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Bipartite Stable Poisson Graphs on R
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2012 (English)In: Markov processes and related fields, ISSN 1024-2953, Vol. 18, no 4, 583-594 p.Article in journal (Refereed) Published
Abstract [en]

Let red and blue points be distributed on R according to two independent Poisson processes R and B and let each red (blue) point independently be equipped with a random number of half-edges according to a probability distribution nu (mu). We consider translation-invariant bipartite random graphs with vertex classes defined by the point sets of R, and B, respectively, generated by a scheme based on the Gale - Shapley stable marriage for perfectly matching the half-edges. Our main result is that, when all vertices have degree 2, then the resulting graph almost surely does not contain an infinite component. The two-color model is hence qualitatively different from the one-color model, where Deijfen, Holroyd and Peres have given strong evidence that there is an infinite component. We also present simulation results for other degree distributions.

Place, publisher, year, edition, pages
2012. Vol. 18, no 4, 583-594 p.
Keyword [en]
Poisson process, random graph, degree distribution, percolation, matching
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:su:diva-87670ISI: 000313221100001OAI: oai:DiVA.org:su-87670DiVA: diva2:606082
Note

AuthorCount:2;

Available from: 2013-02-18 Created: 2013-02-14 Last updated: 2014-08-21Bibliographically approved
In thesis
1. Spatial Marriage Problems and Epidemics
Open this publication in new window or tab >>Spatial Marriage Problems and Epidemics
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers covering three different topics on the modeling of large real networks and phenomena thereon. In Papers I and II, we propose and study the properties of a bipartite version of the model introduced by Deijfen, Holroyd and Häggström for generating translation-invariant spatial random graphs with prescribed degree distribution. In particular, we focus our attention on spatial random graphs generated by a matching scheme based on the Gale-Shapley stable marriage problem. In paper III, we propose a random graph model for generating edge-weighted graphs with prescribed degree and weight distributions, and tunable degree-degree correlation. We then study a simple inhomogeneous epidemic model on such graphs, where the infection probabilities are functions of the edge-weights, and investigate how the epidemic threshold is affected by the degree-degree correlation. In paper IV, we study a simple stochastic model aimed at representing a competition between two virus strains in a population. A longstanding principle in ecology known as the competitive exclusion principle predicts that when one of the strains has even the slightest advantage over the other, the one with the advantage will either drive the competitor to extinction or lead to a transformation in the ecological niche. We investigate how long it will take for the strain to drive its competitor to extinction.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2014. 25 p.
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-106796 (URN)978-91-7447-970-6 (ISBN)
Public defence
2014-09-24, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: In press. Paper 4: Manuscript.

Available from: 2014-09-02 Created: 2014-08-20 Last updated: 2015-03-27Bibliographically approved

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