# SentenceLDA- and ConNetClus-Based Heterogeneous Academic Network Analysis for Publication Ranking
## 3 Methodology
### Construction of the Heterogeneous Academic Network  
The network is generally composed of vertexes and edges. Vertexes represent the objects in the network, and edges represent the relationship between objects. In order to explain the construction of heterogeneous academic networks, the notations and their interpretations are shown in Table 1.  
For the homogeneous network, \(G=(V,E)\) is composed of the set of vertexes and set of edges, where \(V\) represents the vertex set, \(V=\{v_{1},v_{2},\cdots v_{N}\}\), \(E\) represents the edge set, and \(E=\{e_{1},e_{2},\cdots e_{M}\}\). Each edge represents the connection between two vertexes,\(e_{ij}=<v_{i},v_{j}>\), where \(N\) represents the total number of vertexes and \(M\) represents the total number of edges. In this paper, we put the focus on the heterogeneous network, \(G=<V,E,C>\), where \(C\) represents the vertex-type set, \(C=\{c_{1},c_{2},\cdots c_{S}\}\), \(S\) represents the total number of vertex types. If \(S=1\), there is only one type of vertex in the network, which belongs to the homogeneous network. If \(S>1\), there are two or more vertex-types, and the network is a heterogeneous network. In the heterogeneous network, for each vertex, \(v_{i}{}^{c_{s}}\in V\), and its type is \(c_{s}\in C\). For each edge, if \(e_{ij}=\left(v_{i}{}^{c_{s}},v_{j}{}^{c_{s}}\right)\in E\), which means the vertex \(v_{i}{}^{c_{s}}\) is connected to the vertex \(v_{j}{}^{c_{s}}\), and these two vertexes have the same vertex type, \(c_{s}\). If \(e_{ij}{}^{c_{s}}=\left(v_{i}{}^{c_{s}},v_{j}{}^{c_{s}}\right)\in E\), which means the vertex \(v_{j}{}^{c_{s}}\) is connected to the vertex \(v_{j}{}^{c_{s}}\), because the vertex types of \(v_{j}{}^{c_{s}}\) and \(v_{j}{}^{c_{s}}\) are \(c_{s}\) and \(c_{s}\) respectively. \(v_{j}{}^{c_{s}}\) and \(v_{j}{}^{c_{s}}\) have the different vertex type. For a heterogeneous network \(G\), we can disassemble it into several homogeneous sub-networks, \(G=\{G^{c_{1}},G^{c_{2}},\ldots,G^{c_{s}}\}\). The number of sub-networks disassembled from the heterogeneous network is determined by the number of vertex types. Therefore, the heterogeneous network dividing into a collection of sub-networks:  
\[V(G)=V(G^{c_{1}})\cup V(G^{c_{2}})\cup V(G^{c_{3}})\cup\ldots\cup V(G^{c_{S}}) \tag{5}\]  
\[E(G)=E(G^{c_{1}})\cup E(G^{c_{2}})\cup E(G^{c_{3}})\cup\ldots\cup E(G^{c_{S}})\] \[C(G)=C(G^{c_{1}})\cup C(G^{c_{2}})\cup C(G^{c_{3}})\cup\ldots\cup C (G^{c_{S}})\]  
As a simple example, in the heterogeneous network (shown in Figure 1), we assumed there are two edges, \(e_{ij}=\left(v_{i}{}^{c_{s}},v_{j}{}^{c_{s}}\right)\) and \(e_{ik}=\left(v_{i}{}^{c_{s}},v_{k}{}^{c_{s}}\right)\); three vertexes,\(p_{i}{}^{c_{s}}\), \(v_{j}{}^{c_{s}}\) and \(v_{k}{}^{c_{s}}\); and two vertex types, \(c_{s}\) and \(c_{s}\). We can obtain two sub-networks from this heterogeneous network, \(G^{c_{s}}\) and \(G^{c_{s}}\). Where \(V(G)=V(G^{c_{s}})\cup V(G^{c_{s}})\), and \(V(G^{c_{s}})=\{v_{i}{}^{c_{s}}\), \(v_{j}{}^{c_{s}}\}\), \(V(G^{c_{s}})=\{v_{i}{}^{c_{s}}\), \(v_{k}{}^{c_{s}}\)}. \(E(G)=E(G^{c_{s}})\cup E(G^{c_{s}})\), and \(E(G^{c_{s}})=\{e_{ij}\}\), \(E(G^{c_{s}})=\{e_{ik}\}\). \(C(G)\)=\(C(G^{c_{s}})\cup C(G^{c_{s}})\), and \(C(G^{c_{s}})=\{c_{s}\}\), \(C(G^{c_{s}})=\{c_{s}\}\). \begin{table}
\begin{tabular}{l l} \hline \multicolumn{1}{c}{**Notation**} & \multicolumn{1}{c}{**Interpretation**} \\ \hline \(G=(V,E)\) & the homogeneous network \\ V & the vertex set, \(V=\{v_{1},v_{2},\cdots v_{N}\}\), N is the total number of vertexes \\ E & the edge set, \(E=\{e_{1},e_{2},\cdots e_{M}\}\), M is the total number of edges \\ \(G=V,E,C\) & the heterogeneous network \\ C & the vertex-type set, \(C=\{c_{1},c_{2},\cdots c_{S}\}\), S is the total number of vertex types \\ \(v_{i}{}^{c_{s}}\) & the type of vertex \(v_{i}\) is \(c_{s}\) \\ \(e_{ij}=\left(v_{i}{}^{c_{s}},v_{j}{}^{c_{s}}\right)\) & the edge between \(v_{i}\) and \(v_{j}\), the type of vertex \(v_{i}\) is \(c_{s}\)... and \(v_{j}\) is \(c_{s}\) \\ \hline \end{tabular}
\end{table}