Discrete Ricci curvature-based statistics for soft sets

Abstract

Soft sets are efficient mathematical structures to model systems in multiple relations. Since a soft set is basically set system, it is possible to endow them with a proper distance function to obtain a metric space. By this embedding, we propose a discretization of the Ricci curvatures that stresses the relational character of universe elements in a soft set through the analysis of parameters rather than the elements themselves. The Forman and Ollivier-type Ricci curvatures we propose here quantifies the trade-off between parameter size and the cardinality of participation of parameterized universe elements in other parameters. Such discretizations of the Ricci curvature have already been applied to complex systems; however, it has not yet been formulated for soft sets. In this study, our main question is whether the defined geometric concept determines statistics for soft set models. Two examples are discussed for the answer to this question. The first example Ricci on soft sets model of occupational accidents occurred in Turkey in 2013-2014 is compared with the Wasserstein distance of the curvature distributions. The second example is the use of Ricci curvatures as an indicator in the soft sets model of a financial system while the system is in stress. These real world examples show that discrete Ricci curvatures for soft sets offer effective statistics.