Centrality in Networks, and How It Is Measured
Centrality is a key concept in network studies. As the everyday use of the term implies, it means that a person or organization is in some way a focal point or main figure in whatever group of people or organizations is being considered. Based on studies of small groups and the flow of information in hypothetical networks of different shapes and sizes, some network analysts hypothesize that centrality may be an indicator of power if it is assumed that the person or organization is a "gatekeeper" and/or a gathering point for information, with the information contributing to power because of its importance.
For our purposes, we need not make any assumptions about power or greater amounts of information being tied to centrality in our large-scale organizational networks. For example, the organizations at the center of large real-world networks may be ceremonial or social gathering places. Then, too, there are many sources of information for organizational leaders, including phone calls and stories in a wide range of media outlets. It's likely that peripheral organizations in our networks, with one or two ties to other peripheral organizations, are minor in the scheme of things, but it's not immediately certain that central organizations are powerful.
As three prominent network analysts note, "People refer to central nodes as prominent, or influential, or leaders, or gatekeepers, or as having great autonomy, control, visibility, involvement, prestige, power and so on." But, they add, "It is important to realize that these are not definitions or inherent properties of centrality but rather hypotheses about the potential consequences of centrality, either for the node [the organization in the case of this power elite document ] or for the group in which they are embedded" (Borgatti, Everett, and Johnson 2013, p. 164).
Still, we are nonetheless interested in centrality, if for no other reason than that organizations at the center may have any number of interesting facets. Thus, the six primary business organizations discussed in the main document may or may not have power or be sources of information, but they are at the least places where powerful people, who already have considerable information, meet to discuss common interests or develop policy alternatives. They also may be symbols of prestige within the corporate community. More generally, the role of each central organization needs to be determined through studying the organizations themselves.
Among network analysts, there are varying opinions on how centrality in a network should be measured (Borgatti, Everett, and Johnson 2013, Chapter 10). (For another good textbook overview that is available online, see http://faculty.ucr.edu/~hanneman/nettext/C10_Centrality.html.)
Four frequently used centrality measures indicators include "degree," which is based on the number of direct links the organization has to others in the network; "betweenness," which is based on the number of times the organization is part of the shortest pathway between two other organizations; and "reach," which is based on the number of organizations that an organization is linked to through two steps. (The measure can be used for "three steps" or "four steps," but for our purposes the frequently employed "2step-reach" is the most useful.) The fourth centrality measure, "eigenvector," requires a bit more explaining.
Eigenvector centrality is based on the idea that the centrality of any given organization is determined by the centrality of the various organizations to which it is connected (Bonacich 1972, for information on how eigenvectors are calculated). To make this point clear with an example closer to our life experience, think of two children, Bob and Sam, both of whom have five close friends, more than anyone else on the playground. However, Bob's five friends have no other friends besides Bob, whereas Sam's five friends all have several other friends in addition to Sam. Intuitively, we tend to assume that Sam is more central in the overall network of playground friendships than Bob. Similarly, an organization connected to organizations that have many other connections is assumed to be more central. It is in effect a kind of "friends of friends" analysis.
To decide which measure might work best for this database, Staples determined the size of the correlations among these four measures. Generally speaking, the correlations among them were high as far as correlations go, and two of them were very high, .880 or above. The results of this analysis are shown in Table 1, along with a rough measure of each measure's ranking in the overall matrix, as determined by adding up all of its "correlation points."
After determining that Eigenvector (2499) and Degree (2460) had the highest number of correlational points, and noting that they also correlated .880 with each other, Staples then compared Eigenvector and Degree on the centrality results they provided for the very top of our Fortune 500/business group/think tank database, with a special interest in how they compared for the large business policy-discussion groups. As shown in Table 2, both measures ranked the Business Roundtable, Business Council, and Committee for Economic Development 1-2-3. They also agreed on 11 organizations in the top 15.
Based on the similarity of the results with the Degree and Eigenvector measures, along with the simplicity of determining degree, we decided to use degree for purposes of this document.
Although we decided to focus on degree, two experts in sociology on the use of network analysis to study large corporate databases explained to us that there is a mathematical procedure that provides a correction for the fact that there are sometime large differences among organizations in the number of directors, trustees, or members they have. It involves dividing the number of ties between two groups by the geometric mean of the two board sizes, which is determined by finding the square root of the product of the sizes of Board A and Board B. (For example, the geometric mean for an organization with two members and an organization with eight members is 4, because 2 x 8 = 16, and the square root of 16 is 4.) This calculation has to be done among the organizations in the association matrix, which is then entered into UCINET.
Note: Our thanks to Mark Mizruchi for his many helpful suggestions that greatly improved the clarity and accuracy of this centrality document. Any remaining mistakes are due to our shortcomings, not his.
Bonacich, Phillip. 1972. "Technique for analyzing overlapping memberships." in Sociological Methodology, edited by H. Costner. San Francisco: Jossey-Bass.
Borgatti, Stephen, Martin Everett, and Jeffrey Johnson. 2013. Analyzing social networks. Los Angeles: Sage Publications.
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