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.