# Link Prediction with Hypergraphs via Network Embedding
## 1 Introduction  
Link prediction [1; 2; 3] has been widely applied in many fields with extensive research studies, especially in society networks, such as community detection [4] and recommendation [5]. It aims to predict the potential links between nodes based on existing links, and has a wide range of applications in many fields, from bioinformatics [6; 7] and social science [8] to computer science [9]. Existing traditional methods for link prediction [10; 11; 12; 13] focus on simple graphs mostly and less on the interactions between pairs of nodes present in real-world systems, while research on high-order interactions is of great significance for modeling complex systems. For instance, in scientific collaboration networks, several researchers work together on a research project; in the brain network, a human behavior usually involves multiple neurons. Link prediction on high-order interactions leads to some challenges, while a hypergraph [14; 15; 16] provides a useful way to modeling such interactions. A hypergraph can reflect multiple nodes' relations with hyperlinks, and can be used in evaluating vital nodes [17], describing protein interaction [18] and so on. Hyperlink prediction on hypergraph has been investigated to predict higher-order links such as a user releasing a tweet containing a hashtag [19]. Hyperlink prediction [20] has also been helpful to predict multiactor collaborations [21]. By formulating various kinds of nodes and associations into a hypergraph, link prediction on heterogeneous networks has developed increasingly. [PERSON] [19] modeled various types of objects and relations of networks as hypergraphs and used link proximities to construct a cost function to predict users' links. [PERSON] [22] constructed relations between pairs of drugs into a hypergraph to predict multidrug interactions. [PERSON] et al. [23] proposed a Metapath-aware HyperGraph Transformer (Meta-HGT) for node embedding to capture the high-order relations. [PERSON] et al. [24] proposed dynamic hypergraph neural networks based on key hyperedges (DHKH) to considera dynamic hypergraph structure. [PERSON] et al. [25] presented a method named heterogeneous hypergraph variational autoencoder (HeteHG-VAE) for link prediction in heterogeneous information networks (HINs) mapped to a heterogeneous hypergraph with a certain kind of semantics to capture both the high-order semantics and complex relations among nodes, while preserving the low-order pairwise topology information of the original HIN. Network embedding [26; 27] combining machine learning or deep learning with network science has made it possible to automatically learn and preserve network properties by representing nodes in a low-dimensional space. It is usually assumed that the distance between the representation vectors of nodes reflects the similarity of the nodes in networks [28]. Network embedding typically realizes a network representation through matrix factorization, random walk and neural network methods. The matrix factorization methods select an adjacency matrix, an incident matrix, a Laplacian matrix and their variant forms to factorize and obtain the embeddings, such as M-NMF [29] and Laplacian eigenmaps [30]. The random walk methods generate embeddings through a random walk of nodes on graphs and training node sequences in models; representative methods include the Deepwalk [31], Node2 vec [32] and Graphwave models [33]. The methods based on a neural network realize an embedding by the nonlinear function of deep models to map the networks in a vector space, such as HeGan [34], VERSE [35] and SiNE models [36]. Furthermore, deep-learning-based link prediction methods on hypergraphs have achieved rapid development. [PERSON] et al. [37] proposed a neural hyperlink predictor (NHP) adapting graph convolutional networks (GCNs) [38] for link prediction in hypergraphs. Node2 vec [32] with a single-layer perceptron (Node2 vec-SLP) was an improved version of Node2 vec for hyperlink prediction, which employed a one-layer neural network to compute hyperlink scores [39].