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Partitioning Signed Networks

Structural balance theory has proven useful for specifying blockmodel types that permit the delineation of the optimal blockmodel structure for signed social networks. Even so, most of the observed signed networks are not perfectly balanced. One possibility is that in examining the dynamics underlying the generation of signed social networks, insufficient attention has been given to other processes that might be operative. In particular, limited attention has been given to actors who have positive ties to pairs of other actors linked by a negative relation or who belong to two mutually hostile subgroups. Rather than view these situations as violations of structural balance, they can be seen as belonging to another relevant process called mediation. Formalizing this idea leads to a relaxed structural balance blockmodel as a proper generalization of structural balance blockmodels. Some formal properties concerning the relation between these two models are presented along with the properties of the fitting method proposed for the new blockmodel type. The new method is applied to three known empirical data sets where improved fits and interpretations are obtained. The idea of relaxed structural balance is then extended to signed two-mode data. Just as generalized blockmodeling has been extended to analyze two- mode unsigned data, it is straightforward to extend it to analyze signed two-mode network data and a formalization of the extension is provided. A motivating example of actors and beliefs, as two-mode signed networks, is used before using this new blockmodel type to delineate the structure of the voting patterns for the Supreme Court justices for all of their non-unanimous decisions for the 2006-7 term. Some interpretations are presented together with a statement of further problems meriting attention for partitioning signed two-mode data that include a proposal to abandon the idea of a single blockmodel applying to all of a network structure. Generalized blockmodeling is based on a model specification of an anticipated structure reflecting the operation of network processes and not on a statistical model. Combining the two types of models and generating a new fitting method seems a fruitful avenue to explore.