Case-Based Entropy: A New Approach to Measuring Diversity in Communities and Systems

The purpose of this blog entry is to introduce readers to our new approach to measuring diversity in communities and systems, called case-based complexity. 

Rajeev Rajaram and Brian Castellani (and Anna Wilson for studying energy distributions)

Why Diversity?
Diversity in systems is a major topic of concern across the sciences.  From Shannon Entropy in communication systems to the GINI Index to True Diversity measures in evolutionary biology and ecology, the issue of diversity is a major concern.  Here, for example, are some key topics concerned with diversity:
  • Economic Inequality
  • Cultural diversity in society
  • Species diversity in an ecological community 
  • Diversity of information in cybernetic systems 
  • Diversity of major and minor trends in longitudinal data
  • Diversity of health and well-being
  • Trade or political diversity of a country
  • Network diversity
  • Diversity of complexity in systems
  • Power laws
  • Pareto distributions
  • Scale-free networks
  • Energy distributions at the atomic and subatomic level
  • Genetic and biological diversity

Measuring Diversity
For us, case-based entropy is a measure of the diversity within a complex system (be it physical, biological, psychological, social, ecological, etc) that incorporates both species richness (the number of cases in a system) and the evenness of species' abundances (the diversity of types, be they discrete or continuous).

Case-based entropy is based on the work of Lou Jost and Claude Shannon -- see links to Jost below! However, while it deals with probability distributions, it is not simply another measure of probability in systems.  Instead, it has to do with diversity in systems.  To make sense of our approach, we have provided the following below, including three introductory videos as well as links to our recent publications on the topic. 


The goal of the below three videos, Prezi and additional materials it to introduce readers to case-based entropy, a new approach to measuring diversity in communities and systems.




1. CLICK HERE to see the Prezi Presentation that goes with our three videos.
2. CLICK HERE to read or download our publications related to case-based entropy and case-based complexity.

3. CLICK HERE to read our Physica A Paper on the Mathematics of Case-Based Entropy
-->2016    Rajaram, R., and B. Castellani. “An Entropy Based Measure for Comparing Distributions of Complexity.” Physica A: Statistical Mechanics and Its Application.  
 4. CLICK HERE to read our paper in Complexity on Case-Based Complexity and the 60/40 Rule
2016    Castellani, B., and R. Rajaram. “Past the Power Law: Complex Systems and the Limiting Law of Restricted Diversity.” Complexity.

Our impetus for focusing on true diversity comes from evolutionary biology and ecology, where it is employed to measure the true diversity of species (types) in a given ecological system of study.

 Specifically, it comes from the brilliant work of Lou Jost -- see his website.


CECAN and Blogging on Applied Complexity and Policy

I hear the complaint all the time, "Okay, so you folks do all of this complexity stuff, but what is its value?  In other words, how does it help to make the world a better place?  Can you show us, for example, how to apply it?  Can you show us how it helps to improve policy and its evaluation?"

With the latest development of CECAN the answer is "Yes, we can!"

 CECAN stands for Centre for the Evaluation of Complexity Across the Nexus, located in the UK.  As stated on their website, "Nexus issues are complex, with many diverse, interconnected factors involved. This presents a major challenge to policy making because changing one factor can often have unexpected knock-on effects in seemingly unrelated areas. We need new ways to evaluate policy in these situations."

Now CECAN has a blog, which is very good.  Various invited scholars have written some excellent pieces, as well as covering the latest events taking place at or around the goals of CECAN.

For those interested in policy and application, I highly recommend both CECAN and its BLOG.



Bringing the Complexity Sciences to AcademyHealth -- This Year's 2016 Conference in Boston

Washington, D.C. (June 8, 2016)— Today AcademyHealth announced five new Systems Science Scholars who will bring critical research expertise to bear on systems-level challenges affecting the nation’s public health.
“There is great potential for systems science to inform positive changes to the nation’s health care system,” said Kate Papa, director of AcademyHealth’s public and population health program. “This program gives us a unique opportunity to coordinate with other disciplines in health and create solutions for the multifaceted health care and public health issues we face as a nation. Systems science methodologies are well suited to address a range of health issues, and health services research would benefit from integrating systems science methodologies into its toolbox.”      
The Systems Science Scholarship is designed to stimulate systems-level approaches to complex social, behavioral, and environmental influences that contribute to ill health in the U.S. and ultimately contribute innovative solutions in overcoming them. While the field of health services and policy research has contributed a vast evidence base, systems science has not been used to its potential to inform changes to the nation’s health care system.
With support from The Robert Wood Johnson Foundation and a selection committee composed of leaders in both systems science and population health, five scholars were selected from a competitive and diverse pool of applicants.


