Probability-turbulence divergence: A tunable allotaxonometric instrument for comparing heavy-tailed type-probability distributions

Peter Sheridan Dodds, Joshua R Minot, Michael V Arnold, Thayer Alshaabi, Jane Lydia Adams, Andrew J Reagan and Christopher M Danforth

PLOS Complex Systems, 2026, vol. 3, issue 7, 1-24

Abstract: Real-world complex systems often comprise many distinct types of elements as well as many more types of networked interactions between elements. When the relative abundances of types can be measured well, we often observe heavy-tailed distributions for type probabilities (or relative rates). For the comparison of type-probability distributions of two systems or a system with itself at different points in time—a facet of allotaxonometry—a great range of probability divergences are available.Here, we introduce and explore ‘probability-turbulence divergence’, a tunable, straightforward, and interpretable instrument for comparing normalizable type-probability distributions. We model probability-turbulence divergence (PTD) after rank-turbulence divergence (RTD). While probability-turbulence divergence is more limited in application than rank-turbulence divergence, it is more sensitive to changes in type probability. We show how probability-turbulence divergence either explicitly or functionally generalizes many existing kinds of distances and measures, including, as special cases, Lq norms, the Sørensen-Dice coefficient (the F1 statistic), and the Hellinger distance. We discuss similarities with the generalized entropies of Rényi and Tsallis, and the diversity indices (or Hill numbers) from ecology. We then build allotaxonographs to display probability turbulence, incorporating a way to visually accommodate zero probabilities for ‘exclusive types’ which are types that appear in only one system. Using flipbooks, we show how tuning PTD’s single parameter informs the user how two systems diverge for types that are rare, common, and at all scales in between. We demonstrate that PTD can be tuned to a ‘scale-equalizing’ view that is non-universal and dependent on the systems being compared. We explore comparisons of example distributions taken from literature, social media, and ecology. We close with thoughts on open problems concerning the optimization of the tuning of rank- and probability-turbulence divergence.Author summary: Probability-turbulence divergence (PTD) is an allotaxonometric instrument designed to compare complex systems comprised of many element types which follow heavy-tailed abundance distributions. Allotaxonographs for probability-turbulence divergence provide map-and-list visualizations that: (1) Properly accommodate zero probabilities, and (2) Surface which elements most differentiate the composition of any two systems. Constructed with a single parameter α, probability-turbulence divergence can, in the manner of a physical instrument, be ‘tuned’, foregrounding rare (α=0) or common types (α=∞) at its limits. For comparable complex systems that exhibit type turbulence, tuning to a ‘scale-equalizing’ value of α gives equal weighting to types of all abundances which in turn leads to a meaningful, summarizing ranking of types. Probability-turbulence divergence generalizes and unifies a wide range of existing distance measures.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:plo:pcsy00:0000077

DOI: 10.1371/journal.pcsy.0000077

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