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

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.