A mathematical definition of complex adaptive system as interaction space

TL;DR

This paper introduces a rigorous mathematical framework for complex adaptive systems based on interaction spaces, generalizing Zipf's principle of least effort and Mandelbrot's ideas, and demonstrates that many such systems follow power laws and exhibit emergent patterns.

Contribution

It formalizes a new mathematical definition of complex adaptive systems via interaction spaces, extending existing theories and providing theorems on power laws and emergent patterns.

Findings

Large class of systems satisfy power law behavior.

Theorems describing emergence of patterns in complex systems.

Framework applicable to various complex systems.

Abstract

We define a mathematical notion of complex adaptive system by following the original intuition of G.K. Zipf about the principle of least effort, an intuitive idea which is nowadays informally widespread in complex systems modeling. We call generalized evolution principle this mathematical notion of interaction spaces theory. Formalizing and generalizing Mandelbrot's ideas, we also prove that a large class of these systems satisfy a power law. We finally illustrate the notion of complex adaptive system with theorems describing a Von Th\"unen-like model. The latter can be easily generalized to other complex systems and describes the appearance of emergent patterns. Every notion is introduced both using an intuitive description with lots of examples, and using a modern mathematical language.