Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings

The growth dynamics of complex systems often exhibit statistical regularities involving power-law relationships. For real finite complex systems formed by countable tokens (animals, words) as instances of distinct types (species, dictionary entries), an inverse power-law scaling <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>S</a:mi> <a:mo>∼</a:mo> <a:msup> <a:mi>r</a:mi> <a:mrow> <a:mo>−</a:mo> <a:mi>α</a:mi> </a:mrow> </a:msup> </a:mrow> </a:math> between type count <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>S</b:mi> </b:math> and type rank <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>r</c:mi> </c:math> , widely known as Zipf’s law, is widely observed to varying degrees of fidelity. A secondary, summary relationship is Heaps’ law, which states that the number of types scales sublinearly with the total number of observed tokens present in a growing system. Here, we propose an idealized model of a growing system that (1) deterministically produces arbitrary inverse power-law count rankings for types and (2) allows us to determine the exact asymptotics of the type-token relationship. Our argument improves upon and remedies earlier work. We obtain a unified asymptotic expression for all values of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>α</d:mi> </d:math> , which corrects the special cases of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>α</e:mi> <e:mo>=</e:mo> <e:mn>1</e:mn> </e:mrow> </e:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>α</f:mi> <f:mo>≫</f:mo> <f:mn>1</f:mn> </f:mrow> </f:math> . Our approach relies solely on the form of count rankings, avoids unnecessary approximations, and does not involve any stochastic mechanisms or sampling processes. We thereby demonstrate that a general type-token relationship arises solely as a consequence of Zipf’s law.