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

Mathematical transparency enables AI accountability: The paper demonstrates that deriving explicit mathematical formulas for how graph neural networks make decisions (through backpropagation) can make these systems more interpretable and auditable, which is essential for responsible AI deployment in high-stakes applications.

Explainability is computationally feasible: By providing closed-form mathematical solutions rather than relying solely on "black box" automatic differentiation, the authors show that sensitivity analysis and explainability features can be built into graph neural networks without prohibitive computational overhead, making responsible AI practices more practical.

Rigorous validation matters for trustworthiness: The paper validates their mathematical derivation against standard methods with extremely high precision (errors of $10^{-18}$ to $10^{-14}$), demonstrating that responsible AI requires careful verification that systems work as intended before deployment.

Graph neural networks need specialized scrutiny: Since graph neural networks are increasingly used for sensitive applications like drug discovery (as shown in their drug-drug interaction experiments) and social network analysis (Zachary's karate club), having mathematically rigorous understanding of how they function is crucial for identifying potential biases or errors in these high-impact domains.