# User Identity Linkage across Social Networks by Heterogeneous Graph Attention Network Modeling
## 3 Preliminary  
In this section, we define the preliminary concepts used in this paper. A heterogeneous graph is a particular type of graph. Compared with a traditional graph, a heterogeneous graph contains many types of nodes and edges. In this paper, user profiles, user-generated content, structural information, and their features represent nodes in a heterogeneous graph from different social networks, respectively, and we build edges between these nodes and users. These node types include the user node, the user profile node, the user-generated content node,the structural information node, the category node, the feature node, and the structural embedding node. The category nodes are subdivisions of user profiles and user-generated content, such as the username in profile. The feature nodes are unique and universal, extracted from user profiles and user-generated content. The structural embedding nodes dynamically generate after the embedding of every user-to-user edge. **Definition 1** (Heterogeneous Social Network Graph). : _A heterogeneous social network graph denotes as \(\mathcal{G}=\{\mathcal{V},\mathcal{E}\}\), where \(\mathcal{V}=\{v_{i}|i=1,\dots,N\}\) is a node set, and \(\mathcal{E}=\{e_{ij}|(i,j)=1,\dots,N\}\) is an edge set. A node set \(\mathcal{V}=\{\mathcal{V}^{\mathcal{U}},\mathcal{V}^{P},\mathcal{V}^{G}, \mathcal{V}^{S},\mathcal{V}^{C},\mathcal{V}^{F},\mathcal{V}^{E}\}\) consists of several types of nodes: The user node \(\mathcal{V}^{\mathcal{U}}\), the user profile node \(\mathcal{V}^{P}\), the user-generated content node \(\mathcal{V}^{G}\), the structural information node \(\mathcal{V}^{S}\), the category node \(\mathcal{V}^{C}\), the feature node \(\mathcal{V}^{F}\), and the structural embedding node \(\mathcal{V}^{E}\)._  
The edge pattern between the user node \(v_{i}\) and the user node \(v_{j}\) is \(v_{i}\xleftarrow{e_{ij}}v_{j}\), where \(v_{i},v_{j}\in\mathcal{V}^{\mathcal{U}}\). The edge pattern among the user node \(v_{m}\) and the feature node \(v_{i}\) is \(v_{i}\xleftarrow{e_{ij}}v_{j}\xleftarrow{e_{k}}v_{k}\xleftarrow{e_{km}}v_{m}\), where \(v_{i}\in\mathcal{V}^{F}\), \(v_{j}\in\mathcal{V}^{C}\), \(v_{k}\in\mathcal{V}^{P}\cup\mathcal{V}^{G}\), and \(v_{m}\in\mathcal{V}^{U}\). The edge pattern among the user node \(v_{k}\) and the structural embedding node \(v_{i}\) is \(v_{i}\xleftarrow{e_{ij}}v_{j}\xleftarrow{e_{jk}}v_{k}\), where \(v_{i}\in\mathcal{V}^{E}\), \(v_{j}\in\mathcal{V}^{S}\), and \(v_{k}\in\mathcal{V}^{U}\). An example of a heterogeneous social network graph is shown in Figure 1, which includes three user nodes, two user profile nodes, one user-generated content node, three structural information nodes, four category nodes, five feature nodes, and six structural embedding nodes. The user-to-user edges include two types, the edge between user A and user B is the \"friend\" edge, and the edge between user A and user C is the \"blacklist\" edge. For user profiles, the \"gender\" category nodes linked with user A and user B connect to the \"male\" feature node, and the \"name\" category node linked with user A connects to three feature nodes. For user-generated content, the \"interest\" category node linked with user C connects to a feature node. For structural information, every structural embedding node dynamically generates after the embedding of each user-to-user edge. The number of generated nodes is related to the type of user-to-user edges. Without loss of generality, we focus on user identity linkage in two social networks, denoted as \(\mathcal{G}_{s}\) and \(\mathcal{G}_{t}\) respectively. As shown in Figure 2, for users in the two social networks, we intend to predict linkage pairs where these users belong to the same identity. **Definition 2** (User Identity Linkage). : _Given two social networks \(\mathcal{G}_{s}\) and \(\mathcal{G}_{t}\), user identity linkage aims to predict whether a pair of entity \(v_{i}\in G_{s}\) and \(v_{j}\in G_{t}\) belong to the same user identity, i.e.,_  
\[F(v_{i},v_{j})=\left\{