Usefulness of eigenvalue centrality Question for the network theorists/computer scientists: I heard that the eigenvalue centrality is "useless" and "too sensitive to jump behavior" for directed graphs. 
$\bullet$ What does this mean exactly? 
$\bullet$ Can one say for which graphs eigenvalue centrality work or doesn't work, and in what sense precisely?
$\bullet$ What are then the alternatives that avoid these problems?
Question is necessarily vague due to my ignorance of network theory but I am getting the impression that this is part of network theory folklore. Any reference is appreciated.
 A: Centrality measures have very different uses. You cannot really generally say that one is "better" than another. This is always context dependent.
Eigenvector centrality, in general, is certainly not a useless concept, it is the most simple "reputation based" centrality metric, in the sense that you earn reputation/centrality, if you are connected to other high centrality nodes. As such it is the basis for PageRank, being an eigenvector centrality of a modified adjacency matrix (basically allowing a random walker to jump).
For general information and improvement of "literacy" on network centrality measures (i.e. for what situations is what metric suitable) see e.g.
[1] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.131.8175&rep=rep1&type=pdf
[2] Mark Newman, Networks: an introduction
[3] http://www.analytictech.com/borgatti/papers/centflow.pdf
On EVC and PageRank specifically (sorry, seems like I cannot post more than two links):
[4] Wikipedia: PageRank
[5] Page, L. and Brin, S., The PageRank citation ranking: bringing order to the web, 1999
[6] ... google scholar has more ...
Where did you read it is useless, and in which context?
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I have no idea what exactly was meant by "too sensitive to jump behavior", without details about the context. Was it some CS seminar, talking about importance ranking of webpages? I mean, the only thing I can imagine right now that could have been meant (in a strange way), is that PageRank is often used instead of EVC, as it allows a random walker to "jump", i.e. to visit a not-neighboring node. In a web page ranking scenario this is meant to account for random encounter of new webpages, hence avoiding a random walker to get stuck in some dense cluster. If this was meant, the Brin Page paper is a good explanation on why PageRank performs better than EVC, in the webpage ranking scenario. Otherwise see Newman book, for a more pedagogical intro.
