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 Barabási-Albert model for scale-free network modeling, as has been proposed by some authors. This work was supported in part by NSF Grant CCF-1217997. The paper was completed during the academic year 2014-2015 while A.M. Makowski was a Visiting Professor with the Department of Statistics of the Hebrew University of Jerusalem with the support of a fellowship from the Lady Davis Trust. This document does not contain technology or technical data controlled under either the U.S. International Traffic in Arms Regulations or the U.S. Export Administration Regulations. Parts of the material were presented in the 53rd IEEE Conference on Decision and Control (CDC 2015), Osaka (Japan), December 2015. S. Pal was with the Department of Electrical and Computer Engineering, and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA. He is now with Raytheon BBN Technologies (email: [email protected]). A. M. Makowski is with the Department of Electrical and Computer Engineering, and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]).