A derivation of a measure of relative balance for social structures and a characterization of extensive ratio systems

Abstract

As an extension of the Heider and Cartwright-Harary theory of balance in social structures (signed digraphs), a measure of relative balance which is derived from some plausible axioms is needed. Such a measure should reflect the balance ordering, i.e., the relation 'one social structure is more balanced than another'. Axioms for such a balance ordering are stated below in terms of the signed digraph representing the social structure as in the Cartwright-Harary theory. The axioms give rise to a measure which determines the balance ordering uniquely, subject to choice of a parameter f(m) which represents the relative importance of a semicycle of length m in the signed digraph. Special values of the parameter f(m) give rise to measures previously suggested as plausible in the literature. Techniques for estimating the values f(m) are described. A second axiomatization for the balance ordering is obtained by translating the requirements on balance into a typical problem of measurement theory in the Suppes-Zinnes sense. The resulting axioms define a so-called extensive ratio system.