NEW! Routledge Complexity in social science series

My colleagues, David Byrne (Durham University, UK) and Emma Uprichard (University of Warwick, UK) and I are exited to announce the launch of our new Complexity in Social Science series with Routledge.


And, best of all, we are looking for new manuscripts!!!!

For all inquiries and further information, please contact Emily Briggs, Editor for Social Science Research, Sociology, Criminology and Health Research (Social Science Research)


A New Law for Complex Systems! Past the Power Law: The Limiting Law of Restricted Diversity and the 60/40 Rule

Over the past year we have worked on a series of articles which seek to provide a new way to measure the distribution of diversity in complex systems.  The result is a new way to measure complexity, called case-based entropy.  We published an article on this measure in Physica A.

In turn, we used this new measure to uncover a new limiting law that we found hidden in plain sight in a wide variety of skewed-right complex systems.  We call this limiting law restricted diversity, arguing that this law governs the distribution of diversity in skewed-right complex systems the same way the law of central tendency governs the distribution of diversity in normal distributions.

Even more compelling, restricted diversity can be rigorously measured in ways that far exceed the limits of the power law, revealing a 60/40 rule for skewed-right complex systems.  We just published these second two insights in a article in Complexity.

Below are the abstracts and links to the two papers:

Past the power law: Complex systems and the limiting law of restricted diversity

Probability distributions have proven effective at modeling diversity in complex systems. The two most common are the Gaussian normal and skewed-right. While the mechanics of the former are well-known; the latter less so, given the significant limitations of the power-law. Moving past the power-law, we demonstrate that there exists, hidden-in-full-view, a limiting law governing the diversity of complexity in skewed-right systems; which can be measured using a case-based version C of Shannon entropy, resulting in a 60/40 rule. For our study, given the wide range of approaches to measuring complexity (i.e., descriptive, constructive, etc), we examined eight different systems, which varied significantly in scale and composition (from galaxies to genes). We found that skewed-right complex systems obey the law of restricted diversity; that is, when plotted for a variety of natural and human-made systems, as the diversity of complexity --> infinity (primarily in terms of the number of types; but also, secondarily, in terms of the frequency of cases) a limiting law of restricted diversity emerges, constraining the majority of cases to simpler types. Even more compelling, this limiting law obeys a scale-free 60/40 rule: when measured using C, 60%(or more) of the cases in these systems reside within the first 40% (or less) of the lower bound of equiprobable diversity types—with or without long-tail and whether or not the distribution fits a power-law. Furthermore, as an extension of the Pareto Principle, this lower bound accounts for only a small percentage of the total diversity; that is, while the top 20% of cases constitute a sizable percentage of the total diversity in a system, the bottom 60% are highly constrained. In short, as the central limit theorem governs the diversity of complexity in normal distributions, restricted diversity seems to govern the diversity of complexity in skewed-right distributions.


An entropy based measure for comparing distributions of complexity

This paper is part of a series addressing the empirical/statistical distribution of the diversity of complexity within and amongst complex systems. Here, we consider the problem of measuring the diversity of complexity in a system, given its ordered range of complexity types i and their probability of occurrence pi, with the understanding that larger values of i mean a higher degree of complexity. To address this problem, we introduce a new complexity measure called case-based entropy  Cc — a modification of the Shannon–Wiener entropy measure H. The utility of this measure is that, unlike current complexity measures–which focus on the macroscopic complexity of a single system–Cc can be used to empirically identify and measure the distribution of the diversity of complexity within and across multiple natural and human-made systems, as well as the diversity contribution of complexity of any part of a system, relative to the total range of ordered complexity types.