Models of core/periphery structures
Introduction
A common image in social network analysis and other fields is that of the core/periphery structure. The notion is quite prevalent in such diverse fields of inquiry as world systems (Snyder and Kick, 1979; Nemeth and Smith, 1985; Smith and White, 1992), economics (Krugman, 1996) and organization studies (Faulkner, 1987). In the context of social networks, it occurs in studies of national elites and collective action (Laumann and Pappi, 1976; Alba and Moore, 1978), interlocking directorates (Mintz and Schwartz, 1981), scientific citation networks (Mullins et al., 1977; Doreian, 1985), and proximity among Japanese monkeys (Corradino, 1990).
Given its wide currency, it comes as a bit of a surprise that the notion of a core/periphery structure has never been formally defined. The lack of definition means that different authors can use the term in wildly different ways, making it difficult to compare otherwise comparable studies. Furthermore, a formal definition provides the basis for statistical methods of testing whether a given dataset has a hypothesized core/periphery structure, and for computational methods of discovering core/periphery structures in data. Without such a definition, we cannot proceed with developing these kinds of tools.
In this paper, we develop two families of core/periphery models, based on intuitive conceptions of the structure. Any formalization of an intuitive concept needs to identify, in a precise way, the essential features of a particular concept. This part of the process involves a certain degree of conceptual clarification and interpretation that can (and many would argue should) be challenged by others. In view of this, we see this paper as a starting point in a methodological debate on what constitutes a core/periphery structure.
Section snippets
Intuitive conceptions
One intuitive view of the core/periphery structure is the idea of a group or network that cannot be subdivided into exclusive cohesive subgroups or factions, although some actors may be much better connected than others. The network, to put it another way, consists of just one group to which all actors belong to a greater or lesser extent. This is the sense in which Pattison (1993, p. 97) uses the term. This conception is rooted in the cohesive subsets literature (for a review, see Scott, 1991,
Discrete model
In this section we explore the idea that the core periphery model consists of two classes of nodes, namely a cohesive subgraph (the core) in which actors are connected to each other in some maximal sense and a class of actors that are more loosely connected to the cohesive subgraph but lack any maximal cohesion with the core.
Consider the graph in Fig. 1, which intuitively seems to have a core/periphery structure. The adjacency matrix for the graph is given in Table 1.
The matrix has been blocked
Continuous model
One limitation of the partition-based approach presented above is the excessive simplicity of defining just two classes of nodes: core and periphery. To remedy this, we could introduce a three-class partition consisting of core, semiperiphery, and periphery, as world system theorists have done, or try partitions with even more classes. This approach is feasible, but specifying the ideal blockmodel that best captures the notion of a core/periphery structure is relatively difficult, as there are
Conclusion
This paper sets forth a set of ideal images of core/periphery structures, then develops measures of the extent to which real networks approximate these images. These measures are used as the basis for tests of a priori hypotheses and for optimization algorithms to detect core/periphery structures.
What is missing in this paper is a statistical test for the significance of the core/periphery structures found by the algorithms. We know how well the models fit, but we do not know how easy it is to
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