## Eigenvector centrality

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I was wondering if you can calculate eigenvector centrality with undirected graphs and if you can, what is the best means of doing so. I understand how to calculate the adjacency matrix and how to calculate its eigenvector (spectral) decomposition, I just am unaware as to how to combine this parts in order to calculate eigenvector centrality. Thanks in advance!

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What's the matter with the "using the adjacency matrix to compute eigenvector centrality" section of the wikipedia article? en.wikipedia.org/wiki/Centrality  Perhaps you are considering a more general notion than this? – Jon Bannon Jul 20 2010 at 14:46
Looking at en.wikipedia.org/wiki/… it seems that it suffices to find an eigenvector of the largest eigenvalue of the adjacency matrix.  Then the eigenvector centrality of the $i$-th vertex is the $i$-th coordinate of such a vector. – Daniel Litt Jul 20 2010 at 14:46

## 1 Answer

The wikipedia article quoted by Jon Bannon mentions using the power-iteration method as readily applicable -- and this is in my experience (for connected graphs, with degrees <5)quite efficient, say starting with the vector with weight 1 for every site. And this wikipedia article mentions several other choices for measuring centrality, besides the "eigenvector centrality". But it does not mention some choices indicated in D. J. Klein, "Centrality Measure in Graphs", J. Math. Chem. 47 (2010) 1209-1223. There centrality measure is suggested to be related to choice of metric or semimetric D on the graph. A couple choices for D yield centrality measures very similar to common measures, and a new "resistive centrality" is noted to result in connection with the choice of D as the "resistance distance" metric.

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