Asymptotic properties of a random graph with duplications

TL;DR

This paper analyzes a random graph model evolving through vertex duplication and deletion, establishing almost sure convergence of the degree distribution with a stretched exponential decay rate.

Contribution

It proves the almost sure existence of a limiting degree distribution with a specific asymptotic decay in a graph model involving duplication and deletion.

Findings

Degree distribution converges almost surely.

Asymptotic degree distribution exhibits stretched exponential decay.

Explicit asymptotic formula for degree proportions as degree grows large.

Abstract

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an a.s. asymptotic degree distribution, with streched exponential decay; more precisely, the proportion of vertices of degree $d$ tends to some positive number $c_d>0$ almost surely as the number of steps goes to infinity, and $c_d\sim (e\pi)^{1/2} d^{1/4} e^{-2\sqrt d}$ holds as $d\to\infty$.