Probability-turbulence divergence: A tunable allotaxonometric instrument for comparing heavy-tailed categorical distributions
Comparing frequency distributions across systems or over time under heavy-tailed regimes poses challenges due to sensitivity to zero-probability events and insufficient discrimination of subtle shifts. Method: We propose Probability Turbulence Divergence (PTD), a tunable, robust, and interpretable normalized divergence measure. PTD unifies classical metrics—including $L^p$ norms, the Sørensen–Dice coefficient, and the Hellinger distance—by integrating rank-based turbulence modeling and zero-probability embedding. It is theoretically linked to Rényi/Tsallis entropies and ecological Hill numbers. Contribution/Results: PTD exhibits zero-probability robustness and fine-grained frequency sensitivity, smoothly degenerating into multiple standard distances. We introduce the allotaxonograph—a novel multi-scale visualization framework—for granular analysis of frequency dynamics. Experiments on bibliometric, social media, and ecological datasets demonstrate PTD’s superior sensitivity to minor frequency perturbations. An open-source implementation enables cross-domain, interpretable comparisons.