I could not find the answer to this question online. In a graph, what is the range of possible values of eigenvector centrality scores?


Here is an example:

x <- sample_gnm(100,50)
plot(x, vertex.label = eigen_centrality(x)$vector)

enter image description here

  • $\begingroup$ I think it's safe to assume that these scores are between 0 and 1. Can you please add more details? Or even a reproducible example? $\endgroup$ – Vincent Guillemot Jan 9 at 10:30
  • $\begingroup$ Hi Vincent, I added an example. Values can go beyond one. $\endgroup$ – NBK Jan 9 at 11:10
  • $\begingroup$ Hello! In your example, the maximum value is equal to 1. Do you have an example where it goes beyond 1? $\endgroup$ – Vincent Guillemot Jan 9 at 11:16
  • $\begingroup$ You're right. Could you make it an answer? $\endgroup$ – NBK Jan 9 at 11:31
  • $\begingroup$ I think the answer goes a little bit beyond what we discussed, I would like to take a little bit of time to write a more complete answer if you don't mind $\endgroup$ – Vincent Guillemot Jan 9 at 12:03

Since the eigenvector centrality is given by the weights of the eigenvector corresponding to the largest eigenvalue of the adjacency matrix (the Perron–Frobenius theorem specifies that the only eigenvector with non-negative weights will correspond to the largest eigenvalue), it is only defined up to a common factor.

A unique solution can be ensured by adding an additional constraint (such as the normalized eigenvector), but it is not necessary. Note that if you're using igraph's eigen_centrality, there is an explicit option to set the scale, where the default normalizes the vector to have norm 1.

Ultimately, the ratio of the centrality scores for a particular network is what really matters, not the raw values themselves.


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