Suppose that I want to influence you by sending you information, or make a deal to exchange some resources. But, in order to talk to you, I must go through an intermediary. For example, let's suppose that I wanted to try to convince the Chancellor of my university to buy me a new computer. According to the rules of our bureaucratic hierarchy, I must forward my request through my department chair, a dean, and an executive vice chancellor. Each one of these people could delay the request, or even prevent my request from getting through. This gives the people who lie "between" me and the Chancellor power with respect to me. To stretch the example just a bit more, suppose that I also have an appointment in the school of business, as well as one in the department of sociology. I might forward my request to the Chancellor by both channels. Having more than one channel makes me less dependent, and, in a sense, more powerful.
For networks with binary relations, Freeman created some measures of the centrality of individual actors based on their betweenness, as well as overall graph centralization. Freeman, Borgatti, and White extended the basic approach to deal with valued relations.
Freeman's Approach to Binary Relations
With binary data, betweenness centrality views an actor as being in a favored position to the extent that the actor falls on the geodesic paths between other pairs of actors in the network. That is, the more people depend on me to make connections with other people, the more power I have. If, however, two actors are connected by more than one geodesic path, and I am not on all of them, I lose some power. Using the computer, it is quite easy to locate the geodesic paths between all pairs of actors, and to count up how frequently each actor falls on each of these pathways. If we add up, for each actor, the proportion of times that they are "between" other actors for the sending of information in the Knoke data, we get a measure of actor centrality. We can norm this measure by expressing it as a percentage of the maximum possible betweenness that an actor could have had. Network>Centrality>Betweenness>Nodes can be used to calculate Freeman's betweenness measures for actors. The results for the Knoke information network are shown in Figure 10.17.
Figure 10.17: Freeman node betweenness for Knoke information network
We can see that there is a lot of variation in actor betweenness (from zero to 17.83), and that there is quite a bit of variation (std. dev. = 6.2 relative to a mean betweenness of 4.8). Despite this, the overall network centralization is relatively low. This makes sense, because we know that fully one half of all connections can be made in this network without the aid of any intermediary - hence there cannot be a lot of "betweenness". In the sense of structural constraint, there is not a lot of "power" in this network. Actors #2, #3, and #5 appear to be relatively a good bit more powerful than others by this measure. Clearly, there is a structural basis for these actors to perceive that they are "different" from others in the population. Indeed, it would not be surprising if these three actors saw themselves as the movers-and-shakers, and the deal-makers that made things happen. In this sense, even though there is not very much betweenness power in the system, it could be important for group formation and stratification.
Another way to think about betweenness is to ask which relations are most central, rather than which actors. Freeman's definition can be easily applied: a relation is between to the extent that it is part of the geodesic between pairs of actors. Using this idea, we can calculate a measure of the extent to which each relation in a binary graph is between. In UCINET, this is done with Network>Centrality>Betweenness>Lines (edges). The results for the Knoke information network are shown in Figure 10.18.
Figure 10.18: Freeman edge betweenness for Knoke information network
A number of the relations (or potential relations) between pairs of actors are not parts of any geodesic paths (e.g. the relation from actor 1 to actor 3). Betweenness is zero if there is no tie, or if a tie that is present is not part of any geodesic path. There are some quite central relations in the graph. For example, the tie from the board of education (actor 3) to the welfare rights organization (actor 6). This particular high value arises because without the tie to actor 3, actor 6 would be largely isolated.
Suppose A has ties to B and C. B has ties to D and E; C has ties to F and G. Actor A will have high betweenness, because it connects two branches of ties, and lies on many geodesic paths. Actors B and C also have betweenness, because they lie between A and their "subordinates". But actors D, E, F, and G have zero betweenness.
One way of identifying hierarchy in a set of relations is to locate the "subordinates". These actors will be ones with no betweenness. If we then remove these actors from the graph, some of the remaining actors won't be between any more - so they are one step up in the hierarchy. We can continue doing this "hierarchical reduction" until we've exhausted the graph; what we're left with is a map of the levels of hierarchy.
Network>Centrality>Betweenness>Hierarchical Reduction is an algorithm that identifies which actors fall at which levels of a hierarchy (if there is one). Since there is very little hierarchy in the Knoke data, we've illustrated this instead with a network of large donors to political campaigns in California, who are "connected" if they contribute to the same campaign. A part of the results is shown in Figure 10.19.
Figure 10.19: Hierarchical reduction by betweenness for California political donors (truncated)
In these data, it turns out that a three-level hierarchy can be identified. The first portion of the output shows a partition (which can be saved as a file, and used as an attribute to color a graph) of the node's level in the hierarchy. The first two nodes, for example, are at the lowest level (1) of the hierarchy, while the third node is at the third level. The second portion of the output has rearranged the nodes to show which actors are included at the lowest betweenness (level one, or everyone); which drop out at level 2 (that is, are the most subordinate; e.g. actors 1, 2, 52); and successive levels. Our data has a hierarchical depth of only three.
The betweenness centrality measure we examined above characterizes actors as having positional advantage, or power, to the extent that they fall on the shortest (geodesic) pathway between other pairs of actors. The idea is that actors who are "between" other actors, and on whom other actors must depend to conduct exchanges, will be able to translate this broker role into power.
Suppose that two actors want to have a relationship, but the geodesic path between them is blocked by a reluctant broker. If there exists another pathway, the two actors are likely to use it, even if it is longer and "less efficient". In general, actors may use all of the pathways connecting them, rather than just geodesic paths. The flow approach to centrality expands the notion of betweenness centrality. It assumes that actors will use all pathways that connect them, proportionally to the length of the pathways. Betweenness is measured by the proportion of the entire flow between two actors (that is, through all of the pathways connecting them) that occurs on paths of which a given actor is a part. For each actor, then, the measure adds up how involved that actor is in all of the flows between all other pairs of actors (the amount of computation with more than a couple actors can be pretty intimidating!). Since the magnitude of this index number would be expected to increase with sheer size of the network and with network density, it is useful to standardize it by calculating the flow betweenness of each actor in ratio to the total flow betweenness that does not involve the actor.
The algorithm Network>Centrality>Flow Betweenness calculates actor and graph flow betweenness centrality measures. Results of applying this to the Knoke information network are shown in Figure 10.20.
Figure 10.20: Flow betweenness centrality for Knoke information network
By this more complete measure of betweenness centrality, actors #2 and #5 are clearly the most important mediators. Actor #3, who was fairly important when we considered only geodesic flows, appears to be rather less important. While the overall picture does not change a great deal, the elaborated definition of betweenness does give us a somewhat different impression of who is most central in this network.
Some actors are clearly more central than others, and the relative variability in flow betweenness of the actors is fairly great (the standard deviation of normed flow betweenness is 8.2 relative to a mean of 9.2, giving a coefficient of relative variation). Despite this relatively high amount of variation, the degree of inequality, or concentration in the distribution of flow betweenness centralities among the actors is fairly low - relative to that of a pure star network (the network centralization index is \(25.6\%\)). This is slightly higher than the index for the betweenness measure that was based only on geodesic distances.