Approximate positional analysis of fuzzy social networks

Abstract

In order to facilitate the processing of vast amounts of data that emerges in the study of fuzzy social networks, scholars have developed various procedures for reducing these networks. Some procedures use regular and structural fuzzy relations to reduce such networks. In this article, we generalize the notions of regular and structural fuzzy relations to obtain even better reductions of fuzzy networks. More precisely, for a fuzzy social network given by the set of social entities and the family of fuzzy relations between them, we define μ-approximate regular fuzzy relations, where μ is the degree taken from the underlying set of truth values, which is a complete Heyting algebra. Using these specific fuzzy relations, we show that it is possible to reduce a fuzzy social network in some cases when previously developed algorithms fail to reduce it. We investigate the properties of μ-approximate regular fuzzy relations. We show that the blockmodel of a fuzzy social network, which is its reduced fuzzy social network built via μ-approximate regular fuzzy preorder, retains specific structure-preserving properties. We give a method for calculating the greatest μ-approximate regular fuzzy relation on a given fuzzy social network. For fuzzy social networks defined over the real-unit interval [0,1], we give a procedure that determines all subintervals of [0,1] that share the greatest μ-approximate regular fuzzy relation. Analogous results are provided for μ-approximate structural fuzzy relations.