This paper provides a comprehensive and detailed derivation of the backpropagation algorithm for graph convolutional neural networks using matrix calculus. The derivation is extended to include arbitrary element-wise activation functions and an arbitrary number of layers. The study addresses two fundamental problems, namely node classification and link prediction. To validate our method, we compare it with reverse-mode automatic differentiation. The experimental results demonstrate that the median sum of squared errors of the updated weight matrices, when comparing our method to the approach using reverse-mode automatic differentiation, falls within the range of $10^{-18}$ to $10^{-14}$. These outcomes are obtained from conducting experiments on a five-layer graph convolutional network, applied to a node classification problem on Zachary's karate club social network and a link prediction problem on a drug-drug interaction network. Finally, we show how the derived closed-form solution can facilitate the development of explainable AI and sensitivity analysis.

翻译：本文利用矩阵微积分，对图卷积神经网络的反向传播算法进行了全面而详细的推导。该推导可扩展至包含任意逐元素激活函数及任意数量的网络层。本研究针对两个基本问题展开，即节点分类与链接预测。为验证所提方法，我们将其与反向模式自动微分进行了比较。实验结果表明，在将本文方法与基于反向模式自动微分的方法进行比较时，更新后权重矩阵的误差平方和中位数落在 $10^{-18}$ 至 $10^{-14}$ 的范围内。这些结果通过在五层图卷积网络上进行实验获得，实验分别应用于Zachary空手道俱乐部社交网络的节点分类问题以及药物相互作用网络的链接预测问题。最后，我们展示了所推导的闭式解如何促进可解释人工智能与敏感性分析的发展。