# A Hierarchical Generative Embedding Model for Influence Maximization in Attributed Social Networks
## 3 Notations and Problem Formulation  
In this section, we first introduce the definitions and notations, and then formulate the problem studied in this paper. **Definition 1** (**attributed network**). : _An attributed network is an undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\), where \(\mathcal{V}\) is the set of network nodes, \(\mathcal{E}\) is the set of edges and \(\mathcal{A}\in\{0,1\}^{|\mathcal{V}|\times F}\) is a binary-valued attribute matrix recording node-attribute associations, with \(F\) being the number of attributes. The element \(A_{i,a}\in\mathcal{A}\) is a binary-valued attribute weight, indicating that attribute a associates with node \(i\) if \(A_{i,a}=1\)._  
**Definition 2** (**the tensility of community**). : _A community is a set composed of nodes with similar attributes. These nodes form a community due to similar attributes and repel each other in the community due to differences in attributes. The tensility of the community is used to measure the difference of node attributes in the community._  
**Definition 3** (**the granularity of hierarchical characteristics**). : _In hierarchical networks, vertexes at different levels have different granularities of characteristics. As shown in Figure 4, the university in the top level has the characteristics of name, address etc., and the colleges in the second level have the characteristics of majors, number of students etc. The characteristics of nodesat different levels have different granularities, while the granularity of the characteristics is the same for the nodes at the same level. It is the granularity of hierarchical characteristics that causes the vertical constraint of hierarchical structure properties (see Section 4.1.1)._  
The hierarchical clustering tree of \(G\) is denoted as \(T\) with a depth of \(L\). \(V^{l}\) is the vertex set at the \(l\)-th level of \(T\), and \(v_{i}^{l}\in\mathcal{V}^{l}\) is the \(i\)-th vertex at the \(l\)-th level. \(Ch(v)\) denotes the child vertex set of the vertex \(v\), \(fa(v)\) denotes the father vertex of \(v\) and \(\mathbf{\Phi}(v)\) denotes the embedding of vertex \(v\). In our paper, we use'vertex' to represent a user or community in the hierarchical network, while using 'node' to represent a user in the general network, especially when \(l=L\), \(v_{i}^{l}\) represents a leaf vertex (user) at the bottom of the clustering tree. For ease of reference, we summarize the basic notations in Table 1.  
With the definitions and notations described above, we formulate the problem studied. **Problem 1** (Node Embedding). : _Given an undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\), the goal is to learn the latent representation of nodes with the mapping function \(\Xi\):_  
\[\mathcal{G}\overset{\Xi}{\longrightarrow}\mathbf{\Phi}_{\mathcal{V}} \tag{1}\]  
_such that the information of hierarchical network structure, general network and node attributes can be preserved as much as possible by \(\mathbf{\Phi}_{\mathcal{V}}\in\mathcal{R}^{N\times D}\), where \(\mathbf{\Phi}_{\mathcal{V}}\) is the latent embedding matrix for all nodes and \(D\) is the dimension size of the embeddings._  
**Problem 2** (Influence Maximization in Attributed Social Networks). : _Given an IC diffusion model, an undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\) and the latent embedding matrix for all nodes \(\mathbf{\Phi}_{\mathcal{V}}\), the goal is to find the node set \(\mathcal{S}\) with an algorithm \(\mathcal{F}\):_ \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(\mathcal{G}\) & An undirected graph \\ \(\mathcal{V}\) & Set of nodes \\ \(\mathcal{E}\) & Set of edges \\ \(T\) & A hierarchical clustering tree \\ \(L\) & The depth of the clustering tree \\ \(Ch(v)\) & The child vertex set of vertex \(v\) \\ \(fa(v)\) & The father vertex of \(v\) \\ \(v_{i}^{l}\) & The \(i\)-th vertex at the \(l\)-th level \\ \(\mathbf{\Phi}_{v}\) & The embedding of vertex \(v\) \\ \hline \hline \end{tabular}
\end{table}