Cumulated social roles: The duality of persons and their algebras

Abstract

The study of social roles from the perspectives of individual actors, and the relation of graph homomorphisms to semigroup homomorphisms, have been the two most prominent topics to emerge from the recent resurgence of progress made on the algebraic analysis of social networks. Through our central construction, the cumulated person hierarchy, we present a framework for elaborating and extending these two lines of research. We focus on each actor in turn as ego, and we articulate what we believe to be the fundamental duality of persons and their algebras. We derive graph and semigroup homomorphisms for three algebras containing 81, 43, and 93 elements, respectively. Throughout, our discussion of theoretical issues is oriented toward an empirical application to the Padgett data set on conspiracy and faction in Renaissance Florence.