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We extend the direct approach for blockmodeling one-mode data to two-mode data. The key idea in this development is that the rows and columns are partitioned simultaneously but in different ways. Many (but not all) of the generalized block types can be mobilized in blockmodeling twomode network data. These methods were applied to some 'voting' data from the 2000-2001 term of the Supreme Court and to the classic Deep South data on women attending events. The obtained partitions are easy to interpret and compelling. The insight that rows and columns can be partitioned in different ways can be applied also to one-mode data. This is illustrated by a partition of a journal-tojournal citation network where journals are viewed simultaneously as both producers and consumers of scientific knowledge.Blockmodeling tools were developed to partition network actors into clusters, called positions, and, at the same time, to partition the set of ties into blocks that are defined by the positions. (See Lorrain and White (1971), Breiger et al. (1975), andBurt (1976) for the foundational statements.) For these authors, and those using their methods, the foundation for the partitioning was structural equivalence. White and Reitz (1983) generalized structural equivalence to regular equivalence as another principle for blockmodeling networks. For all of these authors, the use of blockmodeling tools was inductive in the sense of specifying an equivalence type and searching for partitions that approximated those equivalence types 1 . The procedures were indirect in the sense of converting network data into a (dis)similarity matrix and using some clustering algorithm. Batagelj et al. (1992a,b) suggested an alternative strategy where the partitioning was done by using the network data directly. In essence, their approach was built upon the recognition that both structural and regular equivalence define certain block types if a partition of actors and ties is exact and consistent with the type of equivalence. For structural equivalence, the ideal blocks are null and complete (Batagelj et al. 1992a), and for regular equivalence, the ideal block types are null and regular (Batagelj et al. 1992b). Subsequently, blockmodeling was generalized to permit many new types of blocks. See Batagelj, 1997 andDoreian et al. (1994). The notion of constructing blockmodeling in terms of a larger set of block types, together with the use of optimization methods mobilized within a direct approach has been called generalized blockmodeling (Doreian et al. (2004). Hitherto, these methods have been applied only to one-mode network data. Here, we consider another extension of blockmodeling by including two-mode network data.

Structural (or social) balance is regarded as a fundamental social process. It has been used to explain how the feelings, attitudes and beliefs, which the social actors have towards each other, promotes the formation of stable (but not necessarily conflict free) social groups. While balance theory has a rich and long history, it has lost favor in recent times. The empirical work has taken one of two forms. Most empirical work on social balance has focused on dyads and triples, and findings have been inconsistent. The remaining studies focus on the structure of the group as a whole. Results here have been inconsistent also. One major problem is that the first line of work is based only on the source ideas of Heider while the second has been based only on the ideas of Cartwright and Harary. Some of the inconsistencies may be due to this empirical split where the two streams of ideas do not inform each other. We propose a new theoretical model for social balance in the form of an agent-based simulation model. The results we present account for several of the inconsistencies found in the literature. The model simulates distinct but interdependent social actors making positive and negative selections of each other in efforts to reach balanced cognitive states. The design variables for the simulations are group size, degree of contentiousness of a group and the mode of communicating choices regarding the existence and sign of social ties. The group level balance mechanism used by the dynamic model is based on the idea of partition balance, as proposed by Doreian and Mrvar [Soc. Netw. 18 (1996) 149]. Actor selections, over time, generate networks that partition group members into stable, balanced subsets at equilibrium or near equilibrium. The design variables have complicated impacts on the number of actor choices made to reach balance, the level of group imbalance, the number of actors with balanced images and the number of plus-sets formed. © 2002 Elsevier Science B.V. All rights reserved. 18 N.P. Hummon, P. Doreian / Social Networks 25 (2003) 17-49

We use balance theoretic ideas to study the dynamics of the international system of nations in a network of signed relations from 1946 through 1999. Using the Correlates of War data for this period, we apply pre-specified signed blockmodeling to characterize the fundamental structure of this network. Even though the system expanded greatly with many ties being created and/or destroyed, the basic structure remained the same but with new positions being added over time. The blockmodels generated temporal measures of imbalance, as did the counts of imbalanced triples. Regardless of using the line index of imbalance or the number of imbalanced 3-cycles, the results provided decisive evidence contradicting the balance theoretic hypothesis of signed networks moving towards balanced states. Structural balance theory remains very useful by pointing to the more important study of how and why signed networks move towards and away from balance at different points over time. Some major methodological problems for studying signed networks, regardless of whether they involve nations or human actors, were raised and addressed. Proposals for future research are suggested for modeling and understanding the dynamics of signed networks.

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