作为交互空间的复杂自适应系统的数学定义
A mathematical definition of complex adaptive system as interaction space
AI总结：
本文基于 Zipf 最小努力原则提出复杂自适应系统的数学定义，形式化并推广 Mandelbrot 思想证明其满足幂律，并通过 Von Thünen 类模型说明涌现模式。
摘要：
我们遵循 G.K. Zipf 关于最小努力原则的原始直觉，定义了一个复杂自适应系统的数学概念，这一直觉思想如今在复杂系统建模中被非正式地广泛使用。我们将这一交互空间理论的数学概念称为广义演化原理。通过对 Mandelbrot 的思想进行形式化和推广，我们还证明了一大类此类系统满足幂律。最后，我们通过描述一个类似 Von Thünen 模型的定理来阐释复杂自适应系统的概念。后者可以容易地推广到其他复杂系统，并描述了涌现模式的出现。每一个概念都同时使用带有大量示例的直观描述和现代数学语言来引入。
英文摘要：
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 Thünen-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.
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