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Bootstrap Percolation on Random Geometric Graphs
Abstract
Bootstrap Percolation is a discrete-time process that models the spread of information or disease across the vertex set of a graph. It was introduced in 1979 by Chalupa, Leith and Reich as a simple model of dynamics of ferromagnetism. We consider the following version of this process:Initially, each vertex of the graph is set active with probability p or inactive otherwise. At each time step, every inactive vertex with at least k active neighbors becomes active. Active vertices will always remain active. The process ends when it reaches a stationary state. If all the vertices eventually become active, then we say we achieve percolation.We analyze the Bootstrap Percolation process on a Random Geometric Graph. Random Geometric Graphs provide a simplified abstract model of spatial networks, and are particularly suitable to describe wireless ad-hoc networks.More precisely, a Random Geometric Graph is obtained by choosing n vertices uniformly at random from the unit d-dimensional cube or torus, and joining any two vertices by an edge if they are within a certain distance, r, from each other. Until now, very little was known about Bootstrap Percolation on Random Geometric Graphs, other than some initial results in a paper by Bradonjić and Saniee (2012).We obtain precise results that characterize the final state of the Bootstrap Percolation process in terms of the parameters p and r asymptotically almost surely as the number n of vertices tends to infinity.We show that, a.a.s., the process is either stationary from the very beginning (i.e. no inactive vertex ever changes to active) or almost all vertices eventually become active. Moreover, we prove that in the latter case the only obstacle to achieve full percolation is the presence of vertices of degree less than k. Indeed, as soon as r is large enough to guarantee that the minimum degree is at least k, a.a.s., the process is either stationary or attains percolation.Finally, we study a version of the model with a restricted focus of infection (i.e. active points can initially occur in a small region of the torus), and obtain analogous results for that case.
Subject Area
Mathematics
Recommended Citation
Whittemore, Alyssa, "Bootstrap Percolation on Random Geometric Graphs" (2021). ETD collection for University of Nebraska-Lincoln. AAI28713105.
https://digitalcommons.unl.edu/dissertations/AAI28713105