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Just to add to Mason's correct answer and to be clear.

If the eigenvector is negative just ignore the sign.

But Ion for some reason thought the centre would have a smaller centrality score than the peripheral nodes. This is not right. Recall that eigenvector centrality means that anyone's score is proportional to the scores of the actors they are connected to. Hence in this example the peripheral scores will be proportional to the central score and hence will all be equal. But the central score will be proportional to the sum of all the peripheral scores and so will be larger than the peripheral scores.

Martin





________________________________________
From: Social Networks Discussion Forum [[log in to unmask]] on behalf of Mason Alexander Porter [[log in to unmask]]
Sent: 14 January 2015 05:21
To: [log in to unmask]
Subject: [SOCNET] Star network and eigenvector centrality

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Ion,

An eigenvector is only defined up to a constant, so if I have an equation
Av = rv (where A is a matrix, r is an eigenvalue, and v is its associated
eigenvector), then -v solves this equation if and only if v does.

The components of the eigenvector give you the eigenvector centralities of
the nodes.

To see what's going on with your example, note that the network is small
enough to do the calculation by hand.  That will be the easiest way to
see how it works. You can construct the adjacency matrix of the star and
calculate the eigenvector centrality vector directly from it.

Also, in addition to the book below that you already own, another source
is this listserv's archives. (The question about getting negative values
for eigenvector centrality has come up before.)

-----
Mason


>
> I am using ORA Software (see McCulloh, Armstrong and Johnson's book =
> "Social
> Network Analysis With Applications", or
> http://www.casos.cs.cmu.edu/projects/ora/index.php).=20
>
> =20
>
> I input a 6-node, star network.=20
>
> I ask for eigenvector centrality scores for all vertices.
>
> Given the general description of what eigenvector centrality does, I =
> expect
> the peripheral vertices to have high scores, and the central vertex to =
> have
> a lower score.
>
> =20
>
> The results are as follows:
>
> Central vertex: -1.0=20
>
> All other vertices: -0.4472136
>
> =20
>
> My questions:
>
> Why are the results negative?
>
> Why does the central vertex have a higher absolute value (should the
> absolute values even be considered)?
>
> Are the results at all interpretable in terms of eigenvector centrality;
> and, if so, how?
>
> =20
>
> I am grateful for any assistance in resolving this conundrum.
>
> =20
>
> Ion Georgiou
>
> =20
>

----
Mason

--------------------------------

  Mason A. Porter
  Professor of Nonlinear and Complex Systems
  Oxford Centre for Industrial and Applied Mathematics
  Mathematical Institute, University of Oxford

  Homepage: http://people.maths.ox.ac.uk/porterm
  Blog: http://masonporter.blogspot.com
  Twitter: @masonporter
  Skype: tepid451
  ----------------------------------------------------------------------------
  "I will be the lion." (Me)
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