The role assignment model nearly fits most social networks

Role assignments, introduced by Everett and Borgatti [Mathematical Social Sciences 26 (1991) 183], who called them role colorings, formalize the idea, arising in the theory of social networks, that individuals of the same social role will relate in the same way to individuals playing counterpart roles. If G is a graph, a k-role assignment is a surjective function mapping each vertex into a positive integer 1,2,...,k, so that if x and y have the same role, then the sets of roles assigned to their neighbors are the same. We show that all graphs G having no astronomical discrepancies between the minimum and the maximum degree have a k-role assignment. Furthermore, we introduce and study a natural measure expressing how close an onto map f:V(G)→{1,...,k} is to being a k-role assignment of a graph G=(V,E), and show that almost all graphs nearly have a k-role assignment.