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 p(i), 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 C-c - 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 - C-c 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. (C) 2016 Elsevier B.V. All rights reserved.