Academic journal article Journal of Social Structure

Eigenvector Centrality: Illustrations Supporting the Utility of Extracting More Than One Eigenvector to Obtain Additional Insights into Networks and Interdependent Structures

Academic journal article Journal of Social Structure

Eigenvector Centrality: Illustrations Supporting the Utility of Extracting More Than One Eigenvector to Obtain Additional Insights into Networks and Interdependent Structures

Article excerpt

Scholars who study social networks often begin with an analysis to determine which actors are the most important, or the most central to a network. For example, in an office environment, to understand a network of colleagues, it would be important to identify the most important players in that office environment. A challenge in social network analysis comes in trying to figure out the best way to understand importance or centrality. It could be that the actors with the largest number of ties are the most important (as reflected by degree centrality), for example the colleague who has meetings with the largest number of other colleagues, yet presumably quantity does not equal quality, and frequent meetings can be a waste of time. On the other hand, it could be that the colleagues who seem to be the link between other colleagues are important or powerful, given that their connections imply a means of access among others in the network (reflected by betweenness centrality). However, neither of these measures would take into account the simple fact that there is more power in being connected to powerful people than there is in being connected to a lot of people with limited access or resources. Eigenvector centrality is a centrality index that calculates the centrality of an actor based not only on their connections, but also based on the centrality of that actor's connections.

Thus, eigenvector centrality can be important, and furthermore, social networks and their study are more popular than ever. Eigenvector centralities have become a staple centrality index, along with degree, closeness, and betweenness (recall: degrees reflect volumes and strengths of ties, closeness captures the extent to which relations traverse few "degrees of separation," and betweenness highlights actors who connect sections of the network; Freeman, 1979). All four centrality indices are included in social network texts (cf., Knoke and Yang, 2007; Scott, 2012; Wasserman and Faust, 1994), and in research articles that compare the performance of centrality indices (cf. Borgatti 2005; Borgatti, Carley, and Krackhardt, 2006; Costenbader and Valente, 2003; Friedkin, 1991; Rothenberg et al., 1995; Smith and Moody, 2013; Stephenson and Zelen, 1989), as well as in the major social network analysis software packages (cf., UCINet, Pajek, NetMiner, NetworkX and LibSNA, NodeXL and SNAP, even Mathematica and StatNet).

When using eigenvector-based centrality, early definitions and current practice are focused on the first eigenvector of the sociomatrix that contains the ties among the actors. The reasoning is sound in that the first eigenvector is associated with the largest eigenvalue, thus capturing the majority of the variance contained in the network. However, there often remains further information about the network structure that subsequent eigenvectors can explain. For example, where the first eigenvector is likely to reflect volumes and strengths of connections among the actors, a second or third eigenvector can delineate those in separate groups within the network who behave in somewhat equivalent manners, or other elements of network structure that can be informative in understanding the actors and the patterns that link them. The research in this paper is conducted to demonstrate that the extraction of only the first eigenvector can be, and in even modest-sized networks typically will be, insufficient for a more comprehensive understanding of the network.

This research is not intended to produce a new centrality measure; rather to evaluate the status of the eigenvector centrality, and suggest that extending it beyond the extraction of only the first eigenvector can be insightful, as illustrated with several examples. To this end, this paper demonstrates that network scholars who consider additional eigenvectors (second, third, and subsequent) will typically be rewarded in obtaining richer insights about additional aspects of network interdependencies. …

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