Analyzing Social Networks by Johnson Jeffrey C. & Everett Martin G. & Borgatti Stephen P

Author:Johnson, Jeffrey C. & Everett, Martin G. & Borgatti, Stephen P
Language: eng
Format: epub
Publisher: SAGE Publications
Published: 2018-01-07T16:00:00+00:00

Figure 9.6 Correspondence analysis of triads in Chesapeake Bay ecosystem data.

Table 9.2 shows the raw counts of the number of triads of each type in each season. It is readily noticeable that the proportions of each kind of triad are basically similar across the seasons, but there are differences. For example, winter features quite a few more 003 triads, where no nodes interact, and correspondingly fewer of most other kinds of triads. It is worth pausing to consider what the different triads mean in this context. A transitive triad (030T), represents omnivory – eating at multiple levels in the food chain. That is, species A eats species B, which eats C, but A also eats C, so it is eating at two separate levels of the food chain. A triad containing a mutual dyad, such as 102, reflects a pair of species that eat each other. This is not as rare as it sounds, but is also due to aggregating different species together into a single node.

In order to see the pattern of differences more clearly, we can use correspondence analysis (see Chapter 6). Figure 9.6 shows the results of a correspondence analysis of the triads in Table 9.2, omitting the three rows that have any zeros. It can be seen that seasons trace an arc through the space, starting with spring at the bottom right and moving counterclockwise to winter. This shows that adjacent seasons are particularly similar to each other, as we would expect. Another pattern we see is that on the right-hand side of the plot, corresponding to warmer months, we have triads that begin with 0, meaning they have no mutual dyads. On the left, corresponding to colder months, are triads that have 1s and even 2s as the first number. These are triads in which there are pairs that eat each other. One explanation is that when the weather is warmer, there are more species available and there is no need to resort to reciprocal trophic interactions. In winter, there is a kind of contraction of the ecosystem, with less variety available and more reciprocal interactions.3

9.6 Centralization and core–periphery indices

We began this chapter by alluding to the concept of centralization. Here, we flesh it out a little more. Centralization refers to the extent a network is dominated by a single node. A maximally centralized graph looks like a star: the node at the center of the network has ties to all other nodes, and no other ties exist (see Figure 9.7). A measure of centralization, then, is a measure of the extent to which a network resembles a star.

There are many ways one could think of to construct such a measure, but the one that has become standard is the approach by Freeman (1979). In his approach, we begin by computing a measure of node centrality (see Chapter 10) for each node in the network. For example, we might compute degree centrality, which is simply the number of ties a node has. To calculate centralization, we sum the difference between each node’s centrality and the centrality of the most central node.

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