# From Graphs to Gradients: Physics-Inspired Structural Attribution for Cyber-Physical IoT Systems and Beyond

arXiv: https://arxiv.org/abs/2607.05563 · HTML: https://mlsanity.com/paper/2607.05563/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [Explainable Machine Learning in Deployment](https://mlsanity.com/paper/1909.06342/) (2019)
- [Counterfactual Explanations and Algorithmic Recourses for Machine Learning: A Review](https://mlsanity.com/paper/2010.10596/) (2020)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [Explainable Artificial Intelligence Applications in Cyber Security: State-of-the-Art in Research](https://mlsanity.com/paper/2208.14937/) (2022)
- [A Survey of Monte Carlo Methods for Parameter Estimation](https://mlsanity.com/paper/2107.11820/) (2021)
- [Evaluation Methods and Measures for Causal Learning Algorithms](https://mlsanity.com/paper/2202.02896/) (2022)
- [Thermodynamics-inspired Explanations of Artificial Intelligence](https://mlsanity.com/paper/2206.13475/) (2022)
- [Exploring Energy Landscapes for Minimal Counterfactual Explanations: Applications in Cybersecurity and Beyond](https://mlsanity.com/paper/2503.18185/) (2025)

## Abstract (arXiv)

Interpretable explanation methods in Artificial Intelligence aim to uncover the underlying causes and their effects, enabling a deeper understanding of why a system behaves in a certain way under different inputs. Unlike traditional explainability methods, which mainly highlight correlations between input and output variables, causal explanation focuses on interventional questions. By doing so, it provides more robust insights, helping users understand automated decisions, especially in high-risk domains. Recovering an explicit directed causal structure, however, is often impractical in large-scale, hybrid cyber-physical systems with feedback loops and partial observability. This paper introduces a novel framework inspired by statistical mechanics that instead models variable dependencies through an undirected, energy-based representation of cyber-physical IoT systems. Our approach enables rigorous dependency-aware attribution by analysing how variations in the energy landscape reflect the influence of individual components, without recovering a directed causal graph. It also supports reasoning about perturbation effects across hybrid interactions, providing reliable explanations of abnormal behaviours. We empirically examined our framework through simulations on an industrial IoT testbed with hybrid continuous and discrete variables, demonstrating higher attribution accuracy, improved robustness and better scalability than state-of-the-art graph-based approaches. While the attributions are not intended to fully recover the system's generative dynamics, they provide valuable, dependency-aware explanations supporting both human interpretation and downstream predictive and diagnostic tasks. Although demonstrated in industrial IoT security, our framework also applies to other high-dimensional cyber-physical and socio-technical systems requiring principled, structural explanations.