  The properties of certain networks are determined by hidden variables that
are not explicitly measured. The conditional probability (propagator) that a
vertex with a given value of the hidden variable is connected to k of other
vertices determines all measurable properties. We study hidden variable models
and find an averaging approximation that enables us to obtain a general
analytical result for the propagator. Analytic results showing the validity of
the approximation are obtained. We apply hidden variable models to
protein-protein interaction networks (PINs) in which the hidden variable is the
association free-energy, determined by distributions that depend on
biochemistry and evolution. We compute degree distributions as well as
clustering coefficients of several PINs of different species; good agreement
with measured data is obtained. For the human interactome two different
parameter sets give the same degree distributions, but the computed clustering
coefficients differ by a factor of about two. This shows that degree
distributions are not sufficient to determine the properties of PINs.
