Randomised reproducing graphs

TL;DR

This paper introduces a probabilistic model for growing random graphs based on vertex reproduction, revealing phase transitions in degree distribution and growth behavior depending on model parameters.

Contribution

It generalizes existing reproducing graph models to include randomness and analyzes phase transitions and degree distribution behaviors.

Findings

Degree distribution converges to a stationary distribution with power law tail under certain conditions.

Vertex degree grows unbounded when parameters exceed a threshold.

Number of edges and spectral gap are also characterized.

Abstract

We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, $\alpha$, $\beta$ and $\gamma$, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If $(1+\alpha)(1+\gamma)<1$ then the degree distribution converges to a stationary distribution, which in most cases has an approximately power law tail with an index which depends on $\alpha$ and $\gamma$. If $(1+\alpha)(1+\gamma)>1$ then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree $d$ tends to zero. We also give some results on the number of edges and on the spectral gap.