Predecessors and successors in random mappings with exchangeable in-degrees

Jennie Charlotte Hansen, Jerzy Jaworski

Research output: Contribution to journalArticle

Abstract

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, TnD^ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D^1,D^2,...,D^n). 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 D^1,D^2,...,D^n. As an application of these results, we consider two special examples of TnD^ 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.
Original languageEnglish
Pages (from-to)721-740
Number of pages23
JournalJournal of Applied Probability
Volume50
Issue number3
DOIs
Publication statusPublished - Sep 2013
Event15th International Conference on Random Structures and Algorithms 2011 - Atlanta, United States
Duration: 24 May 201128 May 2011

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Random Mapping
Preferential Attachment
Degree Sequence
Exact Distribution
Discrete Distributions
Expected Value
Asymptotic Behavior
Graph in graph theory

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title = "Predecessors and successors in random mappings with exchangeable in-degrees",
abstract = "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, TnD^ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D^1,D^2,...,D^n). 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 D^1,D^2,...,D^n. As an application of these results, we consider two special examples of TnD^ 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.",
author = "Hansen, {Jennie Charlotte} and Jerzy Jaworski",
note = "The research in this paper was supported by the Marie Curie Intra-European Fellowship No. 236845 (RANDOMAPP) within the 7th European Community Framework Programme.",
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language = "English",
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Predecessors and successors in random mappings with exchangeable in-degrees. / Hansen, Jennie Charlotte; Jaworski, Jerzy.

In: Journal of Applied Probability, Vol. 50, No. 3, 09.2013, p. 721-740.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Predecessors and successors in random mappings with exchangeable in-degrees

AU - Hansen, Jennie Charlotte

AU - Jaworski, Jerzy

N1 - The research in this paper was supported by the Marie Curie Intra-European Fellowship No. 236845 (RANDOMAPP) within the 7th European Community Framework Programme.

PY - 2013/9

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N2 - 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, TnD^ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D^1,D^2,...,D^n). 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 D^1,D^2,...,D^n. As an application of these results, we consider two special examples of TnD^ 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.

AB - 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, TnD^ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D^1,D^2,...,D^n). 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 D^1,D^2,...,D^n. As an application of these results, we consider two special examples of TnD^ 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.

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