# Edge-Nodes Representation Neural Machine for Link Prediction
## 1 Introduction  
Many parts of the world can be considered as an enormous network where all the people and objects are the nodes, the relationships between people and objects are the edges. In the big world network, there are plenty of small networks such as social network [1], power network [2] and protein network [3]. In recent years, researchers pay abundant effort on the evolution of the network, where the most fundamental and vital task is link prediction [1] that predicts whether there is a link between two nodes in a network. Link prediction has appealed large amounts of attention in the data mining and machine learning fields, since link prediction can apply in lots of situations such as friend recommendation in social network [4], product recommendation in e-commerce system [5], relationship discovery in knowledge graphs [6], finding some interesting interactions among proteins [7]. In the past few decades, many methods are developed to improve the performance of link prediction, some methods are very complicit such as probabilistic matrix factorization [8] and stochastic block models (SBM) [9]. Other simple methods of link prediction tend to learn a heuristic score to measure the similarity or connectivity of two nodes, namely two nodes have a high probability to beconnected if they are similar to each other. For instance, common neighbors (CN) [1] heuristic assumes that the more common neighbors two nodes have, the more likely there is a link between them. It is obvious that this heuristic is easy-understanding and simple, but surprisingly it performs very well on social networks. Besides, other heuristics including Adamic-Adar (AA) [4], preferential Attachment (PA) [10], resource allocation (RA) [11], Katz [12], PageRank (PR) [1], SimRank (SR) [13], resistance distance (RD) [14] also have good performance on different networks. However, none of these methods are suitable for all kinds of networks with different structures and characteristics. For example, the common neighbors [1] heuristic has good performance when predicting the co-authorships in collaboration networks and the friendships in social networks, but it performs poorly when predicting links on the biological networks and power networks [15]. Meanwhile, the resistance distance [14] heuristic works surprisingly well on the power grids and router-Level networks but has low accuracy when predicting links on social networks. A paper [15] compares 20 heuristics and concludes that none of these heuristics can have good performance on all kinds of networks. In that case, people need to choose these methods manually based on experience and prior knowledge when predicting links on different networks. Due to the limitation of above methods, [PERSON] et al. proposed a meaningful and universal method named Weisfeiler-Lehman Neural Machine (WLNM) [16], which learns topological features in the form of graph patterns that promote the formation of the link. WLNM firstly extracts an enclosing subgraph for each target link and encodes the subgraph to an adjacency matrix as link's representation. Then WLNM trains a neural network to predict whether the target link exists. To speak more clearly, WLNM tends to use the link's surrounding environment to represent the link. Actually, the link is equal to the edge to some extents. So WLNM can be considered as generating edge's representation to represent the link. However, as shown in Figure 1, a link's formation may be related to edge \(E_{1}\) and two nodes \(A\) and \(B\) on the two sides of the edge. Actually, WLNM only employs the edge's representation of the target link when predicting the link's existence. Thus, the method may lose the most critical node's information, leading to poor performance on some networks. Recently, many graph embedding methods are proposed to represent the graph more effectively, such as deepwalk [17], Node2 vec [18] and LINE [19]. These methods generate the node's embedding, all the nodes are represented in the form of vectors produced by the learning models such as skip-gram model.