Learning positional information of nodes in a graph is important for link prediction tasks. We propose a representation of positional information using representative nodes called landmarks. A small number of nodes with high degree centrality are selected as landmarks, which serve as reference points for the nodes' positions. We justify this selection strategy for well-known random graph models and derive closed-form bounds on the average path lengths involving landmarks. In a model for power-law graphs, we prove that landmarks provide asymptotically exact information on inter-node distances. We apply theoretical insights to practical networks and propose Hierarchical Position embedding with Landmarks and Clustering (HPLC). HPLC combines landmark selection and graph clustering, where the graph is partitioned into densely connected clusters in which nodes with the highest degree are selected as landmarks. HPLC leverages the positional information of nodes based on landmarks at various levels of hierarchy such as nodes' distances to landmarks, inter-landmark distances and hierarchical grouping of clusters. Experiments show that HPLC achieves state-of-the-art performances of link prediction on various datasets in terms of HIT@K, MRR, and AUC. The code is available at \url{https://github.com/kmswin1/HPLC}.

翻译：学习图中节点的位置信息对于链路预测任务至关重要。我们提出了一种利用代表性节点（称为地标）来表示位置信息的方法。选取少量度中心性较高的节点作为地标,这些地标充当节点位置的参考点。我们针对经典的随机图模型论证了该选取策略的合理性,并推导了包含地标的平均路径长度的闭式边界。在幂律图模型中,我们证明地标能提供节点间距离的渐近精确信息。我们将理论洞见应用于实际网络,提出了基于地标与聚类的层次化位置嵌入方法(HPLC)。HPLC结合了地标选取与图聚类技术：将图划分为紧密连接的簇,并在每个簇中选择度最高的节点作为地标。该方法利用节点在不同层级上的地标位置信息,包括节点到地标的距离、地标间距离以及簇的层次分组。实验表明,在HIT@K、MRR和AUC指标上,HPLC在多种数据集上均取得了链路预测的最新性能。代码开源在\url{https://github.com/kmswin1/HPLC}。