Traditional link prediction methods of sparse networks have low accuracy. In order to capture the possibility of link establishment among sparse network nodes, we propose the concept of simplest subgraph based on the shortest path between a pair of nodes, which reflects the tight topology relationship between nodes. Based on the node2vec node vectorization method, the link representation based on the shortest path is implemented. To complete the link classification task, we use the long short-term memory (LSTM) recurrent neural network to learn the characteristics of long-link node sequences. Compared with existing methods, the proposed method can increase the AUC value on 4 different datasets by 11.6% averagely, and the AP value by an average of 13.3%.