Partitioning Signed Two-Mode Networks

Structural balance theory forms the foundation for a generalized blockmodel method useful for delineating the structure of signed social one-mode networks for social actors (for example, people or nations). Heider's unit formation relation was dropped. We re-examine structural balance by formulating Heider's unit formation relations as signed two-mode data. Just as generalized blockmodeling has been extended to analyze two-mode unsigned data, we extend it to analyze signed two-mode network data and provide a formalization of the extension. The blockmodel structure for signed two-mode networks has positive and negative blocks, defined in terms of different partitions of rows and columns. These signed blocks can be located anywhere in the block model. We provide a motivating example and then use the new blockmodel type to delineate the voting patterns of the Supreme Court justices for all of their nonunanimous decisions for the 2006–07 term. Interpretations are presented together with a statement of further problems meriting attention for partitioning signed two-mode data.     

An Exact Algorithm for Blockmodeling of Two-Mode Network Data

We consider problems where relationships between two sets (or modes) of objects are available in the form of a binary matrix with elements of 1 (0) indicating a bond (lack of a bond) between corresponding row and column objects. The goal is to establish a partition of the row objects and, simultaneously, a partition of the column objects to form blocks that consist of either exclusively 1s or exclusively 0s to the greatest extent possible. This two-mode blockmodeling problem arises in several scientific domains. In the social sciences, two-mode blockmodeling is particularly relevant for social network analysis, where the goal is to simultaneously partition a set of individuals and another set of objects (e.g., events they attended, organizations they are affiliated with, etc.). The inherent computational challenge of simultaneously constructing partitions for two distinct sets of objects has fostered a reliance on heuristics for two-mode blockmodeling. We offer an exact algorithm and demonstrate its efficacy in a simulation study. Two applications to real-world networks are also provided.