47 research outputs found
Contagious sets in a degree-proportional bootstrap percolation process
We study the following bootstrap percolation process: given a connected graph
, a constant and an initial set of
\emph{infected} vertices, at each step a vertex~ becomes infected if at
least a -proportion of its neighbours are already infected (once
infected, a vertex remains infected forever). Our focus is on the size
of a smallest initial set which is \emph{contagious}, meaning that
this process results in the infection of every vertex of .
Our main result states that every connected graph on vertices has
or (note that allowing the latter
possibility is necessary because of the case , as every
contagious set has size at least one). This is the best-possible bound of this
form, and improves on previous results of Chang and Lyuu and of Gentner and
Rautenbach. We also provide a stronger bound for graphs of girth at least five
and sufficiently small , which is asymptotically best-possible.Comment: 14 pages, 1 figur
Two Phase Transitions in Two-Way Bootstrap Percolation
Consider a graph G and an initial random configuration, where each node is black with probability p and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least r black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a d-dimensional torus and identify the threshold values
The sharp threshold for bootstrap percolation in all dimensions
In r-neighbour bootstrap percolation on a graph G, a (typically random) set A
of initially 'infected' vertices spreads by infecting (at each time step)
vertices with at least r already-infected neighbours. This process may be
viewed as a monotone version of the Glauber dynamics of the Ising model, and
has been extensively studied on the d-dimensional grid . The elements of
the set A are usually chosen independently, with some density p, and the main
question is to determine , the density at which percolation
(infection of the entire vertex set) becomes likely.
In this paper we prove, for every pair , that there is a
constant L(d,r) such that as , where denotes an r-times iterated
logarithm. We thus prove the existence of a sharp threshold for percolation in
any (fixed) number of dimensions. Moreover, we determine L(d,r) for every pair
(d,r).Comment: 37 pages, sketch of the proof added, to appear in Trans. of the AM