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Received Jan 31, 2017; Revised Apr 5, 2017; Accepted Apr 23, 2017
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1. Diversity in Systems
Statistical distributions play an important role in any branch of science that studies systems comprised of many similar or identical particles, objects, or actors, whether material or immaterial, human or nonhuman. One of the key features that determines the characteristics and range of potential behaviors of such systems is the degree and distribution of diversity, that is, the extent to which the components of the system occupy states with similar or different features.
As Page outlined in a series of inquiries [1, 2], including The Difference and Diversity and Complexity, diversity within systems is an important concern for science, be it making sense of economic inequality, expanding the trade portfolio of countries, measuring the collapse of species diversity in various ecosystems, or determining the optimal utility/robustness of a network. However, an important major challenge in the literature on diversity and complexity, which Page also points out [1, 2], remains: the issue of measurement. Although statistical distributions that directly reflect the spread of key parameters (such as mass, age, wealth, or energy) provide descriptions of this diversity, it can be difficult to compare the diversity of different distributions or even the same distribution under different conditions, mostly because of differences in scales and parameters. Also, many of the measures currently available compress diversity into a single score or are not intuitive [1–4].
At the outset, motivated by examples of measuring diversity in ecology and evolutionary biology from [3, 4], we sought to address these challenges. We begin with some definitions and a review of our previous research.
First, in terms of definitions, we follow the ecological literature, defining diversity as the interplay of “richness” and “evenness” in a probability distribution. Richness refers to the number of different diversity types in a system. Examples include (a) the different levels of household income in a city, (b) the number of different species in an ecosystem, (c) the diversity of a country’s exports, (d) the distribution of different nodes in a...