A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety
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.
Keywords
Cite
@article{arxiv.2206.04896,
title = {A Formal Definition of Scale-dependent Complexity and the Multi-scale Law of Requisite Variety},
author = {Alexander F. Siegenfeld and Yaneer Bar-Yam},
journal= {arXiv preprint arXiv:2206.04896},
year = {2025}
}