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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: milan@research.bell-labs.com; iis@research.bell-labs.com
Iraj Saniee
Affiliation:
Mathematics of Networks and Communications Bell Labs, Alcatel-Lucent, 600 Mountain Avenue, Murray Hill, NJ 07974, USA. E-mail: milan@research.bell-labs.com; iis@research.bell-labs.com
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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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