In the area of large-scale graph data representation and semi-supervised learning, deep graph-based convolutional neural networks have been widely applied. However, typical graph convolutional network (GCN) aggregates information of neighbor nodes based on binary neighborhood similarity (adjacency matrix). It treats all neighbor nodes of one node equally, which does not suppress the influence of dissimilar neighbor nodes. In this paper, we investigate GCN based on similarity matrix instead of adjacency matrix of graph nodes. Gaussian heat kernel similarity in Euclidean space is first adopted, which is named EGCN. Then biologically inspired manifold similarity is trained in reproducing kernel Hilbert space (RKHS), based on which a manifold GCN (named MGCN) is proposed for graph data representation and semi-supervised learning with four different kernel types. The proposed method is evaluated with extensive experiments on four benchmark document citation network datasets. The objective function of manifold similarity learning converges very quickly on different datasets using various kernel functions. Compared with state-of-the-art methods, our method is very competitive in terms of graph node recognition accuracy. In particular, the recognition rates of MGCN (Gaussian kernel) and MGCN (Polynomial Kernel) outperform that of typical GCN about 3.8% on Cora dataset, 3.5% on Citeseer dataset, 1.3% on Pubmed dataset and 4% on Cora_ML dataset, respectively. Although the proposed MGCN is relatively simple and easy to implement, it can discover local manifold structure by manifold similarity learning and suppress the influence of dissimilar neighbor nodes, which shows the effectiveness of the proposed MGCN.
