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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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> Scientist, Musician, and Philosopher-At-Tall
>    It's that moment of dawning comprehension that I live for--Bill
> Watterson
>    My opinions are too rational and insightful to be those of any
> organization.
>
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