Graph Neural Networks (GNNs) are increasingly becoming the favorite method for graph learning. They exploit the semi-supervised nature of deep learning, and they bypass computational bottlenecks associated with traditional graph learning methods. In addition to the feature matrix $X$, GNNs need an adjacency matrix $A$ to perform feature propagation. In many cases the adjacency matrix $A$ is missing. We introduce a graph construction scheme that construct the adjacency matrix $A$ using unsupervised and supervised information. Unsupervised information characterize the neighborhood around points. We used Principal Axis trees (PA-trees) as a source of unsupervised information, where we create edges between points falling onto the same leaf node. For supervised information, we used the concept of penalty and intrinsic graphs. A penalty graph connects points with different class labels, whereas intrinsic graph connects points with the same class label. We used the penalty and intrinsic graphs to remove or add edges to the graph constructed via PA-tree. This graph construction scheme was tested on two well-known GNNs: 1) Graph Convolutional Network (GCN) and 2) Simple Graph Convolution (SGC). The experiments show that it is better to use SGC because it is faster and delivers better or the same results as GCN. We also test the effect of oversmoothing on both GCN and SGC. We found out that the level of smoothing has to be selected carefully for SGC to avoid oversmoothing.

翻译：图神经网络（GNNs）正日益成为图学习领域备受青睐的方法。它们利用深度学习的半监督特性，并避免了传统图学习方法中的计算瓶颈。除了特征矩阵 $X$ 外，GNNs 需要邻接矩阵 $A$ 来执行特征传播。在许多情况下，邻接矩阵 $A$ 是缺失的。我们提出了一种图构造方案，利用无监督和有监督信息来构建邻接矩阵 $A$。无监督信息用于刻画点周围的邻域特征。我们采用主成分轴树（PA-trees）作为无监督信息的来源，将落在同一叶节点上的点之间创建边。对于有监督信息，我们引入了惩罚图和内禀图的概念。惩罚图连接不同类标签的点，而内禀图连接同类标签的点。我们利用惩罚图和内禀图对基于 PA-tree 构造的图进行边的删除或添加。该图构造方案在两种著名的 GNNs 上进行了测试：1) 图卷积网络（GCN）和 2) 简单图卷积（SGC）。实验表明，使用 SGC 效果更优，因为它速度更快且能获得与 GCN 相同或更优的结果。我们还测试了过平滑对 GCN 和 SGC 的影响。研究发现，SGC 的平滑程度需要仔细选择以避免过平滑。