# Hypernetwork Representation Learning with the Set Constraint
## 4 Preliminaries  
Since the study on traditional graphs is relatively mature, it can be regarded as a desirable way to study the hypergraph to deepen the understanding of the corresponding hypergraph by the study on the traditional graph (if the hypergraph can be transformed into the traditional graph). In the literature [13], it is proposed that the hypergraph can be transformed into three types of traditional graphs, namely a line graph, two-section graph, and incidence graph, where the line graph has only the nodes converted from the hyperedges and no nodes are associated with the hyperedges (which indicates that the relationship between the nodes associated with the hyperedges is lost, leading to a large loss of the hypernetwork topology structure). The incidence graph introduces the nodes converted from the hyperedges to weaken the strong correlation relationships between the nodes associated with the hyperedges, which leads to a certain amount of loss of hypernetwork topology structure. The two-section graph makes all of the nodes  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Network Type** & **Methods** & **Pros** & **Cons** \\ \hline \multirow{6}{*}{traditional network} & \multirow{2}{*}{Deepwalk} & to introduce Skip-Gram into & random walk without \\  & & network representation & constraints \\  & & learning firstly & \\ \cline{2-4}  & \multirow{2}{*}{LINE} & to preserve first-order and & direct concatenation of \\  & & second-order proximities & first-order vector and \\  & & second-order vector & second-order vector \\ \cline{2-4}  & \multirow{2}{*}{SDNE} & to capture highly nonlinear & not applicable to some \\  & & network structures & networks \\ \cline{2-4}  & \multirow{2}{*}{CANE} & to learn context-aware & no inter-relation \\  & & representation vectors of the & between \\  & & nodes with mutual attention & structure-based vector \\  & & mechanism & and text-based vector \\ \hline \multirow{6}{*}{hypernetwork} & \multirow{2}{*}{HPHG} & to preserve hypernetwork & \multirow{2}{*}{not to consider \\  & & topology structure by a & random walk based on \\  & & hyper-path & \\ \cline{1-1} \cline{3-4}  & \multirow{2}{*}{DHNE} & a solution to model & limited to \\ \cline{1-1}  & & hyperedges without & heterogeneous \\ \cline{1-1}  & & hyperedge decomposition & hyperedges of fixed \\ \cline{1-1} \cline{2-4}  & \multirow{2}{*}{HRSC} & to consider hyperedges & to lose partial \\ \cline{1-1}  & & sufficiently & hypernetwork structure \\ \cline{1-1} \cline{3-4}  & & information \\ \hline \hline \end{tabular}
\end{table}
Table 1: The statistics of the pros and cons.  
associated with each hyperedge joined into a complete graph, which leads to less loss of hypernetwork topology structure than other two graphs. Therefore, a two-section graph that approximates the hypernetwork topology structure has the characteristic of preserving the hypergraph topology structure better than other transformation strategies. Hence, in this paper, the hypergraph is transformed into a two-section graph, and then the traditional network topology structure abstracted as two-section graph and the hyperedges in the hypernetwork abstracted as the hypergraph are comprehensively utilized to carry out the study of the hypernetwork representation learning. The detailed strategy for transforming the hypergraph into a two-section graph is as follows.  
Given a hypergraph \(H=(V,E)\), two-section graph \(S=(V^{\prime},E^{\prime})\) corresponding to the hypergraph \(H\) is a traditional graph that meets the following conditions:  
* \(V^{\prime}=V\), that is, the node set of 2-section graph \(S\) is the same as the node set of the hypergraph \(H\).
* One edge is linked between any two different nodes if and only if they are simultaneously contained in at least one hyperedge (that is, all of the nodes in each hyperedge are joined into a complete graph).  
A hypergraph and two-section graph of its transformation are shown in Figure 1.