  We develop a statistical theory to characterize correlations in weighted
networks. We define the appropriate metrics quantifying correlations and show
that strictly uncorrelated weighted networks do not exist due to the presence
of structural constraints. We also introduce an algorithm for generating
maximally random weighted networks with arbitrary $P(k,s)$ to be used as null
models. The application of our measures to real networks reveals the importance
of weights in a correct understanding and modeling of these heterogeneous
systems.
