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  • Open Access

Predicting Physical Links in Networks

Shuai Li1,2, Wei Chen3,*, and Jan Nagler4,†

  • *Contact author: chwei@bnu.edu.cn
  • †Contact author: jan.nagler@gmail.com
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Phys. Rev. X 16, 021048 – Published 2 June, 2026

DOI: https://doi.org/10.1103/kkwg-kv4c

Abstract

Physical links are fundamental to the structure, function, and ultimately the understanding of many networked systems. Erroneously identified, such direct cause-effect relations undermine the understanding of the system’s dynamics and can impede effective intervention. To address the long-standing challenge of distinguishing direct from indirect links in observational data from complex dynamical networks, we introduce a general inference framework tailored to networked dynamical systems. Our approach combines cross mapping with a novel, physics-informed technique for incremental variable construction that leverages temporal delay effects to identify relevant variables and eliminate indirect links. We validate our method across a diverse set of noisy benchmark and real-world systems, including brain networks, gene regulation, synthetic gene circuits, and biosphere-atmosphere interactions. The method achieves particularly strong gains on mesoscale networks (n≈5–20), consistently outperforming existing approaches under diverse noise conditions. Although designed for small- to medium-sized systems, it continues to outperform competing methods even in larger networks, demonstrating both robustness and broad applicability.

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Popular Summary

Many complex systems, from the communications among neurons in the brain to the tangled feedback between Earth’s atmosphere and biosphere, hide the direct forces that shape their behavior. We developed a physics-informed cross mapping method, Invaxmap, that combines a way to reconstruct a system’s hidden history with a physics-guided process for creating meaningful variables. This approach teases apart genuine interactions from indirect effects. Applied to brain activity, genetic networks, and climate studies, our approach reliably reveals which parts of the system truly drive changes. By identifying the actual drivers behind complex dynamics, we open the door to more precise control—whether guiding neural therapies, manipulating gene circuits, or forecasting environmental shifts. Our methodology is a powerful tool useful for understanding and steering intricate complex systems.

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Outline

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Phys. Rev. X 16, 021048– Published 2 June, 2026
  • Received 7 August 2025
  • Revised 16 March 2026
  • Accepted 8 May 2026
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