Maximizing diversity in biology and beyond

TL;DR

This paper proves the existence of a universal distribution that maximizes diversity across all viewpoints, with broad applications in ecology, graph theory, and metric geometry, and provides a finite-time method to compute it.

Contribution

It establishes a single distribution that maximizes diversity for all parameters q simultaneously and offers a finite-time algorithm to find it.

Findings

A universal maximizing distribution exists for all diversity viewpoints.

Any distribution maximizing diversity at some q also maximizes it for all q.

The maximum diversity value is uniquely determined and computable in finite time.

Abstract

Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or lesser importance to rare species. Leinster and Cobbold proposed a one-parameter family of diversity measures taking into account both this variation and the varying similarities between species. Because of this latter feature, diversity is not maximized by the uniform distribution on species. So it is natural to ask: which distributions maximize diversity, and what is its maximum value? In principle, both answers depend on q, but our main theorem is that neither does. Thus, there is a single distribution that maximizes diversity from all viewpoints simultaneously, and any list of species has an unambiguous maximum diversity value. Furthermore, the maximizing distribution(s) can be computed in finite time, and any distribution maximizing diversity from some particular viewpoint q > 0 actually maximizes diversity for all q. Although we phrase our results in ecological terms, they apply very widely, with applications in graph theory and metric geometry.