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.

在大规模图形数据表示和半监督学习领域，基于深度图的卷积神经网络得到了广泛的应用。但是，典型的图卷积网络（GCN）基于二进制邻域相似度（邻接矩阵）来聚合邻居节点的信息。它平等地对待一个节点的所有邻居节点，这不会抑制异种邻居节点的影响。在本文中，我们研究基于相似矩阵而不是图节点邻接矩阵的GCN。首先采用欧氏空间中的高斯热核相似性，称为EGCN。然后在复制核希尔伯特空间（RKHS），在此基础上，提出了流形GCN（名为MGCN）用于图形数据表示和具有四种不同内核类型的半监督学习。在四个基准文档引用网络数据集上进行了广泛的实验，对所提出的方法进行了评估。流形相似性学习的目标函数使用各种核函数在不同数据集上非常迅速地收敛。与最先进的方法相比，我们的方法在图节点识别精度方面非常有竞争力。尤其是，MGCN（高斯核）和MGCN（多项式内核）的识别率分别在Cora数据集上约为3.8％，在Citeseer数据集上为3.5％，在Pubmed数据集上为1.3％，在Cora_ML数据集上为4％，优于典型GCN。尽管建议的MGCN相对简单易行，