I am working with about 300 disconnected of different sizes. I calculate different graph-level centralization measures for these networks using the STATNET and iGraph packages in R.
However, I find that the nodes in subgraphs of N=2 get assigned the highest value of 1 for the Eigenvector centrality measure with iGraph. As a result, networks with a lot of isolated dyads get very high graph-level Eigenvector centralization scores.
In my networks this is not a valid result, because these networks are poorly connected and thus should, theoretically, have a low centralization score.
Does anyone know how these measures handle disconnected graphs? And are there ways to deal with this? Also, are there other ways to assess the structure of these networks?
Any help is welcome. Thank you!
Best Answer
Of the four big centrality measurements (degree, nodal betweenness, eigenvector, and closeness), closeness is the only one that is not well defined for disconnected networks. From the igraph documentation on closeness, it states, "if there is no (directed) path between vertex 'v' and 'i' then the total number of vertices is used in the formula instead of the path length."
Aside from questions that may arise regarding the absence of paths when measuring closeness, the traditional centrality measures are all perfectly valid for disconnected networks. They're commonly used in the literature to describe nodes in disconnected networks.
Bear in mind that centralization is about inequality among nodal centrality measurements, typically against the most centralized network possible. For degree, betweenness, and closeness centralization, the most centralized network possible is a connected star. Eigenvector centralization differs in that the most centralized network possible is a network with just one edge (i.e., a network with n-2 isolates).
If you want to control for these effects, compare your observed networks to random graphs. (For starters, see BS Anderson, C Butts, K Carley [1999] "The interaction of size and density with graph-level indices," Social Networks.)