  In random graph models, the degree distribution of an individual node should
be distinguished from the (empirical) degree distribution of the graph that
records the fractions of nodes with given degree. We introduce a general
framework to explore when these two degree distributions coincide
asymptotically in large homogeneous random networks. The discussion is carried
under three basic statistical assumptions on the degree sequences: (i) a weak
form of distributional homogeneity; (ii) the existence of an asymptotic (nodal)
degree distribution; and (iii) a weak form of asymptotic uncorrelatedness. We
show that this asymptotic equality may fail in homogeneous random networks for
which (i) and (ii) hold but (iii) does not. The counterexample is found in the
class of random threshold graphs. An implication of this finding is that random
threshold graphs cannot be used as a substitute to the Barab\'asi-Albert model
for scale-free network modeling, as has been proposed by some authors. The
results can also be formulated for non-homogeneous models by making use of a
random sampling procedure over the nodes.
