Asymptotic Degree Distributions in Large (Homogeneous) Random Networks: A Little Theory and a Counterexample

Abstract

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 a sequence of homogeneous random networks of increasingly large size. The discussion is carried under three basic statistical assumptions on the degree sequence: (i) a weak form of distributional homogeneity; (ii) the existence of an asymptotic (nodal) degree distribution; and (iii) a weak form of asymptotic uncorrelatedness. It follows from the discussion that under (i)-(ii) the asymptotic equality of the two degree distributions occurs if and only if (iii) holds. We use this observation to show that the asymptotic equality may fail in some homogeneous random networks. The counterexample is found in the class of random threshold graphs with exponentially distributed fitness for which (i) and (ii) hold but where (iii) does not. An implication of this finding is that these random threshold graphs cannot be used as a substitute to the Barabási-Albert model for scale-free network modeling, as was proposed by some authors. Results can also be formulated for non-homogeneous models with the help of a random sampling procedure over the nodes. This approach is applicable to many classes of network models, including preferential attachment models and locally weak convergent sequence of random graphs.