Transition of the Degree Sequence in the Random Graph Model of Cooper, Frieze, and Vera

Abstract

We study the transition of the expected degree sequence from power law to exponential decay in the random graph process introduced by Cooper, Frieze, and Vera. We prove that there is a threshold on the probabilities of introduction of vertices (with probability π1), edges (π2) and deletion of edges (π3) or vertices (π4). If π1 + 2π2 − 2π3 − π4 > 0, the expected fraction of vertices of degree k follows a power law, whereas for π1 + 2π2 − 2π3 − π4 < 0, it decays exponentially fast. This work extends the previous results of Wu et al. and Deijfen and Lindholm to the whole model defined by Cooper et al.