# Deep Cross-Network Alignment with Anchor Node Pair Diverse Local Structure
## 3 Method
### Network Alignment Based on Node Representations  
After acquiring the final distribution representation of each node in \(G^{s}\) and \(G^{t}\), we predict the potential anchor nodes. As introduced in Section 3.1, we aim to learn a predictive function \(f:(G^{s},G^{t},E^{a})\to Y,Y\in\{0,1\}^{|V^{s}|\times|V^{t}|}\), \(Y_{ij}=1\) if \(v_{i}^{s}\) and \(v_{j}^{t}\) are predicted to be a pair of potential anchor nodes across networks \(G^{s}\) and \(G^{t}\), and 0 otherwise. Thus, for a pair of nodes \((v_{i}^{s},v_{j}^{t})\), \(v_{i}^{s}\in G^{s}\), \(v_{j}^{t}\in G^{t}\), to build a predictive function, we use a fully connected network  
\[f\Big{(}(v_{i}^{s},v_{j}^{t})\in Y|\mathbf{h}_{i}^{s},\mathbf{h}_{j}^{t} \Big{)} =ReLU(\mathbf{W}[\mu_{i}^{s}||\sigma_{i}^{s}||d_{v_{i}^{s}}||N_{a} (v_{i}^{s})]| \tag{11}\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mu_{j}^{t}||\sigma_{j }^{t}||M_{a}(v_{j}^{t})|+\mathbf{b}),\]  
where \([\cdot||\cdot]\) denotes the concatenation of embeddings, \(\mathbf{W}\) and \(\mathbf{b}\) are trainable parameters. \(d_{v_{i}^{s}}\) and \(d_{v_{j}^{t}}\) are the degree of \(v_{i}^{s}\) and \(v_{j}^{t}\), respectively. \(N_{a}(v_{i}^{s})\) and \(N_{a}(v_{j}^{t})\) are the number of labeled anchor nodes in the neighborhood of \(v_{i}^{s}\) and \(v_{j}^{t}\), respectively. Intuitively, node degree can reflect its local topology to some extent, and the number of neighborhood labeled anchor nodes can reflect the ability to receive labeled anchor node information. The final result of the prediction is then obtained using the labeled anchor links set \(E^{a}\) to calculate the cross-entropy as the loss function.  
In general, only a small portion of the entire network is made up of labeled anchor nodes. The following guidelines can be used to build some pseudo-anchor links in order to effectively train the prediction function mentioned above  
\[sim\Big{(}v_{i}^{s},v_{j}^{t}\Big{)}=\frac{\left|\mathcal{N}_{a}(v_{i}^{s}) \cap\mathcal{N}_{a}(v_{j}^{t})\right|}{\left|\mathcal{N}(v_{i}^{s})\cup \mathcal{N}(v_{j}^{t})-\mathcal{N}_{a}(v_{i}^{s})\cap\mathcal{N}_{a}(v_{j}^{t} )\right|}e^{-\left|d(v_{i}^{s})-d(v_{j}^{t})\right|}, \tag{12}\]  
where \(\mathcal{N}_{a}(v_{i}^{s})\) and \(\mathcal{N}_{a}(v_{j}^{t})\) are the labeled anchor nodes in the neighborhood of \(v_{i}^{s}\) and \(v_{j}^{t}\), respectively. \(\mathcal{N}(v_{i}^{s})\) and \(\mathcal{N}(v_{j}^{t})\) are the neighbors of \(v_{i}^{s}\) and \(v_{j}^{t}\), respectively. As a result, if two nodes in separate networks have more neighbors who are designated anchor nodesand their node degree differences are lower, they have a high likelihood of being possible anchor node pairs.  
Then, we construct a pseudo-anchor link generator by using pairs of labeled anchor nodes:  
\[f_{S}\left(v_{i}^{s},v_{j}^{t}\right)=\begin{cases}1&\text{if }sim\left(v_{i}^{s},v_{j}^{t} \right)\geq\frac{1}{|E^{s}|}\sum_{\left(v_{i}^{s},v_{i}^{t}\right)\in E^{s}} sim\left(v_{i}^{s},v_{i}^{t}\right)\\ 0&\text{otherwise}\end{cases} \tag{13}\]  
Pseudo-anchor links generated by the generator are easier to find the anchor node pairs that satisfy the equivalence connection status and containment connection status as introduced in Section 1.