Complex systems often comprise many kinds of components which vary over many
orders of magnitude in size: Populations of cities in countries, individual and
corporate wealth in economies, species abundance in ecologies, word frequency
in natural language, and node degree in complex networks. Comparisons of
component size distributions for two complex systems---or a system with itself
at two different time points---generally employ information-theoretic
instruments, such as Jensen-Shannon divergence. We argue that these methods
lack transparency and adjustability, and should not be applied when component
probabilities are non-sensible or are problematic to estimate. Here, we
introduce `allotaxonometry' along with `rank-turbulence divergence', a tunable
instrument for comparing any two (Zipfian) ranked lists of components. We
analytically develop our rank-based divergence in a series of steps, and then
establish a rank-based allotaxonograph which pairs a map-like histogram for
rank-rank pairs with an ordered list of components according to divergence
contribution. We explore the performance of rank-turbulence divergence for a
series of distinct settings including: Language use on Twitter and in books,
species abundance, baby name popularity, market capitalization, performance in
sports, mortality causes, and job titles. We provide a series of supplementary
flipbooks which demonstrate the tunability and storytelling power of rank-based
allotaxonometry.Comment: 23 pages, 7 figures, 1 table; online appendices:
http://compstorylab.org/allotaxonometry