Ambivalence and indifference in generalizations of structural balance

Despite the demonstrated usefulness of signed digraphs in treating the structural properties of configurations of evaluative relationships, they possess certain inherent limitations. Since each relationship is classified merely as positive or negative, gradations in strength must be ignored, and no means is provided for representing evaluations of ambivalence or indifference. It is proposed that the theory of networks constitutes a more suitable conceptualization since it places no restriction on the values assigned to lines. The use of networks for this purpose, however, requires that an appropriate system of values be formulated. The properties of several possible value systems are examined critically, and two, which we call the diamond model and the triangle model, are judged most promising. Both are based on the assumption that in order to state the value of a relationship it is necessary to specify two components: a positive and a negative one. It is then shown that these two systems are generalizations of the symbolic psycho-logic proposed by Abelson and Rosenberg and of conceptualizations of structural balance stated in terms of signed digraphs. It is found, however, that the cycle, path, and partition criteria for balance, which are equivalent in the absence of ambivalence, are no longer equivalent for these generalized systems of values. In view of this result, three types of balance—cycle balance, path balance, and partition balance—are defined, and the empirical implications of each are examined. Finally, several suggestions are made concerning the construction of quantitative indexes for balanced and unbalanced structures.