Bootstrap percolation

Adler, Joan

doi:10.1016/0378-4371(91)90295-n

Cited by 232 publications

(221 citation statements)

References 33 publications

Supporting

Mentioning

216

Contrasting

Order By: Relevance

“…In Bootstrap Percolation (BP) [1] the lattice is occupied randomly with probability p, and then all sites that do not have at least m neighbours are iteratively removed. For m = 1 isolated sites are removed and for m = 2 both isolated sites and dangling ends are removed.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bootstrap Percolation: visualizations and applications

Adler¹,

Lev²

2003

Braz. J. Phys.

Self Cite

209

View full text Add to dashboard Cite

Bootstrap percolation models describe systems as diverse as magnetic materials, fluid flow in rocks and computer storage systems. The models have a common feature of requiring not just a simple connectivity of neighbouring sites, but rather an environment of other suitably occupied sites. Different applications as well as the connection with the mathematical literature on these models is presented. Visualizations that show the compact nature of the clusters are provided. I IntroductionIn Bootstrap Percolation (BP) [1] the lattice is occupied randomly with probability p, and then all sites that do not have at least m neighbours are iteratively removed. For m = 1 isolated sites are removed and for m = 2 both isolated sites and dangling ends are removed. For m = 3 and above the situation becomes interesting and lattice dependent and is currently quite hot amongst probabilists. For sufficiently high m, no occupied sites remain for an infinite lattice, and the interest is in the scaling as a function of system size.The conference talk on which this manuscript is based came about as follows: Uri Lev asked for a project topic just as Joan Adler was thinking about BP because Scott Kirpatrick's [2] new application to computer memory storage had reminded her that BP was an interesting topic. (Uri Lev's visualizations and website [3] are described in the final section of this paper.) Other recent activity, also motivated from ref [2] The models have connections to rigidity percolation, [11]. Time developments according to the local rules of Bootstrap Percolation can be viewed as cellular automata [12] , and there are also metastable bootstrap models, not considered further here. Models closely related to canonical BP have been proposed as descriptions of fluid flow in porous media. In these cases crack development [13] and/or advance of the fluid front [14] are assumed to be strongly dependent on the local microstructure. II Applications of Bootstrap PercolationWe show here the importance of the bootstrap principle for several applications. Adler, Palmer and Meyer, [10] showed that environmental effects of a bootstrap type play a crucial role in the orientational ordering process of quadrupoles in solid molecularOrtho hydrogen and para deuterium are quadrupolar molecules, whereas para hydrogen and or- The recent application by Kirkpatrick [2] is of a completely different nature. It relates to the failure of units in a cubic structured collection of computer memories, called a Dense Storage Array. Because it is too expensive to fix individual memory units that fail in such a system, the units are allowed to "fail in place", and it is necessary to ensure that even with a relatively high density of failed units there will still be percolation of data communications. III VisualizationsSince the Computational Physics Group at the Technion developed animation software (AViz) for atomistic simulations it was interesting to apply this also to Bootstrap Percolation. AViz [16] enables a user to draw and animate atoms (or spins, or...

show abstract

Section: Introductionmentioning

confidence: 99%

“…Other Brazilians who have worked on BP include: N. S. Branco and C. J. Silva [7], V. Sidovarius, R. Sanchis and F. Camis. However the main reason to select this topic is that BP accounts for three of Adler's eleven joint papers with Stauffer, and Stauffer made her write a mini-review about BP [1] some time ago.…”

Section: Introductionmentioning

confidence: 99%

Bootstrap Percolation: visualizations and applications

Adler¹,

Lev²

2003

Braz. J. Phys.

Self Cite

209

View full text Add to dashboard Cite

show abstract

“…But when regarded more generally as a change of state-not just a decision-the model belongs to a larger class of contagion problems that includes models of failures in engineered systems such as power transmission networks (8) or the internet (19,20), epidemiological (21) and percolation (22,23) models of disease spreading, and a multiplicity of cellular-automata models including random-field Ising models (24), bootstrap percolation (25,26), majority voting (27,28), spreading activation (29), and self-organized criticality (8,29).…”

Section: Model Specificationmentioning

confidence: 99%

A simple model of global cascades on random networks

Watts¹

2011

The Structure and Dynamics of Networks

View full text Add to dashboard Cite

The origin of large but rare cascades that are triggered by small initial shocks is a phenomenon that manifests itself as diversely as cultural fads, collective action, the diffusion of norms and innovations, and cascading failures in infrastructure and organizational networks. This paper presents a possible explanation of this phenomenon in terms of a sparse, random network of interacting agents whose decisions are determined by the actions of their neighbors according to a simple threshold rule. Two regimes are identified in which the network is susceptible to very large cascades-herein called global cascadesthat occur very rarely. When cascade propagation is limited by the connectivity of the network, a power law distribution of cascade sizes is observed, analogous to the cluster size distribution in standard percolation theory and avalanches in self-organized criticality. But when the network is highly connected, cascade propagation is limited instead by the local stability of the nodes themselves, and the size distribution of cascades is bimodal, implying a more extreme kind of instability that is correspondingly harder to anticipate. In the first regime, where the distribution of network neighbors is highly skewed, it is found that the most connected nodes are far more likely than average nodes to trigger cascades, but not in the second regime. Finally, it is shown that heterogeneity plays an ambiguous role in determining a system's stability: increasingly heterogeneous thresholds make the system more vulnerable to global cascades; but an increasingly heterogeneous degree distribution makes it less vulnerable.

show abstract

“…Moreover, when R init is such that phases D and F overlap, we witness a case of bootstrap-percolation [16] (upon a square lattice). This has been vastly studied, and an approximation for a finite lattice of LxL individuals yields [18] …”

Section: Terminal Phases Of a Revolutionmentioning

confidence: 99%

“…The explanation is that for a square lattice and a sufficiently large fraction of Z 2 , conditions allowing phase D would result in a total revolution following the dynamics of a bootstrap-percolation [16], [17], [18] rather than our more specific revolution/ percolation model. For a random graph, which has a different geographical representation of clusters than a square lattice, this is not the compulsory behavior of the network.…”

Section: Terminal Phases Of a Revolutionmentioning

confidence: 99%