An entropy based measure for comparing distributions of complexity

•An entropy based measure for comparison of diversity of complexity is proposed.•The measure allows for comparison of diversity both within and across distributions.•The measure is multiplicative i.e., a doubling of value implies a doubling of diversity.

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 ii and their probability of occurrence pipi, with the understanding that larger values of ii mean a higher degree of complexity. To address this problem, we introduce a new complexity measure called case-based entropy  CcCc — a modification of the Shannon–Wiener entropy measure HH. The utility of this measure is that, unlike current complexity measures–which focus on the macroscopic complexity of a single system–CcCc 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.