Logical Analysis of Weighted Social Networks -- An Extended Abstract

Social network analysis is a methodology used extensively in social sciences. While classical social network can only represent the qualitative relationships between actors, weighted social networks can describe the degrees of connection between actors. In classical social network, regular equivalence is used to capture the similarity between actors based on their linking patterns with other actors. Specifically, two actors are regularly equivalent if they are equally related to equivalent others. The definition of regular equivalence have been extended to weighted social networks in two ways. The first definition, called regular similarity, is to consider regular equivalence as an equivalence relation that commutes with the underlying graph edges. The second definition, called generalized regular equivalence, is based on the notion of role assignment. A role assignment is a mapping from the set of actors to a set of roles. The mapping is regular if actors assigned to the same role have the same roles in their neighborhoods. Recently, it was shown that social positions based on regular equivalence can be syntactically expressed as well-formed formulas in a kind of modal logic. Thus, actors occupying the same social position based on regular equivalence will satisfy the same set of modal formulas. In this paper, we will present analogous results for regular similarity and generalized regular equivalence based on many-valued modal logics.