Blockmodeling

A More Detailed Description

The term blockmodeling arises from the fact that when a network is represented by a matrix, and this matrix is rearranged according to a partition (so that the units that are part of the same groups are located together), blocks that represent ties within and between groups appear if one separates the groups by lines. These blocks are seen in the second row of the left column in Figure 1, where nine blocks are visible.

More formally, blockmodeling is an approach for clustering units in a network based on some form of equivalence. Equivalence is something that tells which units can be considered equal in some respect. As units typically are not perfectly equivalent to each other, blockmodeling methods usually attempt to cluster together units that are more equivalent to each other than to other units.

Equivalences

The form of equivalence class that is most commonly used is structural equivalence. Units are structurally equivalent if they are connected in the same way to the same units. If a partition is compatible with structural equivalence—that is, if the units within all the groups are perfectly structurally equivalent—then all ties within each block that are induced by the partition into such groups are all equal. With binary networks, this means that either the induced blocks are null (empty, meaning that no ties are present or, equivalently, that all ties have a value of 0) or they are complete (all possible ties are present or, equivalently, all of the ties have a value of 1).

Similarly, units are regularly equivalent if they are connected in the same way to equivalent others. The definition is circular by design. For binary networks, if a partition is compatible with regular equivalence, then all the blocks induced by that partition are either null or regular. A regular block is a block that has at least one tie in each row and each column. Namely, each unit from a row group must be connected to at least one unit from the column group.

Generalized equivalence is defined by block types and possibly their position in the blockmodel. Block type defines the allowed pattern of ties within a block. While describing structural and regular equivalence, these patterns are described earlier for binary networks for null, complete, and regular block types, yet numerous block types are defined for both binary and valued networks. Therefore, generalized equivalence is not a single equivalence class, but a way of specifying custom equivalences.

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