On the asymptotics of degree distributions

In random graph models, the degree distribution of individual nodes should be contrasted against the degree distribution of the graph, i.e., the usual fractions of nodes with given degrees. We introduce a general framework to discuss conditions under which these two degree distributions coincide asymptotically in large random networks. Somewhat surprisingly, we show that even in strongly homogeneous random networks, this equality may fail to hold. This is done by means of a counterexample drawn from the class of random threshold graphs. An implication of this finding is that random threshold graphs cannot be used as a substitute to the Barabási-Albert model for scale-free network modeling, as proposed by some authors.