Techniques: Dichotomizing a Network

There are many reasons to dichotomize valued network data. It might be for methodological reasons, for example, in order to use a graph-theoretic concept such as a clique or an n-clan, or to use methods such as ERGMs or SAOMs, which largely assume binary data

There are, of course, many methods that do not require dichotomization. For example, we do not need to dichotomize in order to measure eigenvector centrality, nor to apply the relational event model (Butts, 2008).

Whatever the reason, if we are going dichotomize, the question is at what level should we dichotomize? In some cases, the situation is guided by theoretical meaningfulness and the research design. For example, suppose respondents are asked to rate others on a scale of 1 = do not know them, 2 = acquaintance, 3 = friend, and 4 = family. We see there is a loose gradation from “does not know” to “knows well”; however, categories 3 and 4 do not possess so much degrees of closeness as different kinds of social relations. The choice of which to use is determined by the research question. A similar example is provided by questions that ask for a range of effects from negative to positive. If respondents are asked to rate others on a scale of 1 = dislike a lot, 2 = dislike somewhat, 3 = neither like nor dislike, 4 = like somewhat, and 5 = like a lot, for many analyses, it will make sense to choose a cut off of >3 or >4 for positive ties and <3 or <2 for negative ties. Note that in both of the last examples, we are still confronted with a choice of two values to choose from. In addition, if the scale points are more ambiguous than the ones above, or if the data are counts or rankings, then there is likely no a priori way of deciding where to dichotomize.

Here, we propose a two-step approach to dichotomizing. Step 1 is to simply dichotomize at every level (or a collection of k bins) and examine the network produced at each level. Step 2 is to use simple analytics in order to obtain an informed rationale for a specific dichotomization threshold that makes sense for a given data set.

For step 1, input your valued network into your favorite network data management software and dichotomize at every level of the scale (see insert for information about how to do this in R and in UCINET). We recommend always spending some time visualizing the networks, which can be very informative regarding the emergence of clusters at certain levels of dichotomization. For example, consider Davis et al.’s (1941) women-by-events data (often referred to as the Davis data set or the DGG data). We construct a 1-mode women-by-women network by multiplying the original by its transpose. The result is shown in Table 1.

If we dichotomize at >1 and visualize, we get Figure 1.

Figure 1

If we dichotomize at >2, we get Figure 2.

Figure 2

And if we dichotomize at >3, we get Figure 3.

Figure 3

Thus, the successive dichotomizations reveal a 2-group structure, which is illuminating

However, this should not be taken as definitive. Various normalizations of the data, as well as bipartite representations, tend to show a third smaller subgroup. See Freeman (2003) for a related discussion.

Figure 4

If we dichotomize at >2, we get Figure 5.

Figure 5

Dichotomizing at >4, we get Figure 6.

Figure 6

Dichotomizing at >6, we get Figure 7.

Figure 7

And so on. Core-periphery structures have a kind of self-similarity property where the main component always looks the same regardless of what level of dichotomization produced it.

Now, successive dichotomizations are informative, but our original question was about choosing a single dichotomization that would be used in all further analyses, which is where step 2 becomes important. For step 2, we present three potential approaches. The first will horrify some people. This approach is to choose the level of dichotomization that maximizes your results. For example, suppose you are predicting managers’ performance as a function of betweenness centrality. For each possible level of dichotomization, you measure betweenness centrality and regress performance on betweenness, along with any control variables. The level of dichotomization that yields the highest r² is the one you choose.

As we said, some people (scientists, statisticians, and people of good character) will be horrified

On the other hand, these same people are happy to use regression to find the optimal coefficients to show a relationship between their explanatory variable and a dependent. Perhaps, we should ask them to choose the coefficients a priori on the basis of strong theory.

Clearly, we would choose 5, but how to make sense of these results? They rise and fall with no rhyme or reason. In this case, we would strongly advise against taking this approach. On the other hand, if the results were something like those presented in Table 3, we would be comforted by the underlying regularity and feel good about choosing 5, even though we might be hard-pressed to explain why medium density worked best.

A slightly less controversial version of this approach might be to choose the dichotomized version of your network that maximizes the replication of results from past studies. For example, we know from past studies that actors with higher levels of self-monitoring are more likely to receive more friendship nominations. We could choose the dichotomization threshold that maximizes the relationship between self-monitoring and new friendship nominations, even if the test of our hypothesis has to do with betweenness centrality and performance.

That was the first approach. The second approach is less controversial. Dichotomization, by its very nature, is a distortion of the data

Clearly, in some cases, distorting the data is what we are looking for, for example, when distinguishing between negative and positive ties. In this case, we should not expect the dichotomized data to preserve the properties of the original dataset and we should either use a theoretically or literature driven approach or revert to approach 1.

Interestingly, ≥3 is the level just below the one at which the network splits into two large components (along with four isolates). At ≥4, the network looks like this, as shown in Figure 8

Figure 8

The third approach is theory based, and can be harder to implement. There are certain cases where we can use the emergent properties of the dichotomized networks themselves in order to identify the correct dichotomization threshold, just like when we noticed the appearance of clusters while visually inspecting different dichotomization thresholds in the DGG data. As an example, let us consider an approach proposed by Freeman (2003) to distinguish between weak and strong ties. In his piece on the strength of weak ties, Granovetter (1973) argues that an important characteristic of strong ties is that if A is strongly tied to B, and B is strongly tied to C, then A is likely to be at least weakly tied to C. In his analysis of the DGG data, Freeman (2003) refers to Granovetter’s transitivity rule as g-transitivity. A data set is perfectly g-transitive if there are no violations of g-transitivity. Given a valued data set (and selecting a value such as zero as an indicator of no ties), Freeman’s proposal is to dichotomize the data set at every possible cutoff and calculate the number of violations of g-transitivity at each level. The lowest cutoff with an acceptable number of violations (such as zero) identifies the strong tie. For example, applied to the Davis women data, we get Table 5.

The table shows that at ≥4, the number of g-transitive triples is 160 and the number of intransitive triples is 0. Hence, ties 4 or above are strong ties, and ties <4 but >0 are weak ties.

Combining this with our previous approach, we might summarize the situation as follows. Dichotomizing at ≥3 optimally identifies ties of any kind in terms of the least-violence criterion, and maintains a single large component (plus isolates). Dichotomizing at ≥4 identifies strong ties, which strongly fragment the network. The latter is useful for sharply outlining a subgroup structure, while the former enables the calculation of measure that requires connected networks (aside from isolates) (Figure 9).

Figure 9

It is worth noting that Freeman’s approach needs not be limited to maximizing g-transitivity. On theoretical grounds, we may identify a specific mechanism that organizes ties. For example, we may see a status mechanism such at the Matthew effect in which nodes that already have a lot of ties tend to attract even more ties. Now, to dichotomize valued data, we choose the cutoff that maximizes the extent to which there are just a few nodes with many ties and a great many nodes with few ties. Alternatively, we might choose the cutoff to maximize the level of transitivity in the network.

This “How to” guide on dichotomization is intended to provide guidance on how to find a suitable threshold for dichotomization for social network data. We propose that in all cases, we should start by creating multiple versions of the dichotomized network at every possible value of the threshold and inspect them visually. Then, we suggest three separate approaches in order to choose (and justify your choice of) a single threshold based on (i) maximizing expected results, (ii) minimizing distortions, and (iii) identifying specific emergent properties in the network.