Electronic Journal of Probability

Randomised Reproducing Graphs

Jonathan Jordan

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30]
Secondary: 60G99: None of the above, but in this section 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

reproducing graphs random graphs degree distribution phase transition

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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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