Asymptotic degree distribution of a duplication-deletion random graph model

Abstract

We study a discrete–time duplication–deletion random graph model and analyse its asymptotic degree distribution. The random graphs consists of disjoint cliques. In each time step either a new vertex is brought in with probability 0 < p < 1 and attached to an existing clique, chosen with probability proportional to the clique size, or all the edges of a random vertex are deleted with probability 1 − p. We prove almost sure convergence of the asymptotic degree distribution and find its exact values in terms of a hypergeometric integral, expressed in terms of the parameter p. In the regime 0 < p < 1 2 we show that the degree sequence decays exponentially at rate p 1−p , whereas it satisfies a power–law with exponent p 2p−1 if 1 2 < p < 1. At the threshold p = 1 2 the degree sequence lies between a power–law and exponential decay.