Modeling Rank Distribution and the Relative Importance Factor Index in Discrete Power-Law Models: Application to Social Resilience Using the Scopus Database

Prior research on power-law distributions has primarily focused on modeling frequency patterns, with less attention given to rank distributions and how ranked positions reflect relative importance among elements. In discrete power-law distributions, frequency-based metrics often provide limited discrimination in the tail, where elements may exhibit similar counts but differ in relative dominance. These patterns are especially evident, for instance, in academic publishing, where keywords, affiliations, and citations commonly exhibit power-law behavior. To address this limitation, we introduce the Relative Importance Factor (RIF) Index, a statistical measure derived from the estimated discrete power-law rank distribution rather than an additional independent parameter. The RIF Index compares the probability of an element at a given rank with its probabilities at lower ranks, enabling explicit pairwise statistical comparison, particularly within the tail. We formalize the mathematical framework for discrete rank modeling and apply RIF to synthetic data and a Scopus dataset on social resilience. Our results show that RIF clarifies dominance relationships among ranked elements, providing stronger discrimination in the tail than frequency-based measures alone. We further introduce the RIF matrix and RIF network to represent these pairwise relationships structurally, supporting interpretation of prominence patterns. Although demonstrated in academic publishing, the method generalizes to domains where categorical variables follow discrete power-law behavior under appropriate model-fit validation.