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Invariant Bipartite random graphs on R
Stockholm University, Faculty of Science, Department of Mathematics.
2014 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 51, no 3, 769-779 p.Article in journal (Refereed) Published
##### Abstract [en]

Suppose that red and blue points occur in $\mathbb{R}^d$ according to two simple point processes  with finite intensities $\lambda_{\mathcal{R}}$ and $\lambda_{\mathcal{B}}$, respectively. Furthermore, let $\nu$ and $\mu$ be two probability distributions on the strictly positive integers with means $\bar{\nu}$ and $\bar{\mu}$, respectively. Assign independently a random number of stubs (half-edges) to each red (blue) point with law $\nu$ ($\mu$).  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 $\nu$ or $\mu$ depending on its color. For a large class of point processes we show that such translation-invariant schemes matching a.s. all stubs are possible if and only if$\lambda_{\mathcal{R}} \bar{\nu}= \lambda_{\mathcal{B}} \bar{\mu},$also including the case when $\bar{\nu}=\bar{\mu}=\infty$ 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 $\nu$ and $\mu$ for the presence and absence of infinite components. These results are  two-color versions of those obtained by Deijfen, Holroyd and H\"{a}ggstr\"{o}m.

##### Place, publisher, year, edition, pages
2014. Vol. 51, no 3, 769-779 p.
##### Keyword [en]
Poisson process, random graphs, bipartite, stable matching, percolation
Mathematics
##### Research subject
Mathematical Statistics
##### Identifiers
ISI: 000342035400013OAI: oai:DiVA.org:su-106729DiVA: diva2:738448
Available from: 2014-08-18 Created: 2014-08-18 Last updated: 2017-12-05Bibliographically 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.
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)
##### 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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