Success has been obtained using a semi-supervised graph analysis method based on a graph convolutional network (GCN). However, GCN ignores some local information at each node in the graph, so that data preprocessing is incomplete and the model generated is not accurate enough. Thus, in the case of numerous unsupervised models based on graph embedding technology, local node information is important. In this paper, we apply a local analysis method based on the similar neighbor hypothesis to a GCN, and propose a local density definition; we call this method LDGCN. The LDGCN algorithm processes the input data of GCN in two methods, i.e., the unbalanced and balanced methods. Thus, the optimized input data contains detailed local node information, and then the model generated is accurate after training. We also introduce the implementation of the LDGCN algorithm through the principle of GCN, and use three mainstream datasets to verify the effectiveness of the LDGCN algorithm (i.e., the Cora, Citeseer, and Pubmed datasets). Finally, we compare the performances of several mainstream graph analysis algorithms with that of the LDGCN algorithm. Experimental results show that the LDGCN algorithm has better performance in node classification tasks.