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Randomised Reproducing Graphs


 
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1. Title Title of document Randomised Reproducing Graphs
 
2. Creator Author's name, affiliation, country Jonathan H Jordan; University of Sheffield; United Kingdom
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) reproducing graphs, random graphs, degree distribution, phase transition
 
3. Subject Subject classification Primary 05C82; secondary 60G99, 60J10
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) EPSRC
 
7. Date (YYYY-MM-DD) 2011-08-22
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/921
 
10. Identifier Digital Object Identifier 10.1214/EJP.v16-921
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 16
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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