  The hidden variable formalism (based on the assumption of some intrinsic node
parameters) turned out to be a remarkably efficient and powerful approach in
describing and analyzing the topology of complex networks. Owing to one of its
most advantageous property - namely proven to be able to reproduce a wide range
of different degree distribution forms - it has become a standard tool for
generating networks having the scale-free property. One of the most intensively
studied version of this model is based on a thresholding mechanism of the
exponentially distributed hidden variables associated to the nodes (intrinsic
vertex weights), which give rise to the emergence of a scale-free network where
the degree distribution $p(k)\sim k^{-\gamma}$ is decaying with an exponent of
$\gamma =2$. Here we propose a generalization and modification of this model by
extending the set of connection probabilities and hidden variable distributions
that lead to the aforementioned degree distribution, and analyze the conditions
leading to the above behavior analytically. In addition, we propose a
relaxation of the hard threshold in the connection probabilities, which opens
up the possibility for obtaining sparse scale free networks with arbitrary
scaling exponent.
