Graph convolutional networks (GCN) have recently demonstrated their potential in analyzing non-grid structure data that can be represented as graphs. The core idea is to encode the local topology of a graph, via convolutions, into the feature of a center node. In this paper, we propose a novel GCN model, which we term as Shortest Path Graph Attention Network (SPAGAN). Unlike conventional GCN models that carry out node-based attentions, on either first-order neighbors or random higher-order ones, the proposed SPAGAN conducts path-based attention that explicitly accounts for the influence of a sequence of nodes yielding the minimum cost, or shortest path, between the center node and its higher-order neighbors. SPAGAN therefore allows for a more informative and intact exploration of the graph structure and further the more effective aggregation of information from distant neighbors, as compared to node-based GCN methods. We test SPAGAN for the downstream classification task on several standard datasets, and achieve performances superior to the state of the art.