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.
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?
----- Edit 1 ---- 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.