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A Dynamic Network in a Dynamic Population: Asymptotic Properties

Published online by Cambridge University Press:  14 July 2016

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2011 

References

Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509512.Google Scholar
Barabási, A.-L., Albert, R. and Jeong, H. (1999). Mean-field theory for scale-free random networks. Physica A, 272, 173187.Google Scholar
Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3122.Google Scholar
Bollobás, B., Janson, S. and Riordan, O. (2011). Sparse random graphs with clustering. Random Structures Algorithms, 38, 269323.Google Scholar
Britton, T. and Lindholm, M. (2010). Dynamic random networks in dynamic populations. J. Statist. Phys. 139, 518535.Google Scholar
Callaway, D. S. et al. (2001). Are randomly grown graphs really random? Phys. Rev. E 64, 041902, 7 pp.Google Scholar
Haccou, P., Jagers, P. and Vatutin, V. A. (2005). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.Google Scholar
Malyshev, V. A. (1998). Random graphs and grammars on graphs. Discrete Math. Appl. 8, 247262.Google Scholar
Newman, M. E. J. (2002). Assortative Mixing in Networks. Phys. Rev. Lett. 89, 208701, 4 pp.Google Scholar
Turova, T. S. (2002). Dynamical random graphs with memory. Phys. Rev. E 65, 066102, 9 pp.Google Scholar
Turova, T. S. (2003). Long paths and cycles in dynamical graphs. J. Statist. Phys. 110, 385417.Google Scholar
Turova, T. S. (2006). Phase transitions in dynamical random graphs. J. Statist. Phys. 123, 10071032.Google Scholar
Turova, T. S. (2007). Continuity of the percolation threshold in randomly grown graphs. Electron. J. Prob. 12, 10361047.Google Scholar