  We present an analytic formalism describing structural properties of random
uncorrelated networks with arbitrary degree distributions. The formalism allows
to calculate the main network characteristics like: the position of the phase
transition at which a giant component first forms, the mean component size
below the phase transition, the size of the giant component and the average
path length above the phase transition. We apply the approach to classical
random graphs of Erdos and Renyi, single-scale networks with exponential degree
distributions and scale-free networks with arbitrary scaling exponents and
structural cut-offs. In all the cases we obtain a very good agreement between
results of numerical simulations and our analytical predictions.
