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Invariant Bipartite Random Graphs on Rd

Part of: Geometric probability and stochastic geometry Graph theory Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Fabio Lopes
Fabio Lopes*
Affiliation:
Stockholm University
*
∗ Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: [email protected]
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Abstract

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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.

Keywords

Poisson processrandom graphbipartitestable matchingpercolation

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 769 - 779
Copyright
© Applied Probability Trust 

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