Network macrostructure models for the Davis-Leinhardt set of empirical sociomatrices

Abstract

Davis, Holland and Leinhardt have studied various microlevel and macrolevel models for group social structure in terms of certain two-valued relations on the group. These models are essentially defined in terms of permitted and forbidden triad types at the microlevel which imply and are implied by specific ordered clique structures at the macrolevel. Upon testing these models against a large collection of empirical sociomatrices, it was found that these models fit the data moderately well; however, not perfectly, since one or two triads forbidden by the models occur in over half of the sociomatrices at frequencies greater than or equal to chance expectation. These discrepancies have encouraged further theorizing and elaboration of the models, but there has heretofore been no macromodel put forth which fits the total data exactly. In this paper we obtain such a model, which allows hierarchy within cliques. In addition, we give an exact data-fitting macromodel for each class of group sizes as well, except for a somewhat less precise description for the class of largest sizes. In each case the general macromodel exhibits all of the permitted triads and none of the forbidden ones. Similar data from a study of Hallinan not only support the macromodel for the total data when the forbidden triads are taken individually but also when they are taken as a set, thus showing that the data actually support this particular macromodel and not another.