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Re: [igraph] alpha.centrality() and evcent()

From: Gábor Csárdi
Subject: Re: [igraph] alpha.centrality() and evcent()
Date: Wed, 25 Apr 2012 11:35:36 -0400

On Fri, Apr 13, 2012 at 12:09 PM, santiago gil <address@hidden> wrote:
> Hello all! Long time no post...
> I have a question about the relationship between the functions
> alpha.centrality() and evcent(). According to my understanding of these two
> measures of centrality (which may very well be totally wrong) when e ("exo"
> in alpha.centrality) is 0 for all components or when alpha is very large,
> alpha centrality should reduce to eigenvector centrality.

Well, this is not that simple. To make alpha centrality "reduce" to
eigenvector centrality, you need to set e=0, and set alpha close to
1/lambda, where lambda is the largest eigenvalue of the adjacency
matrix. As alpha gets closer to 1/lambda, the solution gets closer to
the eigenvector centrality. (If alpha=1/lambda, then alpha centrality
does not exist, however.)

> However,
> alpha.centrality() doesn't seem to work on undirected graphs,

Why not?

> and evcent()
> doesn't seem to be affected by the directionality of the edges in directed
> graphs (the results are unchanged by transposing the adjacency matrix).

Yes, this was updated in the 0.6 igraph version.

> The
> result of this is that in the implementations of these functions, exo=0
> produces a singularity error, and the results for alpha.centrality() with
> alpha>20 and evcent() in my current problem show no resemblance whatsoever.
> Does anybody have any insight for me on this? Is it possible to calculate
> evcent() on directed graphs and measure the convergence of
> alpha.centrality() to evcent()? Or am I way off here?

I would suggest to read this paper:

P. Bonacich, P. Lloyd. Eigenvector-like measures of centrality for
asymmetric relations. doi:10.1016/S0378-8733(01)00038-7


> Good wishes,
> Santiago
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Gabor Csardi <address@hidden>     MTA KFKI RMKI

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