Abstract
We propose a novel formalization of roles in social networks that unifies the most commonly used definitions of role equivalence. As one consequence, we obtain a single, straightforward proof that role equivalences form lattices. Our formalization focuses on the evolution of roles from arbitrary initial conditions and thereby generalizes notions of relative and iterated roles that have been suggested previously. In addition to the unified structure result this provides a micro-foundation for the emergence of roles. Considering the genesis of roles may explain, and help overcome, the problem that social networks rarely exhibit interesting role equivalences of the traditional kind. Finally, we hint at ways to further generalize the role concept to multivariate networks.