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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:
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> Hi all,
> Does anybody have any experience with / references for eigenvalue
> measures and graphs with multiple components?
> We are starting to work with the standard eigenvector centrality
> but the networks we use contain multiple components. Not surprisingly,
> this seems to cause some problems. The eigenvector calculation (both
> 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
> example) so small when compared with the large component (a 5 node
> 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?
> assuming that in the UCINET-style version where the scores are
> they should only be compared within components?)
> Any suggestions welcome ( I have yet to locate many papers directly
> to this, nor in the socnet archives)
> 1) R. Poulin, M.-C. Boily B.R. Masse (2000) "Dynamical systems to
> centrality in social networks" Social Networks 22 187–220
> ATA S.p.A Lucca Italy
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