In R: How do iGraph and STATNET handle disconnected graphs in measuring network centralization

Question

I am working with about 300 disconnected networks 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!

Answer 1

Eigenvector centrality is not well-defined for disconnected graphs since the centrality scores of the individual components are independent of each other; one can inflate the centrality scores of one component by multiplying the with a large constant (say, 10000), then normalize the centrality scores again to sum up to 1; the resulting vector would still satisfy the eigenvector centrality equation. Therefore, you should calculate eigenvector centralities for connected graphs only. If you have multiple components in your graph, break it down to connected components first and then calculate and compare eigenvector centralities for the individual components only.