Deep models can be made scale-invariant when trained with multi-scale information. Images can be easily made multi-scale, given their grid-like structures. Extending this to generic graphs poses major challenges. For example, in link prediction tasks, inputs are represented as graphs consisting of nodes and edges. Currently, the state-of-the-art model for link prediction uses supervised heuristic learning, which learns graph structure features centered on two target nodes. It then learns graph neural networks to predict the existence of links based on graph structure features. Thus, the performance of link prediction models highly depends on graph structure features. In this work, we propose a novel node aggregation method that can transform the enclosing subgraph into different scales and preserve the relationship between two target nodes for link prediction. A theory for analyzing the information loss during the re-scaling procedure is also provided. Graphs in different scales can provide scale-invariant information, which enables graph neural networks to learn invariant features and improve link prediction performance. Our experimental results on 14 datasets from different areas demonstrate that our proposed method outperforms the state-of-the-art methods by employing multi-scale graphs without additional parameters.