Structural Similarity in Graphs (A Relaxation Approach for Role Assignment)

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.