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Graph Convolutional Networks (GCNs) have achieved impressive performance in a wide variety of areas, attracting considerable attention. The core step of GCNs is the information-passing framework that considers all information from neighbors to the central vertex to be equally important. Such equal importance, however, is inadequate for scale-free networks, where hub vertices propagate more dominant information due to vertex imbalance. In this paper, we propose a novel centrality-based framework named CenGCN to address the inequality of information. This framework first quantifies the similarity between hub vertices and their neighbors by label propagation with hub vertices. Based on this similarity and centrality indices, the framework transforms the graph by increasing or decreasing the weights of edges connecting hub vertices and adding self-connections to vertices. In each non-output layer of the GCN, this framework uses a hub attention mechanism to assign new weights to connected non-hub vertices based on their common information with hub vertices. We present two variants CenGCN\_D and CenGCN\_E, based on degree centrality and eigenvector centrality, respectively. We also conduct comprehensive experiments, including vertex classification, link prediction, vertex clustering, and network visualization. The results demonstrate that the two variants significantly outperform state-of-the-art baselines.