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

Asymptotic degree distributions in large (homogeneous) random networks: A little theory and a counterexample