Regular blockmodels of multiway, multimode matrices

Blockmodels are used to collapse redundant elements in a system in order to clarify the patterns of relationships among the elements. Traditional blockmodels define redundancy in terms of structural equivalence. This choice serves many analytic purposes very well, but is inadequate for others. In particular, role systems would be better modeled by blockmodels based on regular equivalence. The first goal of this paper is to generalize blockmodels to incorporate both structural and regular equivalence. Another limitation of traditional blockmodels is that they are defined only for (collections of) 2-way 1-mode adjacency matrices. This excludes common datasets such as actor-by-event, actor-by-organization, item-by-use and case-by-variable matrices. It also excludes 3-way data such as actor-by-actor-by-time or subject-by-verb-by-object matrices. The second goal of this paper is to define blockmodels for multiway, multimode matrices in general. In so doing, we also shift the focus of attention away from the blocking of actors (or other entities) and toward the blocking of ties (or multiway cells).