
5.3: Matrix permutation, blocks, and images


It is also helpful, sometimes, to rearrange the rows and columns of a matrix so that we can see patterns more clearly. Shifting rows and columns (if you want to rearrange the rows, you must rearrange the columns in the same way, or the matrix won't make sense for most operations) is called "permutation" of the matrix.

Our original data look like Figure 5.6:

 Bob Carol Ted Alice Bob --- 1 1 0 Carol 0 --- 1 0 Ted 1 1 --- 1 Alice 0 0 1 ---

Let's rearrange (permute) this so that the two males and the two females are adjacent in the matrix. Matrix permutation (Data>Permute) simply means to change the order of the rows and columns. Since the matrix is symmetric, if I change the position of a row, I must also change the position of the corresponding column. The result is shown in Figure 5.7.

 Bob Ted Carol Alice Bob --- 1 1 0 Ted 1 --- 1 1 Carol 0 1 --- 0 Alice 0 1 0 ---

Figure 5.7. Permuted matrix

None of the elements have had their values changed by this operation or rearranging the rows and columns, we have just shifted things around. We've also highlighted some sections of the matrix. Each colored section is referred to as a block. Blocks are formed by passing dividing lines through the matrix (e.g. between Ted and Carol) rows and columns. Passing these dividing lines through the matrix is called partitioning the matrix. Here we have partitioned by the actor by their sex. Partitioning is also sometimes called "blocking the matrix," because partitioning produces blocks.

This kind of grouping of cells is often done in network analysis to understand how some sets of actors are "embedded" in social roles or in larger entities. Here, for example, we can see that all occupants of the social role "male" choose each other as friends; no females choose each other as friends, and that males are more likely to choose females (3 out of 4 possibilities are selected) than females are to choose males (only 2 out of 4 possible choices). We have grouped the males together to create a "partition" or "super-node" or "social role" or "block." We often partition social network matrices in this way to identify and test ideas about how actors are "embedded" in social roles or other "contexts."

We might wish to dispense with the individual nodes altogether, and examine only the positions or roles. If we calculate the proportion of all ties within a block that are present, we can create a block density matrix. In doing this, we have ignored self-ties in Figure 5.8.

 Male Female Male 1.00 0.75 Female 0.50 0.00

Figure 5.8. Block density matrix

We may wish to summarize the information still further by using block image or image matrix. If the density in a block is greater than some amount (we often use the average density for the whole matrix as a cut-off score, in the current example the density is .58), we enter a "1" in a cell of the blocked matrix, and a "0" otherwise. This kind of simplification is called the "image" of the blocked matrix, as in Figure 5.9.

 Male Female Male 1 1 Female 0 0

Figure 5.9. Image matrix of sex blocked data, using overall mean density as the cut-off

Images of blocked matrices are powerful tools for simplifying the presentation of complex patterns of data. Like any simplifying procedure, good judgment must be used in deciding how to block and what cut-offs to use to create images -- or we may lose important information.

UCINET includes tools that make permuting and blocking matrices rather easy.

Transform>Block allows you to select a matrix to be blocked, a row and/or column partition, and a method for calculating the entries in the resulting blocks.

To use this command, you need to first create separate files that describe the row partition and the column partition. These files are simply vectors (either one row, or one column) that identify which actors are to fall into which partition. For example, if actors 1, 2, and 5 were to form group A, and actors 3 and 4 were to form group B, my column partition data set would read: 1 1 2 2 1. These partitions or blockings are simply regular UCINET data files with one row or one column.

The command asks for a method of summarizing the information within each block. You may take the average of the values in the block (if the data are binary, taking the average is the same thing as calculating the density), sum the values in the block, select the highest value or the lowest value, or select a measure of the amount of variation among the scores in the block -- either the sums of squares or the standard deviation.

The command outputs two new matrices. The "PreImage" data set contains the original scores, but permuted; the "Reduced image dataset" contains a new block matrix containing the block densities.

Transform>Collapse allows you to combine rows and/or columns by specifying (detailed instructions are given on the command window) which elements are to be combined, and how. We might select, for example, to combine columns 1, 2, and 5, and rows 1, 2, and 5 by taking the average of the values (we could also select the maximum, minimum, or sum). The command creates a new matrix that has collapsed the desired rows or columns using the summary operation you selected.

The data menu also gives you some tools for this kind of work:

Data>Permute allows you to re-arrange the rows and/or columns and/or matrices (if your data set contains multiple matrices representing multiple relations, like the Knoke bureaucracies "information" and "money" relations). You simply specify the new order with a list. If I wanted to group rows 1, 2, and 5 to be new rows 1, 2, and 3; and rows 3 and 4 to be new rows 4 and 5, I would enter 1 2 4 5 3.

Data>Sort re-arranges the rows, columns, or both of the matrix according to a criterion you select. If you data are valued (i.e. represent tie strength) you might want to sort the rows and columns in ascending or descending order (this could make sense for binary data, too). If you want a more complicated sort (say "all the 3's first, then all the 1's, then all the 2's) you can use an external UCINET data file to specify this as a vector (i.e. the data set would just be: 3 1 2).

Data>Transpose re-arranges the data in a way that is very commonly used in matrix algebra -- by taking the "transpose." A transpose is, very simply, switching the rows and columns of a matrix for one another.