Phase Transition on The Degree Sequence of a Mixed Random Graph Process

TL;DR

This paper investigates the degree sequence phase transition in a mixed random graph model combining classical and BA models, revealing exponential and power law behaviors depending on the model composition.

Contribution

It introduces a novel analysis of phase transition in degree sequences for a mixed random graph process, including models with non-uniform edge addition and hard copying.

Findings

Classical model has exponential degree sequence.

BA and mixed models exhibit power law degree sequences.

Phase transition depends on the mixture proportion.

Abstract

This paper focuses on the problem of the degree sequence for a mixed random graph process which continuously combines the {\it classical} model and the BA model. Note that the number of step added edges for the mixed model is random and non-uniformly bounded. By developing a comparing argument, phase transition on the degree distributions of the mixed model is revealed: while the {\it pure} classical model possesses a {\it exponential} degree sequence, the {\it pure} BA model and the mixed model possess {\it power law} degree sequences. As an application of the methodology, phase transition on the degree sequence of {\it another} mixed model with {\it hard copying} is also studied, especially, in the power law region, the inverse power can take any value greater than 1.