Ole C Brudvik's Library tagged methods →  View Popular

An alternative block model is that of "factions."  Factions
are groupings that have high density within the group, and low density of ties
between groups.  Networks>Subgroups>Factions
fits this block model to one-mode data (for any user-specified number of factions). 
Network>2-Mode>2-Mode Factions fits
the same type of model to two-mode data (but for only two factions).

When we apply the core-periphery model to actor-by-event data (Network>2-Mode>Categorical
Core/Periphery) we are seeking the same idealized "image"
of a high and a low density block along the main diagonal.  But, now the
meaning is rather different.

The "core" consists of a partition of actors that are closely
connected to each of the events in an event partition; and simultaneously a
partition of events that are closely connected to the actors in the core
partition.  So, the "core" is a cluster of frequently
co-occurring actors and events.  The "periphery" consists of a partition of actors who are not
co-incident to the same events; and a partition of events that are disjoint
because they have no actors in common.

The core-periphery structure is an ideal typical pattern that divides both
the rows and the columns into two classes.  One of the blocks on the main
diagonal (the core) is a high-density block; the other block on the main
diagonal (the periphery) is a low-density block.  The core-periphery model is indifferent
to the density of ties in the off-diagonal blocks.

When we apply the core-periphery model to actor-by-actor data (see
Network>Core/Periphery), the model seeks to identify a set of actors who have
high density of ties among themselves (the core) by sharing many events in
common, and another set of actors who
have very low density of ties among themselves (the periphery) by having few
events in common.  Actors in
the core are able to coordinate their actions, those in the periphery are
not.  As a consequence, actors in the core are at a structural advantage in
exchange relations with actors in the periphery.

Often all that we know about actors and events is simple co-presence. 
That is, either an actor was, or wasn't present, and our incidence matrix is
binary.  In cases like this, the scaling methods discussed above can be
applied, but one should be very cautious about the results.  This is
because the various dimensional methods operate on similarity/distance matrices,
and measures like correlations (as used in two-mode factor analysis) can be
misleading with binary data.  Even correspondence analysis, which is more
friendly to binary data, can be troublesome when data are sparse.

An alternative approach is block modeling.  Block modeling works
directly on the binary incidence matrix by trying to permute rows and columns to
fit, as closely as possible, idealized images.  This approach doesn't
involve any of the distributional assumptions that are made in scaling analysis.

In principle, one could fit any sort of block model to actor-by-event
incidence data.  We will examine two models that ask meaningful
(alternative) questions about the patterns of linkage between actors and
events.  Both of these models can be directly calculated in UCINET. 
Alternative block models, of course, could be fit to incidence data using more
general block-modeling algorithms.

For binary data, the use of factor analysis and SVD is not recommended. 
Factoring methods operate on the variance/covariance or correlation matrices
among actors and events.  When the connections of actors to events is
measured at the binary level (which is very often the case in network analysis)
correlations may seriously understate covariance and make patterns difficult to discern.

As an alternative for binary actor-by-event scaling, the method of
correspondence analysis (Tools>2-Mode Scaling>Correspondence)
can be used.  Correspondence analysis (rather like Latent Class
Analysis) operates on multi-variate binary cross-tabulations, and its distributional assumptions are better suited to binary data.

Factor analysis provides an alternative method to SVD to the same
goals:  identifying underlying dimensions of the joint space of
actor-by-event variance, and locating or scaling actors and events in that
space.  The method used by factor analysis to identify the dimensions
differs from SVD. 

Singular value decomposition (SVD) is one method of identifying the factors
underlying two-mode (valued) data.  The method of extracting factors
(singular values) differs somewhat from conventional factor and components analysis, so it is a
good idea to examine both SVD and 2-mode factoring results.

The cross-product method takes each entry of the row for actor A, and
multiplies it times the same entry for actor B, and then sums the result. 
Usually, this method is used for binary data because the result is a count of
co-occurrence.  With binary data, each product is 1 only if both actors
were "present" at the event, and the sum across events yields the
number of events in common - a valued measure of strength.

