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Family trees of continuous-time birth-and-death processes

Part of: Markov processes Graph theory

Published online by Cambridge University Press:  14 July 2016

Johannes Müller  and
Martin Möhle
Johannes Müller*
Affiliation:
Munich University of Technology
Martin Möhle*
Affiliation:
University of Tübingen
*
∗Postal address: Mathematics Institute, Munich University of Technology, Boltzmannstraße 3, D-85748 Garching, Germany. Email address: johannes.mueller@gsf.de
∗∗Postal address: Mathematics Institute, University of Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany.
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Abstract

We consider a stochastic graph generated by a continuous-time birth-and-death process with exponentially distributed waiting times. The vertices are the living particles, directed edges go from mothers to daughters. The size and the structure of the connected components are investigated. Furthermore, the number of connected components is determined.

Keywords

Birth-and-death processstochastic graphfamily tree

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 05C80: Random graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 980 - 994
Copyright
Copyright © Applied Probability Trust 2003 

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References

Abramowitz, M., and Stegun, I. A. (eds) (1965). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Feller, W. (1939). Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung. Acta Biometrika 5, 11–40.Google Scholar
Goel, N. S., and Richter-Dyn, N. (1974). Stochastic Models in Biology. Academic Press, New York.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Kendall, D. G. (1949). Stochastic processes and population growth. J. R. Statist. Soc. B 11, 230–264.Google Scholar
Müller, J., Kretzschmar, M., and Dietz, K. (2000). Contact tracing in deterministic and stochastic models. Math. Biosci. 164, 39–64.Google Scholar
Small, P. M. et al. (1994). The epidemiology of tuberculosis in San Francisco. New England J. Med. 330, 1703–1709.Google Scholar