Journal of Applied Probability

Invariant bipartite random graphs on Rd

Fabio Lopes

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 51, Number 3 (2014), 769-779.

Dates
First available in Project Euclid: 5 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1409932673

Digital Object Identifier
doi:10.1239/jap/1409932673

Mathematical Reviews number (MathSciNet)
MR3256226

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G55: Point processes 05C80: Random graphs [See also 60B20]

Keywords
Poisson process random graph bipartite stable matching percolation

Citation

Lopes, Fabio. Invariant bipartite random graphs on R d. J. Appl. Probab. 51 (2014), no. 3, 769--779. doi:10.1239/jap/1409932673. https://projecteuclid.org/euclid.jap/1409932673


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