Maximizing diversity in biology and beyond
By Tom Leinster, Mark W. Meckes. 2015
48 citations recorded by the source corpus.
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.
Subfields: Environmental Engineering, Nature and Landscape Conservation, Paleontology, Statistics and Probability, Computational Biology & Bioinformatics
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Citations in the corpus
- GAIT: A Geometric Approach to Information Theory (2019, background): “Moreover, the exponential of the maximum entropy, exp [ supθ H Kr 1 [pθ] ] = 1 + tanh(r) ∈ [1, 2], measures the effective number of points (Leinster and Meckes, 2016) at scale r.”
- The magnitude of a graph (2014, background): “The magnitude of metric spaces is also closely related to certain measures of entropy and of biological diversity [12], and admits a further interpretation in terms of potential theory [16].”