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Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  30 January 2018

Jennie C. Hansen  and
Jerzy Jaworski
Jennie C. Hansen*
Affiliation:
Heriot-Watt University
Jerzy Jaworski*
Affiliation:
Adam Mickiewicz University
*
∗ Postal address: Actuarial Mathematics and Statistics Department and The Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
∗∗ Postal address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland. Email address: [email protected]
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Abstract

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In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

Keywords

Random mappingexchangeable in-degreespreferential attachmentanti-preferential attachmenturn schemeepidemic process

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 721 - 740
Copyright
© Applied Probability Trust 

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