Randomised reproducing graphs

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, α, β and γ, 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+α)(1+) < 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 α and . If (1+α)(1+γ) > 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. © 2011 Applied Probability Trust.