Partitioning networks based on generalized concepts of equivalence

The idea of partitioning a network in terms of a specific conceptualization of equivalence has taken a powerful hold on the imagination of network analysts. Frequently, an empirically established blockmodel is assessed in terms of its consistency with a particular visualization of a network. We demonstrate that, while a visual representation of a network can be helpful, this also constrains powerfully our image of the structure of that network. This implies that a particular picture of a network is not sufficient for establishing the adequacy of a blockmodel. We argue that once committed to a specific form of equivalence, a network analyst must be committed also to an explicit method of assessing 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 blockmodel in terms of a single form of equivalence reveal a serious weakness in the idea of using only a single form of equivalence to partition a network. It follows that this idea must be reconsidered. An appropriate 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, together with a local optimization procedure, for establishing such a generalized blockmodel. This criterion function also provides an appropriate measure of fit. Finally, we propose partitioning a network into a generalized blockmodel where each block, again in an image, can also have a particular pattern within which each equivalence type is a special case. Again, we provide a method for establishing such a model and assessing its fit.