Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence

TL;DR

This paper derives the backpropagation algorithm for graph convolutional networks using matrix calculus, enabling improved explainability and sensitivity analysis in graph-based AI tasks.

Contribution

It provides a detailed matrix calculus derivation of backpropagation for GCNs, extending to arbitrary activations and layers, and demonstrates its accuracy and application to explainable AI.

Findings

Median squared error of derivation vs. automatic differentiation is between 10^{-18} and 10^{-14}

Validated on node classification and link prediction tasks

Facilitates development of explainable AI and sensitivity analysis

Abstract

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.