Generalized blockmodeling of two-mode network data
Introduction
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), Burt (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, Batagelj et al., 1992b 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) and Doreian 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 is called generalized blockmodeling (Doreian et al., 2005). 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.
Section snippets
Two-mode network data
Wasserman and Faust (1994, Chapter 8) provide a discussion of affiliation networks as two-mode data. In essence, two-mode data are defined for two sets of social units and contain measurements of a relation from the units in one set to units in the other set. Pairs of network actor types and relations include: people attending events, organizations employing people, justices on a court rendering decisions, and nations belonging to alliances. The most used example of a two-mode network is the
Approaches to two-mode network data
In a two-mode network, , one set of social units is denoted by and the second set of units is denoted by . By definition, . The social relation is defined as one between the units in these two sets and is represented by the set of lines (which can be arcs or edges) with initial vertices in the set and terminal vertices in the set . The mapping is a weight. Two examples of weighted two-mode networks are (persons
Blockmodels for two-mode network data
Thinking of applying generalized blockmodeling tools to two-mode data implies making some adjustments to this set of techniques, as well as the thinking behind them. Because the data come in the form of rectangular arrays the language of diagonal and off-diagonal blocks is no longer applicable. Blockmodels applied to (the usual) ‘square’ network data, require that the rows and columns are partitioned simultaneously in exactly the same way. It makes no sense, conceptually and technically, to
A formalization of blockmodeling two-mode data
The theoretical background for two-mode blockmodeling comes from Batagelj et al., 1992a, Batagelj et al., 1992b; Doreian et al. (1994); Batagelj et al. (1998): we view the blockmodeling of two-mode data as a simple extension of one-mode blockmodeling.
The main difference is that in blockmodeling of a two-mode network we are trying to identify a two-clustering where C1 is a partition of and C2 is a partition of —such that they induce blocks of the selected types. We
Three empirical examples
We consider three examples. One features some of the some decisions handed down by the US Supreme Court in their 2000–2001 term. The second comes in the form of the Deep South data discussed above and the third is a journal-to-journal citation network.
Thinking of one-mode data as if they were two-mode data
Recognizing that the row and column partitions may be different makes it possible to view one-mode network data in a different fashion by thinking of them as two-mode data. People in an organization can seek advice from each other and form advice networks. Because seeking advice and providing advice are directional activities, the advice networks that they generate will be asymmetric.7
Summary and discussion
While (generalized) blockmodeling has been used primarily as a powerful way of modeling one-mode network data, we have extended (generalized) blockmodeling to examine two-mode data. To do this, it is necessary to think in terms of partitioning the rows and columns separately as a part of the overall partitioning procedure. Two examples, the Supreme Court data and the Southern Women data, were two-mode data sets. For the former we used structural equivalence in an unrestricted fashion and, for
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