Link prediction in simple graphs is a fundamental problem in which new links between nodes are predicted based on the observed structure of the graph. However, in many real-world applications, there is a need to model relationships among nodes which go beyond pairwise associations. For example, in a chemical reaction, relationship among the reactants and products is inherently higher-order. Additionally, there is need to represent the direction from reactants to products. Hypergraphs provide a natural way to represent such complex higher-order relationships. Even though Graph Convolutional Networks (GCN) have recently emerged as a powerful deep learning-based approach for link prediction over simple graphs, their suitability for link prediction in hypergraphs is unexplored -- we fill this gap in this paper and propose Neural Hyperlink Predictor (NHP). NHP adapts GCNs for link prediction in hypergraphs. We propose two variants of NHP --NHP-U and NHP-D -- for link prediction over undirected and directed hypergraphs, respectively. To the best of our knowledge, NHP-D is the first method for link prediction over directed hypergraphs. Through extensive experiments on multiple real-world datasets, we show NHP's effectiveness. The problem of link prediction in graphs has numerous applications in the fields of social network analysis BID24 , knowledge bases BID30 , bioinformatics BID26 to name a few. However, in many real-world problems relationships go beyond pairwise associations. For example, in chemical reactions data the relationship representing a group of chemical compounds that can react is inherently higher-order and similarly, the co-authorship relationship in a citation network is higher-order etc. Hypergraphs provide a natural way to model such higher-order complex relations. Hyperlink prediction is the problem of predicting such missing higher-order relationships in a hypergraph.Besides the higher-order relationships, modeling the direction information between these relationships is also useful in many practical applications. For example, in the chemical reactions data, in addition to predicting groups of chemical compounds which form reactants and/or products, it is also important to predict the direction between reactants and products, i.e., a group of reactants react to give a group of products. Directed hypergraphs BID12 provide a way to model the direction information in hypergraphs. Similar to the undirected hypergraphs, predicting the missing hyperlinks in a directed hypergraph is also useful in practical settings. Figure 1 illustrates the difference between modeling the chemical reactions data using undirected and directed hypergraphs. Most of the previous work on hyperlink prediction BID43 focus only on undirected hypergraphs. In this work we focus both on undirected and directed hypergraphs.Recently, Graph Convolutional Networks (GCNs) BID21 have emerged as a powerful tool for representation learning on graphs. GCNs have also been successfully applied for link prediction on normal graphs BID34 BID20 . Inspired by the success of GCNs for link prediction in graphs and deep learning in general BID39 , we propose a GCN-based framework for hyperlink prediction which works for both undirected and directed hypergraphs. We make the following contributions:Figure 1: Illustrating the difference between modeling chemical reactions data using undirected and directed hypergraphs. To the left is the undirected hypergraph, in which both the reactants and products are present in the same hyperlink. Whereas in the directed hypergraph (to the right), for a given reaction, the reactants are connected by one hyperlink and products are connected by another hyperlink and both these hyperlinks are connected by a directed link.• We propose a Graph Convolutional Networks (GCN)-based framework called Neural Hyperlink Predictor (NHP) for the problem of hyperlink prediction. To the best of our knowledge, this is the first ever deep learning based approach for this problem.• We extend the proposed NHP for the problem of hyperlink prediction in directed hypergraphs.To the best of our knowledge, this is the first ever attempt at the problem of link prediction in directed hypergraphs.• Through extensive experiments on multiple real-world datasets, we show the effectiveness of proposed NHP for link prediction in both undirected and directed hypergraphs.We have released NHP's source code at this anonymous location: https://anonymous.4open. science/repository/7d86231e-f6ba-4795-ae51-ac28d89f1521/. As we can see in table 9, the standard deviations of random negative sampling are on the higher side. This is expected as the particular choice made for negative samples decides the decision boundary for the binary classifier. The superior AUC values of mixed in table 8 supports our intuition that it provides benefits of both positive unlabeled learning and uniform random negative sampling. The standard deviations of mixed are much lower but still higher than positive-unlabeled learning.In general, summarising the results for all datasets, we believe that positive-unlabeled learning is superior to random negative sampling because of the higher confidence (low standard deviation) predictions. We have introduced NHP, a novel neural approach for hyperlink prediction in both undirected and directed hypergraphs. To the best of our knowledge, this is the first neural method for hyperlink prediction in undirected hypergraphs. NHP is also the first method for hyperlink prediction in directed hypergraphs. Through extensive experiments on multiple real-world datasets, we have demonstrated NHP's effectiveness over state-of-the art baselines. Approaches that augment GCNs with attention BID38 , self-training and co-training with random walks BID23 , edge-feature learning in a dual-primal setup BID28 have been recently proposed on graph-based semi-supervised learning tasks. Our NHP framework provides the flexibility to incorporate these approaches for more improved performance. An interesting future direction is predicting hyperlinks in partial-order hypergraphs (Feng et al., 2018) . We leave extending NHP framework to inductive settings as part of future work.hyperparameter value number of hidden units 16 number of hidden layers 2 dropout rate 0.5 L2 regularisation 5 × 10 −4 learning rate 0.01 non-linearity ReLU TAB1 : Hyperparameters of GCN used for all the datasets• DBLP: We used the DBLP database v4 3 . We filtered out papers without abstracts, and processed each abstract by tokenizing it and removing stop-words removal. Further, we filtered out papers with one author only. This left 540532 papers. In order to ensure that the hypergraph formed would be sufficiently dense, we found the number of papers authored by each author and took the top 1000 authors as 'selected authors'. Then we filtered out the papers that were not authored by at least three of the selected authors. Finally, we were left with 1590 papers by 685 of the original 1000 selected authors. To extract word features from each of these abstracts, we took all words appearing in these abstracts with a frequency greater than 50. Each abstract was thus represented by a 602-dimensional bag-of-words representation.For both datasets, we randomly sample |E| fake papers according to the author distribution of the existing non-fake papers (2708 and 1590 for CORA and DBLP respectively). We randomly generated Gaussian p dimensional features for these fake papers (1433 and 602 for CORA and DBLP respectively).
We propose Neural Hyperlink Predictor (NHP). NHP adapts graph convolutional networks for link prediction in hypergraphs
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