  We derive a class of generalized statistics, unifying the Bose and Fermi
ones, that describe any system where the first-occupation energies or
probabilities are different from subsequent ones, as in presence of thresholds,
saturation, or aging. The statistics completely describe the structural
correlations of weighted networks, which turn out to be stronger than expected
and to determine significant topological biases. Our results show that the null
behavior of weighted networks is different from what previously believed, and
that a systematic redefinition of weighted properties is necessary.
