The Degree Sequence of a Scale-Free Random Graph Process with Hard Copying

TL;DR

This paper introduces a random graph model with hard copying and preferential attachment, analyzing its degree sequence and showing it follows a power-law distribution under certain conditions.

Contribution

It proposes a new random graph process combining preferential attachment and uniform copying, deriving the asymptotic degree distribution.

Findings

The degree sequence follows a power-law distribution with exponent depending on lpha.

The model exhibits scale-free properties for sufficiently large lpha.

The mean degree sequence converges to a specific asymptotic form.

Abstract

In this paper we consider a simple model of random graph process with {\it hard} copying as follows: At each time step $t$, with probability $0<\alpha\leq 1$ a new vertex $v_t$ is added and $m$ edges incident with $v_t$ are added in the manner of {\it preferential attachment}; or with probability $1-\alpha$ an existing vertex is copied uniformly at random. In this way, while a vertex with large degree is copied, the number of added edges is its degree and thus the number of added edges is not upper bounded. We prove that, in the case of $\alpha$ being large enough, the model possesses a mean degree sequence as $ d_{k}\sim Ck^{-(1+2\alpha)}$, where $d_k$ is the limit mean proportion of vertices of degree $k$.