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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,

Frank

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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.)

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  • $\begingroup$ Thanks for your comment @BenjaminLind. I have been going to the literature. However, I still find it odd that the Eigenvector centrality measure on graph-level gives higher centralization scores for very disconnected networks. For example: I have highly connected networks with a giant component and very limited number of disconnected pair of nodes. On the other end, I have networks with only disconnected pair of nodes. Comparing these networks, the disconnected networks have higher Eigenvector centrality scores. I don't understand what is going on. Can you explain? $\endgroup$ – wake_wake Nov 6 '14 at 21:15
  • $\begingroup$ You're welcome. I would recommend reading more into the literature on the measurement. See Borgatti and Everett (2006) Social Networks for comparison: sciencedirect.com/science/article/pii/S0378873305000833 and Bonacich (2007) Social Networks for a relatively recent discussion on its properties: sciencedirect.com/science/article/pii/S0378873307000342 Also, keep in mind that evcent() in igraph rescales the maximum value to one by default. $\endgroup$ – BenjaminLind Nov 10 '14 at 12:42

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