neurons by hidden layer that usually be \(2*d\). Similar for \(\mathbf{W}^{t}_{0}\in R^{n_{t}\times F}\), \(\mathbf{W}^{t}_{\mu}\in R^{F\times d}\), and \(\mathbf{W}^{t}_{\sigma}\in R^{F\times d}\). \(\rho(\cdot)\) is a non-linear activation function, such as ReLU.