The first eigenvector I obtain after decomposing an N x N symmetric correlation matrix (computed using a custom correlation measure) has very little variability in it.
It is also associated to a very high eigenvalue, usually ~50% higher than that of the second eigenvalue, e.g.: first eigenvalue = 1.9, second 1.3.
I've also noticed that the values of the first eigenvector are very closely related to the sum of my input matrix: Pearson's correlation = -0.80.
Second eigenvector's correlation is ~0.15, and the rest around 0.05.
What's the origin behind this (odd?) behavior of the first eigenvector?
Your data must be very correlated and traversed by a main factor (think principal component)
Assuming your custom correl measure is 1 when applied to the same variable, the ratio of the top eigenvalue to N would typically correspond to the percentage of variance explained by the first principal factor... Probably also close to the average correlation coefficient in your matrix...
Typical case would be financial correlation matrix, like stocks which are traversed by a main "market factor" (in PCA this would take a shape close to an equally-weighted portfolio). See Plerou et al for an "early" paper on this topic (there are others)