Title: Are Graph Convolutional Networks Fully Exploiting the Graph Structure?. Abstract: Graph Convolutional Networks (GCNs) represent the state-of-the-art for many graph related tasks. At every layer, GCNs rely on the graph structure to define an aggregation strategy where each node updates its representation by combining information from its neighbours. A known limitation of GCNs is their inability to infer long-range dependencies. In fact, as the number of layers increases, information gets smoothed and node embeddings become indistinguishable, negatively affecting performance. In this paper we formalize four levels of injection of graph structural information, and use them to analyze the importance of long-range dependencies. We then propose a novel regularization technique based on random walks with restart, called RWRReg, which encourages the network to encode long-range information into node embeddings. RWRReg does not require additional operations at inference time, is model-agnostic, and is further supported by our theoretical analysis connecting it to the Weisfeiler-Leman algorithm. Our experimental analysis, on both transductive and inductive tasks, shows that the lack of long-range structural information greatly affects the performance of state-of-the-art models, and that the long-range information exploited by RWRReg leads to an average accuracy improvement of more than $5\%$ on all considered tasks.