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  • Letter
  • Open Access

Nonlinear correlations underlie linear response and causality

Gabriele Di Antonio1,2,3,* and Gianni Valerio Vinci1,†

  • *Contact author: gabriele.diantonio@cref.it
  • †Contact author: gianni.vinci@iss.it
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Phys. Rev. Research 7, L042029 – Published 31 October, 2025

DOI: https://doi.org/10.1103/vjvt-gdn1

Abstract

The inference of causal relationships among observed variables is a pivotal, long-standing problem in the scientific community. An intuitive method for quantifying these causal links involves examining the response of one variable to perturbations in another. The fluctuation-dissipation theorem elegantly connects this response to the correlation functions of the unperturbed system, thereby bridging the concepts of causality and correlation. However, this relationship becomes intricate in nonlinear systems, where knowledge of the invariant measure is required but elusive, especially in high-dimensional spaces. In this study, we establish a link between generalized correlation functions with the response function. This connection provides an alternative method for computing the inference of causal interactions, even when the invariant measure is unknown. We validate our theoretical framework by applying it to several nonlinear systems, showing convergence and consistency for increasing number of measured observables and samples. Finally, we discuss a significant interplay between the resulting causal network and the relevant timescales of the system.

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Phys. Rev. Research 7, L042029– Published 31 October, 2025
  • Received 11 October 2024
  • Accepted 21 July 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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