A theoretical investigation of regular equivalences for fuzzy graphs
International Journal of Approximate Reasoning
Many-Valued modal logic and regular equivalences in weighted social networks
ECSQARU'13 Proceedings of the 12th European conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty
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Social network analysis is a methodology used exten- sively in social and behavioral sciences, as well as in polit- ical 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 so- cial 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 so- cial networks based on two alternative definitions of regu- lar 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 reg- ular 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 sec- ond 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 Also with the Department of Computer Science and Information En- gineering, National Penghu University, Penghu 880, Taiwan Partially supported by the NSC (Taiwan) Grant 95-2221-E-001-029- MY3. Visiting research fellow of Institute of Information Science, Academia Sinica, Taiwan supported by the NSC (Taiwan) Grant 96-2811-E-001-001 of actors to a set of roles (resp. colors). The mapping is reg- ular 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.