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You are wrong.

Eigenvector centrality finds the dominant eigenvector of the matrix you
prescribe as data. If this has values it takes them and uses them.

On 6 Apr 2005 at 21:29, Scott White wrote:

> Also I believe eigenvector centrality is computed on the adjacency
> matrix and not on the transition matrix so that edges are *not* in
> fact weighted. Someone correct me if I'm wrong.
>
> Scott
>
> > *****  To join INSNA, visit http://www.insna.org  *****
> >
> > I feel I must correct what Joshua has said.
> >
> > 1 Both UCINET and MATLAB produce the correct results. They do not
> > test for singularity (as this merely says that zero is an
> > eigenvalue) but use sophisticated (particularly MATLAB) specialist
> > routines.
> >
> > 2 The method Joshua describes is the power method and is a well
> > known technique for finding eigenvectors in special cases. One of
> > those cases is if the graph is connected. If the graph is not
> > connected the method may fail, that is it will not find an
> > eigenvector. Usually it simply does not converge and hence you may
> > get some values but these cannot be interpreted as eigenvectors,
> > they are not.
> >
> > 3 One possible way to use eigenvector centrality on disconnected
> > graphs is to take the eigenvectors corresponding to the smaller
> > eigen values. UCINET reports the dominant eigenvector ie the
> > eigenvector corresponding to the largest eigenvalue. This will have
> > positive values for the largest component and zero for all other
> > components. The other eigenvalues will have eigenvectors that have a
> > zero for all elements of the largest component but have non zero
> > values for one of the others. You could compare the centralities
> > within each component, I guess but care would be needed in
> > interpretation. To do this in UCINET you need to run the svd command
> > in the matrix algebra section. It is easy to do using the eigenvalue
> > command in MATLAB.
> >
> > Personally I would not recommend eigenvector centrality for graphs
> > with multiple components.
> >
> >
> >
> >
> >
> > > *****  To join INSNA, visit http://www.insna.org  *****
> > >
> > > Skye:
> > >
> > > I believe that what's going on here is that eigenvector centrality
> > > is defined in a way that the closed-form calculation of it
> > > requires calculation of the inverse of the network's (weighted)
> > > matrix.  I would guess that UCINET and matlab are both calculating
> > > EVC in this way. Since I'm pretty sure that the matrix of such a
> > > network is singular (i.e., it has no inverse), I'd hazard a guess
> > > that the implementors of EVC on these platforms simply tested for
> > > this condition and ran the calculation on the largest component.
> > >
> > > However, you can calculate eigenvector centrality by iterative
> > > approximation: essentially, you start every node out with an equal
> > > amount of "potential", and that potential is allowed to flow along
> > > edges in proportion to their edge weights; continue until the
> > > potential stops changing.  This is the method that JUNG uses, and
> > > I've just confirmed that it appears to work fine--that is, assigns
> > > non-zero values that seem plausible--on a network with
> > > disconnected components.
> > >
> > > If you want to try this with JUNG, you'll want the PageRank class
> > > with bias set to 0; let me know if you have any questions.
> > >
> > > Hope this helps--
> > >
> > >
> > > On 8 Mar 2005, at 8:28, Skye Bender-Demoll wrote:
> > >
> > > > *****  To join INSNA, visit http://www.insna.org  *****
> > > >
> > > > Hi all,
> > > >
> > > > Does anybody have any experience with / references for
> > > > eigenvalue centrality measures and graphs with multiple
> > > > components?
> > > >
> > > >   We are starting to work with the standard eigenvector
> > > >   centrality
> > > > measure,
> > > > but the networks we use contain multiple components.  Not
> > > > surprisingly, this seems to cause some problems.  The
> > > > eigenvector calculation (both in UCINET and in matlab) gives
> > > > correct results for the largest components, but zero for all the
> > > > other components.  When we run the calculation on one component
> > > > alone, we get very different results.  I'm guessing that the
> > > > eigenvector centrality measure is not defined for multiple
> > > > components? (UCINET seems to suggest this)
> > > >
> > > > Or are the values for the smaller component (a three node chain
> > > > in a test example) so small when compared with the large
> > > > component (a 5 node bow-tie) that they are lost in round off
> > > > during the eigenvector calculation?
> > > >
> > > > If we run the algorithm independently on each component, can we
> > > > compare scores between components, or are they only valid within
> > > > components? (I'm assuming that in the UCINET-style version where
> > > > the scores are normalized they should only be compared within
> > > > components?)
> > > >
> > > > Any suggestions welcome ( I have yet to locate many papers
> > > > directly related to this, nor in the socnet archives)
> > > >
> > > > 1)  R. Poulin, M.-C. Boily  B.R. Masse (2000) "Dynamical systems
> > > > to define centrality in social networks" Social Networks 22
> > > > 187–220
> > > >
> > > > thanks,
> > > >    -skye
> > > > ATA S.p.A Lucca Italy
> > > >
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> >
> >
> > Martin Everett
> > Marylebone Provost
> > University of Westminster
> > London NW1 5LS
> >
> > Tel +44(0)20 7911 5000 Ext 3183
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Martin Everett
Marylebone Provost
University of Westminster
London NW1 5LS

Tel +44(0)20 7911 5000 Ext 3183
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