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

arXiv: https://arxiv.org/abs/2408.01408 · HTML: https://mlsanity.com/paper/2408.01408/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 43 references that are in the map.
- First posted on arXiv 2 Aug 2024.

## Builds on (most cited first, 16 of 16)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Cooperative Game Theory Approaches for Network Partitioning](https://mlsanity.com/paper/1707.03587/) (2017)
- [Neo-GNNs: Neighborhood Overlap-aware Graph Neural Networks for Link Prediction](https://mlsanity.com/paper/2206.04216/) (2022)
- [A Survey on Graph-Based Deep Learning for Computational Histopathology](https://mlsanity.com/paper/2107.00272/) (2021)
- [Permutation Equivariant Graph Framelets for Heterophilous Graph Learning](https://mlsanity.com/paper/2306.04265/) (2023)
- [HAQJSK: Hierarchical-Aligned Quantum Jensen-Shannon Kernels for Graph Classification](https://mlsanity.com/paper/2211.02904/) (2022)
- [Feedforward and Recurrent Neural Networks Backward Propagation and Hessian in Matrix Form](https://mlsanity.com/paper/1709.06080/) (2017)
- [Backpropagation in matrix notation](https://mlsanity.com/paper/1707.02746/) (2017)
- [Derivation of the Backpropagation Algorithm Based on Derivative Amplification Coefficients](https://mlsanity.com/paper/2102.04320/) (2021)
- [Benchmarking Graph Neural Networks on Link Prediction](https://mlsanity.com/paper/2102.12557/) (2021)
- [An induction proof of the backpropagation algorithm in matrix notation](https://mlsanity.com/paper/2107.09384/) (2021)

## Built on by (most cited first, 1 of 1)

- [Predicting Steady-State Behavior in Complex Networks with Graph Neural Networks](https://mlsanity.com/paper/2502.01693/) (2025)

## Abstract (arXiv)

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.