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Real-world complex systems often comprise many distinct types of elements aswell as many more types of networked interactions between elements. When therelative abundances of types can be measured well, we further observeheavy-tailed categorical distributions for type frequencies. For the comparisonof type frequency distributions of two systems or a system with itself atdifferent time points in time -- a facet of allotaxonometry -- a great range ofprobability divergences are available. Here, we introduce and explore`probability-turbulence divergence', a tunable, straightforward, andinterpretable instrument for comparing normalizable categorical frequencydistributions. We model probability-turbulence divergence (PTD) afterrank-turbulence divergence (RTD). While probability-turbulence divergence ismore limited in application than rank-turbulence divergence, it is moresensitive to changes in type frequency. We build allotaxonographs to displayprobability turbulence, incorporating a way to visually accommodate zeroprobabilities for `exclusive types' which are types that appear in only onesystem. We explore comparisons of example distributions taken from literature,social media, and ecology. We show how probability-turbulence divergence eitherexplicitly or functionally generalizes many existing kinds of distances andmeasures, including, as special cases, $L^{(p)}$ norms, the S{\o}rensen-Dicecoefficient (the $F_1$ statistic), and the Hellinger distance. We discusssimilarities with the generalized entropies of R{\'e}nyi and Tsallis, and thediversity indices (or Hill numbers) from ecology. We close with thoughts onopen problems concerning the optimization of the tuning of rank- andprobability-turbulence divergence.

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