## Electronic Journal of Probability

### Randomised Reproducing Graphs

Jonathan Jordan

#### 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 $(1+\alpha)(1+\gamma)&lt;1$ 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 $(1+\alpha)(1+\gamma)&gt;1$ 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.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 57, 1549-1562.

Dates
Accepted: 22 August 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820226

Digital Object Identifier
doi:10.1214/EJP.v16-921

Mathematical Reviews number (MathSciNet)
MR2827470

Zentralblatt MATH identifier
1244.05211

Rights

#### Citation

Jordan, Jonathan. Randomised Reproducing Graphs. Electron. J. Probab. 16 (2011), paper no. 57, 1549--1562. doi:10.1214/EJP.v16-921. https://projecteuclid.org/euclid.ejp/1464820226

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