PARTITIONING NETWORKS BASED ON GENERALIZED CONCEPTS OF EQUIVALENCE

The idea of partitioning a network in terms of a specific conceptualiz ation of equivalence has taken a powerful hold on the imagination of n etwork analysts. Frequently, an empirically established blockmodel is assessed in terms of its consistency with a particular visualization o f a network. We demonstrate that, while a visual representation of a n etwork can be helpful, this also constrains powerfully our image of th e structure of that network. This implies that a particular picture of a network is not sufficient for establishing the adequacy of a blockm odel. We argue that once committed to a specific form of equivalence, a network analyst must be committed also to an explicit method of asse ssing the extent to which a blockmodel is consistent with the selected form of equivalence. We provide a method for doing this. Additionally , and perhaps more importantly, efforts to measure the fit of a blockm odel in terms of a single form of equivalence reveal a serious weaknes s in the idea of using only a single form of equivalence to partition a network. It follows that this idea must be reconsidered. An appropri ate generalization of the equivalence idea is one where each block, of a particular image in a blockmodel, is free to conform to a different form of equivalence. We provide a general criterion function, togethe r with a local optimization procedure, for establishing such a general ized blockmodel. This criterion function also provides an appropriate measure of fit. Finally, we propose partitioning a network into a gene ralized blockmodel where each block, again in an image, can also have a particular pattern within which each equivalence type is a special c ase. Again, we provide a method for establishing such a model and asse ssing its fit.