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Advancing shannon entropy for measuring diversity in systems

Rajaram, Rajeev; Castellani, Brian; Wilson, AN


Rajeev Rajaram

AN Wilson


From economic inequality and species diversity to power laws and the analysis of multiple trends and trajectories, diversity within systems is a major issue for science. Part of the challenge is measuring it. Shannon entropy has been used to rethink diversity within probability distributions, based on the notion of information. However, there are two major limitations to Shannon’s approach. First, it cannot be used to compare diversity distributions that have different levels of scale. Second, it cannot be used to compare parts of diversity distributions to the whole. To address these limitations, we introduce a renormalization of probability distributions based on the notion of case-based entropy as a function of the cumulative probability . Given a probability density , measures the diversity of the distribution up to a cumulative probability of , by computing the length or support of an equivalent uniform distribution that has the same Shannon information as the conditional distribution of up to cumulative probability . We illustrate the utility of our approach by renormalizing and comparing three well-known energy distributions in physics, namely, the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac distributions for energy of subatomic particles. The comparison shows that is a vast improvement over as it provides a scale-free comparison of these diversity distributions and also allows for a comparison between parts of these diversity distributions.


Rajaram, R., Castellani, B., & Wilson, A. (2017). Advancing shannon entropy for measuring diversity in systems. Complexity, 2017, Article 8715605.

Journal Article Type Article
Acceptance Date Apr 23, 2017
Online Publication Date May 24, 2017
Publication Date 2017
Deposit Date Sep 24, 2018
Journal Complexity
Print ISSN 1076-2787
Publisher Hindawi
Peer Reviewed Peer Reviewed
Volume 2017
Article Number 8715605