Asymptotic degree distribution of a duplication-deletion random graph model

TL;DR

This paper analyzes the asymptotic degree distribution of a duplication-deletion random graph model, revealing exponential decay for low attachment probability and power-law decay for higher probabilities, with a precise mathematical characterization.

Contribution

It provides a rigorous proof of the almost sure convergence of the degree distribution and derives its exact form using hypergeometric integrals, including phase transition behavior.

Findings

Degree sequence decays exponentially for p<0.5

Degree sequence follows a power-law for p>0.5

Exact degree distribution expressed via hypergeometric integral

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 $0<p<\frac{1}{2}$ we show that the degree sequence decays exponentially at rate $\frac{p}{1-p}$, whereas it satisfies a power-law with exponent $\frac{p}{2p-1}$ if $\frac{1}{2}<p<1$. At the threshold $p=\frac{1}{2}$ the degree sequence lies between a power-law and exponential decay.