A Formal Definition of Scale-Dependent Complexity and the Multi-Scale Law of Requisite Variety
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entropy-27-00835.pdf
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1.5 MB
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Author(s)
Siegenfeld, Alexander F.
Bar-Yam, Yaneer
Date Issued
August 6, 2025
Journal
Entropy
Publisher
Multidisciplinary Digital Publishing Institute
Citation
Siegenfeld, A.F.; Bar-Yam, Y. A Formal Definition of Scale-Dependent Complexity and the Multi-Scale Law of Requisite Variety. Entropy 2025, 27, 835.
Version
Final published version
Abstract
Ashby’s law of requisite variety allows a comparison of systems with their environments, providing a necessary (but not sufficient) condition for system efficacy: A system must possess at least as much complexity as any set of environmental behaviors that require distinct responses from the system. However, to account for the dependence of a system’s complexity on the level of detail—or scale—of its description, a multi-scale generalization of Ashby’s law is needed. We define a class of complexity profiles (complexity as a function of scale) that is the first, to our knowledge, to exhibit a multi-scale law of requisite variety. This formalism provides a characterization of multi-scale complexity and generalizes the law of requisite variety’s single constraint on system behaviors to a class of multi-scale constraints. We show that these complexity profiles satisfy a sum rule, which reflects a tradeoff between smaller- and larger-scale degrees of freedom, and we extend our results to subdivided systems and systems with a continuum of components.
MIT Department
Massachusetts Institute of Technology. Department of Physics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.3390/e27080835
