September 2014 Invariant bipartite random graphs on Rd
Fabio Lopes
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J. Appl. Probab. 51(3): 769-779 (September 2014). DOI: 10.1239/jap/1409932673


Suppose that red and blue points occur in Rd according to two simple point processes with finite intensities λR and λB, respectively. Furthermore, let ν and μ be two probability distributions on the strictly positive integers with means ν̅ and μ̅, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law ν (μ). We are interested in translation-invariant schemes for matching stubs between points of different colors in order to obtain random bipartite graphs in which each point has a prescribed degree distribution with law ν or μ depending on its color. For a large class of point processes, we show that such translation-invariant schemes matching almost surely all stubs are possible if and only if λRν̅ = λBμ̅, including the case when ν̅ = μ̅ = ∞ so that both sides are infinite. Furthermore, we study a particular scheme based on the Gale-Shapley stable marriage problem. For this scheme, we give sufficient conditions on ν and μ for the presence and absence of infinite components. These results are two-color versions of those obtained by Deijfen, Holroyd and Häggström.


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Fabio Lopes. "Invariant bipartite random graphs on Rd." J. Appl. Probab. 51 (3) 769 - 779, September 2014.


Published: September 2014
First available in Project Euclid: 5 September 2014

MathSciNet: MR3256226
Digital Object Identifier: 10.1239/jap/1409932673

Primary: 60D05
Secondary: 05C80 , 60G55

Keywords: bipartite , percolation , Poisson process , random graph , Stable matching

Rights: Copyright © 2014 Applied Probability Trust


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Vol.51 • No. 3 • September 2014
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