Higher order stability conditions in mathematical models of sociometric or cognitive structure

The conditions are found under which the removal of an intransitivity in one triad of a sociometric or cognitive structure may be accomplished without in the process creating new intransitivities elsewhere in the structure. The results are applied to modify and elaborate the theories holding intransitivity to be unstable so that sociometric or cognitive structures become more nearly transitive over time, and also to shed some light on the question of the extent to which observed intransitivities may properly be attributed to measurement error. Properties of tetrads play a central role in this theory, which thus challenges the widespread notion due to Simmel and others, that once group size exceeds three there are no significant new properties to consider. Empirical data document the relevance of this theory: “correction” of “errors” in the data or change in truly intransitive triads turns out not to significantly reduce the overall amount of observed intransitivity.