  In the paper, we present an incremental approach in the construction of scale
free networks with hidden variables. The work arises from the necessity to
generate that type of networks with a given number of links instead of
obtaining a random configurations for a given set of parameters as in the usual
literature. I propose an analytical approach of network evolution models
gathering information along time based on the construction of a stochastic
process on the space of possible networks. The analytical solution is eact in a
case of multigraph network, meanwhile in simple graph we deal with important
finite size effects. We show the statistical properties of this network such as
number of isolated nodes, degree correlations and multilinks, also discussing
the limitations of such predictions in real networks. Numerical simulations are
used tu support the analytical calculations. On the computational side, such
evolutive network construction allows to regulate the connectivity of the
network obtaining desired values of connectivity density used as control
parameter of the system. As a consequence, an incremental generetive approach
is more suitable in those situation we want to control the number of links in
the system while the number of nodes are kept fixed.
