Classical graph theory represents pairwise relationships using vertices and edges, while hypergraphs extend this model by allowing hyperedges to join any number of vertices, enabling complex multi‐way connections. SuperHyperGraphs further generalize hypergraphs through iterated powerset constructions, capturing hierarchical relationships at multiple layers. Weighted and signed graph models assign numerical weights or positive/negative signs to edges, respectively, and these concepts have been lifted to hypergraphs and, more recently, to SuperHyperGraphs. In this paper, we systematically develop the definitions and core properties of weighted SuperHyperGraphs and signed SuperHyperGraphs. We provide detailed examples to illustrate their structure and discuss potential applications in modeling layered networks with quantitative and polarity annotations. Our results lay a foundation for future theoretical and algorithmic advances in this emerging area.
World Bank macrodata for every country on our planet indicate that national incomes per capita account for a significant portion of population disparity, and these incomes follow well-known distributions documented in the literature across almost all continents. Measuring and comparing disparity is a substantial task that requires assembling the relative nature of both small and large national incomes without distinctions. This is the primary reason we consider the Atkinson inequality index (in the continuous case) in this paper, which was developed towards the end of the 20th century to measure this disparity. Since then, a nonparametric estimator for the Atkinson index has not been developed; instead, a well-known classical discrete form has been utilized. This reliance on the classical form makes the estimation or measurement of economic inequalities relatively straightforward. In this paper, we construct a kernel estimator of the Atkinson inequality index and, by extension, that of its associated welfare function. We then establish their almost sure asymptotic convergence. Finally, we explore the performance of our estimators through a simulation study and draw conclusions about national incomes per capita on each continent, as well as globally, by making comparisons with the classical form based on World Bank staff estimates derived from sources and methods outlined in “The Changing Wealth of Nations”. The results obtained highlight the advantages of kernel-based measures and the sensitivity of the index concerning the aversion parameter.
