Abstract. Communities in social networks are often defined as groups of densely connected actors. However, members of the same dense group are not equal but may differ largely in their social position or in the role they play. Furthermore, the same positions can be found across the borders of dense communities so that networks contain a significant group structure which does not coincide with the structure of dense groups. This papers gives a survey over formalizations of network-positions with a special emphasis on the use of algebraic notions.

Standard methods for role assignment partition the vertex set of a graph in such a way that vertices in the same class can be considered to have equivalent roles in the graph. Several classes of equivalence relations such as regular equivalence and equitable partitions have been proposed for role assignment, but they all suffer from the strictness of classifying vertices into being either equivalent or not. It is an open problem how to allow for varying degrees of similarity. Proposals include ad-hoc algorithmic approaches and optimization approaches which are computationally hard. In this paper we introduce the concept of structural similarity by relaxation of equitable partitions, thus providing a theoretical foundation for similarity measures which enjoys desirable properties with respect to existence, structure, and tractability.