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BOOTSTRAP PERCOLATION ON RANDOM GEOMETRIC GRAPHS

Published online by Cambridge University Press:  13 December 2013

Milan Bradonjić  and
Iraj Saniee
Milan Bradonjić
Affiliation:
Mathematics of Networks and Communications Bell Labs, Alcatel-Lucent, 600 Mountain Avenue, Murray Hill, NJ 07974, USA. E-mail: [email protected]; [email protected]
Iraj Saniee
Affiliation:
Mathematics of Networks and Communications Bell Labs, Alcatel-Lucent, 600 Mountain Avenue, Murray Hill, NJ 07974, USA. E-mail: [email protected]; [email protected]
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Abstract

Bootstrap percolation (BP) has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of collective action and cultural fads in human societies. It is defined on an (arbitrary) network of interacting agents whose state is determined by the state of their neighbors according to a threshold rule. In a typical setting, BP starts by random and independent “activation” of nodes with a fixed probability p, followed by a deterministic process for additional activations based on the density of active nodes in each neighborhood (θ activated nodes). Here, we study BP on random geometric graphs (RGGs) in the regime when the latter are (almost surely) connected. Random geometric graphs provide an appropriate model in settings where the neighborhood structure of each node is determined by geographical distance, as in wireless ad hoc and sensor networks as well as in contagion. We derive bounds pc′, pc″ on the critical thresholds such that for all p > p″c full percolation takes place, whereas for p < p′c it does not. We conclude with simulations that compare numerical thresholds with those obtained analytically.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 2 , April 2014 , pp. 169 - 181
Copyright
Copyright © Cambridge University Press 2013 

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References

1.Aizenman, M. & Lebowitz, J.L. (1988). Metastability effects in bootstrap percolation. Journal of Physics A: Mathematical and General 21(19): 3801–3813.CrossRefGoogle Scholar
2.Amini, H. (2010). Bootstrap percolation and diffusion in random graphs with given vertex degrees. Electronic Journal of Combinatorics, 17 R25.Google Scholar
3.Balogh, J., Bollobás, B., Duminil-copin, H. & Morris, R. (2012). The sharp threshold for bootstrap percolation in all dimensions. Transaction of the American Mathematical Society 364(5): 2667–2701.CrossRefGoogle Scholar
4.Balogh, J., Peres, Y. & Pete, G. (2006). Bootstrap percolation on infinite trees and non-amenable groups. Combinatorics, Probability & Computing 15(5): 715–730.Google Scholar
5.Balogh, J. & Pittel, B. (2007). Bootstrap percolation on the random regular graph. Random Structures & Algorithms 30(1–2): 257–286.Google Scholar
6.Cerf, R. & Cirillo, E.N.M. (1998). Finite size scaling in three-dimensional bootstrap percolation. Annals of Probability 27: 1837–1850.Google Scholar
7.Chalupa, J., Leath, P.L. & Reich, G.R. (1979). Bootstrap percolation on a Bethe lattice. Journal of Physics C 12: L31.Google Scholar
8.Erdős, P. & Rényi, A. (1959). On random graphs. Publication of the Mathematical Institute, Hungarian Academy of Science.Google Scholar
9.Erdős, P. & Rényi, A. (1960). On the evolution of random graphs. Publication of the Mathematical Institute, Hungarian Academy of Science.Google Scholar
10.Gersho, A. & Mitra, D. (1975). A simple growth model for the diffusion of new commuinication services. IEEE Transactions on Systems, Man, and Cybernetics SMC-5, 2 (March), 209–216.Google Scholar
11.Gilbert, E.N. (1959). Random graphs. Annals of Mathematical Statistics 30(4): 1141–1144.Google Scholar
12.Gilbert, E.N. (1961). Random plane networks. Society for Industrial and Applied Mathematics 9(4): 533–543.Google Scholar
13.Gupta, P. & Kumar, P.R. (1998). Critical power for asymptotic connectivity. In Proceedings of the 37th IEEE Conference on Decision and Control vol. 1, pp. 1106–1110.CrossRefGoogle Scholar
14.Holroyd, A.E. (2003). Sharp metastability threshold for two-dimensional bootstrap percolation. Probability Theory and Related Fields 125: 195–224.CrossRefGoogle Scholar
15.Janson, S., Luczak, T., Turova, T. & Vallier, T. (2012). Bootstrap percolation on the random graph G n, p. The Annals of Applied Probability 22(5): 1989–2047.Google Scholar
16.Penrose, M.D. (1997). The longest edge of the random minimal spanning tree. The Annals of Applied Probability 7(2): 340–361.Google Scholar
17.Penrose, M.D. (2003). Random geometric graphs. Oxford: Oxford University Press.Google Scholar
18.Pottie, G.J. & Kaiser, W.J. (2000). Wireless integrated network sensors. Communications of the ACM 43(5): 51–58.Google Scholar
19.Watts, D.J. (2002). A simple model of global cascades in random networks. Proceedings of the National Academy of Sciences 99: 5766–5771.Google Scholar