Our example is a little more complicated because we've applied the
cross-product method to valued data.  Here, if neither actor donated to an
initiative (0 * 0 = 0), or if one donated and the other did not (0 * -1 or 0 *
+1 = 0), there is no tie.  If both donated in the same direction (-1 * -1 =
1 or +1 * +1 = 1) there is a positive tie.  If both donated, but in
opposite directions (+1 * -1 = -1) there is a negative tie.  The sum of the
cross-products is a valued count of the preponderance of positive or negative
ties.

The minimums method examines the entries for the two actors at each
event, and selects the minimum value.  For binary data, the result is the
same as the cross-product method (if both, or either actor is zero, the minimum
is zero; only if both are one is the minimum one).  For valued data, the
minimums method is essentially saying:  the tie between the two actors is
equal to the weaker of the ties of the two actors to the event.  This
approach is commonly used when the original data are measured as valued.

Two-mode data offer some very interesting analytic possibilities for gaining
greater understanding of "macro-micro" relations.  In the Davis
data, for example, we can see how the choices of the individual women
"make" the meaning of the parties by choosing to attend or not. 
We can also see how the parties, as macro structures may affect the choices of
the individual women.

With a little creativity, you can begin to see examples of these kinds of
two-mode, or macro-micro social structures everywhere.  The social world is
one of "nesting" in which individuals (and larger structures) are
embedded in larger structures (and larger structures are embedded in still
larger ones).  Indeed, the analysis of the tension between "structure
and agency" or "macro and micro" is one of the core themes in
sociological theory and analysis.

In this chapter we will take a look at some of the tools that have been
applied (and, in some cases, developed) by social network analysts for examining
two-mode data.  We begin with a discussion of data structures, proceed to
visualization, and then turn our attention to techniques for identifying
quantitative and qualitative patterns in two-mode data.

Most of tools of social network analysis deal with structures defined by
patterns in a single kind of relationship among actors: friendship, kinship,
economic exchange, warfare, etc.  Social relations among actors, however,
are usually more complex, in that actors are connected in multiple ways simultaneously.

Laumann and Guttman studied the links among 55 occupational categories. They asked individuals to report their own occupations and those of seven of their close associates. Then they built a 55 by 55, occupation by occupation, matrix and tabulated the frequencies in which a respondent's occupation was paired with each of the others.

They then used multidimensional scaling to map the data matrix into a three-dimensional array. Multidimensional scaling is like factor analysis in that is a procedure for data reduction. The investigator begins by specifying the number of dimensions desired in a solution, say 1 or 2 or 3. Then the original data are entered and the multidimensional scaling program finds the best possible arrangement of points -- the one that come closest to capturing patterning in the original data -- in the specified number of dimensions.

The three-dimensional solution obtained by Laumann and Guttman is shown in Figure 16. They used that image to define the 17 occupational clusters shown. In general, each of their clusters contains occupations that have similar prestige levels. High prestige occupations are displayed at the top of the diagram and low prestige ones at the bottom.

Figure 16. Clusters of Occupations.	Figure 17. Links between Corporations and Corporate Directors.	Figure 18. Individual and Corporate Proximities (Levine, 1979).

The study included a questionnaire that was administered to the heads of households in two communities. One of the questions was "In case of a death in the family, whom would you notify first?" The answers to that question were examined in detail. Proctor counted the "agreements," the number of times two actors chose one another and/or both chose the same third actor). The matrix of agreements was converted into a matrix of correlations and a factor analysis was performed on that matrix.

Proctor used the results of the factor analysis to partition the families into groups. Each of the first seven factors was strongly associated with a different subset of families. Each of these seven factors was used to define a "clique" of families that were linked together. Figure 14 was produced by arranging the members of each "clique" into a circle and placing the seven circles in arbitrary locations in the page.

Figure 14. "Cliques" of Families in Atirro, Costa Rica.	Figure 15. Clusters of School Children (Bock and Husain, 1952).

Figure 9. The "Lady Bountiful."

Figure 10. Apparatus for the Display of the Data of Figure 9.

Figure 11. A Target Sociogram.

Figure 6. A Nautical Food Web.

Figure 7. Visiting Patterns among Households.

Figure 8A. A Map of Russian River Trade Routes.

Figure 8B. A Graph of Russian River Trade Routes.