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A Dynamic Network in a Dynamic Population: Asymptotic Properties

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Tom Britton ,
Mathias Lindholm  and
Tatyana Turova
Tom Britton*
Affiliation:
Stockholm University
Mathias Lindholm*
Affiliation:
Uppsala University
Tatyana Turova*
Affiliation:
Lund University
*
∗ Postal address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. Email address: [email protected]
∗∗ Current address: AFA Insurance, SE-106 27 Stockholm, Sweden. Email address: [email protected]
∗∗∗ Postal address: Mathematical Center, Lund University, Lund S-221 00, Sweden. Email address: [email protected]
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Abstract

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We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with a social index given at birth. During the life of a node it creates edges to other nodes, nodes with high social index at higher rate, and edges disappear randomly in time. For this model, we derive a criterion for when a giant connected component exists after the process has evolved for a long period of time, assuming that the node population grows to infinity. We also obtain an explicit expression for the degree correlation ρ (of neighbouring nodes) which shows that ρ is always positive irrespective of parameter values in one of the two treated submodels, and may be either positive or negative in the other model, depending on the parameters.

Keywords

Degree correlationdynamic networkphase transitionrandom graphstationary distribution

MSC classification

Secondary: 92D30: Epidemiology 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1163 - 1178
Copyright
Copyright © Applied Probability Trust 2011 

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