Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T14:23:53.244Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Bailey, W. N. (1935). Generalised Hypergeometric Series. Cambridge University Press.Google Scholar
Ball, F., Mollison, D. and Scalia-Tomba, G. (1997). Epidemics with two levels of mixing. Ann. Appl. Prob. 7, 46–89.CrossRefGoogle Scholar
Berg, S. (1981). On snowball sampling, random mappings and related problems. J. Appl. Prob. 18, 283–290.Google Scholar
Berg, S. (1983). Random contact processes, snowball sampling and factorial series distributions. J. Appl. Prob. 20, 31–46.CrossRefGoogle Scholar
Berg, S. and Mutafchiev, L. (1990). Random mappings with an attracting center: Lagrangian distributions and a regression function. J. Appl. Prob. 27, 622–636.Google Scholar
Burtin, Y. D. (1980). On simple formula for random mappings and its applications. J. Appl. Prob. 17, 403–414.CrossRefGoogle Scholar
Consul, P. C. and Famoye, F. (2006). Quasi-probability models. In Lagrangian Probability Distributions, Birkhäuser, Boston, pp. 69–91.Google Scholar
Gertsbakh, I. B. (1977). Epidemic processes on a random graph: some preliminary results. J. Appl. Prob. 14, 427–438.Google Scholar
Gould, H. W. (1972). Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations. Henry W. Gould, Morgantown.Google Scholar
Hansen, J. C. and Jaworski, J. (2008). Local properties of random mappings with exchangeable in-degrees. Adv. Appl. Prob. 40, 183–205.Google Scholar
Hansen, J. C. and Jaworski, J. (2008). Random mappings with exchangeable in-degrees. Random Structures Algorithms 33, 105–126.CrossRefGoogle Scholar
Hansen, J. C. and Jaworski, J. (2009). A random mapping with preferential attachment. Random Structures Algorithms 34, 87–111.Google Scholar
Hansen, J. C. and Jaworski, J. (2013). Structural transition in random mappings. Submitted.Google Scholar
Islam, M. N., O'Shaughnessy, C. D. and Smith, B. (1996). A random graph model for the final-size distribution of household infections. Statist. Medicine 15, 837–843.3.0.CO;2-V>CrossRefGoogle ScholarPubMed
Jaworski, J. (1990). Random mappings with independent choices of the images. In Random Graphs '87, eds Karoński, M., Jaworski, J. and Ruciński, A., John Wiley, Chichester, pp. 89–101.Google Scholar
Jaworski, J. (1998). Predecessors in a random mapping. Random Structures Algorithms 13, 501–519.Google Scholar
Jaworski, J. (1999). Epidemic processes on digraphs of random mappings. J. Appl. Prob. 36, 780–798.Google Scholar
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, 2nd edn. John Wiley, New York.Google Scholar
Kolchin, V. F. (1986). Random Mappings. Optimization Software, New York.Google Scholar
Moon, J. W. (1970). Counting Labelled Trees (Canad. Math. Monog. 1). Canadian Mathematical Congress, Montreal.Google Scholar
Mutafchiev, L. (1981). Epidemic processes on random graphs and their threshold function. Serdica 7, 153–159.Google Scholar
Mutafchiev, L. (1982). A limit distribution related to random mappings and its application to an epidemic process. Serdica 8, 197–203.Google Scholar
Mutafchiev, L. (1984). On some stochastic problems of discrete mathematics. In Mathematics and Mathematical Education (Sunny Beach, 1984), pp. 57–80.Google Scholar
Pitman, J. (2001). Random mappings, forest, and subsets associated with Abel-Cayley-Hurwitz multinomial expansions. Seminaire Lothar. Combin. 46, 45pp.Google Scholar
Pittel, B. (1983). On distributions related to transitive closures of random finite mappings. Ann. Prob. 11, 428–441.Google Scholar
Prüfer, H. (1918). Neuer beweis eines satzes über permutationen. Arch. Math. Phys. 27, 142–144.Google Scholar
Ross, S. M. (1981). A random graph. J. Appl. Prob. 18, 309–315.Google Scholar
Rota, G. C. et al. (1975). Finite Operator Calculus. Academic Press.Google Scholar
Stepanov, V. E. (1971). Random mappings with a single attracting center. Theory Prob. Appl. 16, 155–162.Google Scholar