Symmetric-acyclic decompositions of networks

This paper presents two new developments for partitioning networks, One is the symmetric-acyclic decompostition of a network into clusters of vertices where the vertices in a cluster are linked only by symmetric ties only (with null ties for some pairs of vertices permitted). The induced structure of clusters and ties between clusters is an acyclic graph. A corresponding ideal blockmodel is defined and, given this definition, a generalized blockmodeling method for establishing such decompositions of networks is the second approach introduced here. Both are founded in the Davis and Leinhardt (1972) formulation of a ranked clusters model as a theoretical expectation concerning the structure of human groups and directed affect ties. The decomposition also creates a delineation of the internal structure of identified components but is sensitive to departures from the ideal model. The generalized blockmodeling approach is complementary to the decomposition because it is robust in the presence of such departures and, moreover, identifies them. While initially formulated in a small groups context, the ranked clusters model can be applied to a variety of network phenomena. We illustrate the decomposition and generalized blockmodeling methods with the marriage network of noble families in Ragusa (Dubrovnik) for the 18th Century and early 19th Century.