Equivalence of inequality indices in the three-dimensional model of informetric impact

Inequality is an inherent part of our lives: we see it in the distribution of incomes, talents, citations, to name a few. However, its intensity varies across environments: there are systems where the available resources are relatively evenly distributed but also where a small group of items or agents controls the majority of assets. Numerous indices for quantifying the degree of inequality have been proposed but in general, they work quite differently. We recently observed (Siudem et al., 2020) that many rank-size distributions might be approximated by a time-dependent agent-based model involving a mixture of preferential (rich-get-richer) and accidental (sheer chance) attachment. In this paper, we point out its relationship to an iterative process that generates rank distributions of any length and a predefined level of inequality, as measured by the Gini index. We prove that, under our model, the Gini, Bonferroni, De Vergottini, and Hoover indices are equivalent for samples of similar sizes. Given one of them, we can recreate the value of another measure. Thanks to the obtained formulae, we can also understand how they depend on the sample size. An empirical analysis of a large database of citation records in economics (RePEc) yields a good match with our theoretical derivations.