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)