Journal ArticleDOI
Bootstrap percolation on the random regular graph
József Balogh,Boris Pittel +1 more
TLDR
Here, thanks to a “principle of deferred decisions,” the percolation dynamics is described by a surprisingly simple Markov chain, which is replaced by a deterministic dynamical system, and its integrals are used to show—via exponential supermartingales—that thePercolation process undergoes relatively small fluctuations around the deterministic trajectory.Abstract:
The k-parameter bootstrap percolation on a graph is a model of an interacting particle system, which can also be viewed as a variant of a cellular automaton growth process with threshold k ≥ 2. At the start, each of the graph vertices is active with probability p and inactive with probability 1 − p, independently of other vertices. Presence of active vertices triggers a bootstrap percolation process controlled by a recursive rule: an active vertex remains active forever, and a currently inactive vertex becomes active when at least k of its neighbors are active. The basic problem is to identify, for a given graph, p− and p+ such that for p p+ resp.) the probability that all vertices are eventually active is very close to 0 (1 resp.). The bootstrap percolation process is a deterministic process on the space of subsets of the vertex set, which is easy to describe but hard to analyze rigorously in general. We study the percolation on the random d-regular graph, d ≥ 3, via analysis of the process on the multigraph counterpart of the graph. Here, thanks to a “principle of deferred decisions,” the percolation dynamics is described by a surprisingly simple Markov chain. Its generic state is formed by the counts of currently active and nonactive vertices having various degrees of activation capabilities. We replace the chain by a deterministic dynamical system, and use its integrals to show—via exponential supermartingales—that the percolation process undergoes relatively small fluctuations around the deterministic trajectory. This allows us to show existence of the phase transition within an interval [p−(n),p+(n)], such that (1) p±(n) p* = 1 − miny∈(0,1)y/ℙ(Bin(d − 1,1 − y) < k); (2) p+(n) − p−(n) is of order n−1/2 for k < d − 1, and n, (en 0,en log n ∞), for k = d − 1. Note that p* is the same as the critical probability of the process on the corresponding infinite regular tree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 30, 257–286, 2007read more
Citations
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Journal ArticleDOI
Recent advances in percolation theory and its applications
TL;DR: In this paper, a variety of percolation models have been introduced some of which have completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, dimensionality and the type of the underlying rules and networks.
Journal ArticleDOI
Recent advances in percolation theory and its applications
TL;DR: The basic features of the ordinary model are outlined and a glimpse at a number of selective variations and modifications of the original model are taken and a connection with the magnetic models is established.
Journal ArticleDOI
The sharp threshold for bootstrap percolation in all dimensions
TL;DR: In this paper, it was shown that there is a constant L(d,r) such that the density at which percolation becomes likely in any (fixed) number of dimensions.
Journal ArticleDOI
Resilience to contagion in financial networks
TL;DR: In this article, the authors derived rigorous asymptotic results for the magnitude of contagion in a large counterparty network and gave an analytical expression for the fraction of defaults, in terms of network characteristics, and showed that institutions which contribute most to network instability have both large connectivity and a large fraction of contagious links.
Journal ArticleDOI
Bootstrap percolation on complex networks
TL;DR: In networks with degree distributions whose second moment diverges (but whose first moment does not), the giant active component is robust to damage, and also is very easily activated, and a generalized bootstrap process is formulated in which each vertex can have an arbitrary threshold.
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