Blockmodeling

Complications in Empirical Networks

Empirical networks are complicated and messy. Therefore, what is needed is a good way of establishing block-models while allowing for departures from “perfect” blockmodels, defined as having actors in positions with identical locations. According to Francois Lorrain and Harrison White, actors are structurally equivalent if they have the same set of ties. To deal with empirical messiness, actors are grouped that are “almost structurally equivalent” by grouping locations that are sufficiently alike into positions. Two basic strategies exist for implementing this approximation: indirect or direct. The indirect strategy has two steps: (1) convert the network into a matrix of (dis)similarities computed for all pairs of locations, and (2) clustering these (dis)similarities. For the first step, Ronald Breiger and colleagues used product-moment correlations, while Ronald Burt used Euclidean distances. In principle, any clustering algorithm can be used for the second step. Three problems arise: (1) using correlations and Euclidean distances can lead to different blockmodels, (2) different clustering algorithms often produce different blockmodels, and (3) there is a lack of good measures of how well these indirectly fitted blockmodels fit the data.

Vladimir Batagelj and colleagues propose a “direct” strategy to partition the network data rather than (dis) similarities. The key step is to answer: what types of blocks are consistent with structural equivalence? In essence, there are only two: null blocks containing only null ties (zeroes), and complete blocks containing only ones. Networks are partitioned directly by identifying blocks close to these ideal block types. Douglas White and Karl Reitz generalized structural equivalence to regular equivalence: loosely, actors are regularly equivalent if they are equivalently connected to equivalent others. [Page 78](As an example, think of a hierarchical organization: actors at one level are regularly equivalent.) Regular equivalence poses a major problem for the indirect approach: to date, there are no good measures of the extent to which pairs of actors are regularly equivalent. However, regular equivalence is handled easily under the direct strategy because the only permitted block types are null blocks and one-covered blocks. A one-covered block has at least one one in every row and every column.

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