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Invariant Bipartite Random Graphs on Rd

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

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: flopes@math.su.se
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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 

References

Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, General Theory and Structure, 2nd edn. Springer, New York.Google Scholar
Deijfen, M. (2009). Stationary random graphs with prescribed iid degrees on a spatial Poisson process. Eletron. Commun. Prob. 14, 81–89.Google Scholar
Deijfen, M. and Jonasson, J. (2006). Stationary random graphs on {\BBZ} with prescribed iid degrees and finite mean connections. Eletron. Commun. Prob. 11, 336–346.Google Scholar
Deijfen, M. and Lopes, F. M. (2012). Bipartite stable Poisson graphs on {\BBR}. Markov Process. Relat. Fields 18, 583–594.Google Scholar
Deijfen, M. and Meester, R. (2006). Generating stationary random graphs on {\BBZ} with prescribed independent, identically distributed degrees. Adv. Appl. Prob. 38, 287–298.Google Scholar
Deijfen, M., Häggström, O. and Holroyd, A. E. (2012). Percolation in invariant Poisson graphs with i.i.d. degrees. Ark. Mat. 50, 41–58.Google Scholar
Deijfen, M., Holroyd, A. E. and Peres, Y. (2011). Stable Poisson graphs in one dimension. Eletron. J. Prob. 16, 1238–1253.Google Scholar
Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.Google Scholar
Ferrari, P. A., Landim, C. and Thorisson, H. (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Prob. Statist. 40, 141–152.Google Scholar
Gale, D. and Shapley, L. S. (1962). College admissions and the stability of marriage. Amer. Math. Monthly 69, 9–15.Google Scholar
Häggström, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9, 295–315.Google Scholar
Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Eletron. Commun. Prob. 8, 17–27.Google Scholar
Holroyd, A. E. and Soo, T. (2013). Insertion and deletion tolerance of point processes. Electron. J. Prob. 18, 24pp.Google Scholar
Holroyd, A. E., Pemantle, R., Peres, Y. and Schramm, O. (2009). Poisson matching. Ann. Inst. H. Poincaré Prob. Statist. 45, 266–287.Google Scholar
Jonasson, J. (2009). Invariant random graphs with iid degrees in a general geography. Prob. Theory Relat. Fields 143, 643–656.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Meester, R. and Roy, R. (1994). Uniqueness of unbounded occupied and vacant components in Boolean models. Ann. Appl. Prob. 4, 933–951.Google Scholar