Positional Analysis in Fuzzy Social Networks

Social network analysis is a methodology used extensively in social and behavioral sciences, as well as in political science, economics, organization theory, and industrial engineering. Positional analysis of a social network aims to find similarities between actors in the network. One of the the most studied notions in the positional analysis of social networks is regular equivalence. According to Borgatti and Everett, two actors are regularly equivalent if they are equally related to equivalent others. In recent years, fuzzy social networks have also received considerable attention because they can represent both the qualitative relationship and the degrees of interaction between actors. In this paper, we generalize the notion of regular equivalence to fuzzy social networks based on two alternative definitions of regular equivalence. While these two definitions are equivalent for social networks, they induce different generalizations for fuzzy social networks. The first generalization, called regular similarity, is based on the characterization of regular equivalence as an equivalence relation that commutes with the underlying social relations. The regular similarity is then a fuzzy binary relation that specifies the degree of similarity between actors in the social network. The second generalization, called generalized regular equivalence, is based on the definition of role assignment or coloring. A role assignment (resp. coloring) is a mapping from the set of actors to a set of roles (resp. colors). The mapping is regular if actors assigned to the same role have the same roles in their neighborhoods. Consequently, generalized regular equivalence is an equivalence relation that can determine the role partition of the actors in a fuzzy social network.