Talk:Eigenvector centrality

There seem to be a couple of lapses in the reasoning:

The Perron-Frobenius theorem guarantees that the largest eigenvalue is simple when the coefficients are > 0, but not if they are ≥ 0. It is obvious that this theorem does not hold with ≥ 0; consider the 2×2 identity matrix, it has (1 2) and (2 1) as positive eigenvectors for eigenvalue 1. Thus, there are many eigenvectors that are solutions, including multiple eigenvectors with 1-norm 1.
However, the additional requirement that all the entries in the eigenvector be positive implies (by the Perron–Frobenius theorem) that only the greatest eigenvalue results in the desired centrality measure. I may have missed a point here, but while the Perron-Frobenius theorem guarantees that the largest eigenvalue has a positive eigenvector, it does not seem to guarantee that only the largest eigenvalue has a positive eigenvector. David.Monniaux 00:15, 8 December 2006 (UTC)