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## Re: [igraph] centrality eigenvector vs hub/auth - ARPACK LM vs LA

 From: Gábor Csárdi Subject: Re: [igraph] centrality eigenvector vs hub/auth - ARPACK LM vs LA Date: Fri, 7 Sep 2012 21:59:25 -0400

```On Fri, Sep 7, 2012 at 4:11 PM, Matthew Galati <address@hidden> wrote:
> I was looking at the ARPACK implementation of centrality.
>
> I noticed, that for eigenvector centrality, you are using LA, while for
> hub/authority/page-rank scores, you are using LM. I believe the power method
> will give LM - which is the alternate way to calculate these metrics (under
> some conditions).
>
> What is the reason behind using LA for eigenvector centrality (instead of
> using LM)? The literature seems to be a little ambiguous - using terms like
> "largest" but not clarifying if that is algebraic or magnitude (in the
> presence of negative eigenvalues).

Actually, for eigenvector centrality we also have LR, which is the
largest real part, for the directed version.

AFAIK for ARPACK LM is the easiest. For page rank we can use LM,
because the matrix is strictly positive, and then according to the
Frobenius-Perron theorem, its largest eigenvalue is positive and also
strictly bigger than the absolute value of any other eigenvalue. The
same is true for hub/authority scores. (Well, as long as there are no
isolates, hmmm, we might have a bug here.)

For eigenvector centrality the matrix is not strictly positive any
more, and there might be several eigenvalues, both positive and
negative, that are largest, according to their absolute values. To get
a non-negative eigenvector, one needs a positive largest eigenvalue.
That's why we are using LA, and similarly, LR for the non-symmetric
case.

Does it make sense?

Best,
Gabor

> Matt
>
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>

--
Gabor Csardi <address@hidden>     MTA KFKI RMKI

```