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Bootstrap percolation on geometric inhomogeneous random graphs
[article]

2020
*
arXiv
*
pre-print

*Geometric*

*inhomogeneous*

*random*

*graphs*(GIRGs) are a model for scale-free networks with underlying geometry. ... This is the first time that the role of geometry

*on*

*bootstrap*

*percolation*is analysed mathematically for

*geometric*scale-free networks. ... The notion of localised

*bootstrap*

*percolation*relies heavily

*on*a

*random*

*graph*model which has an underlying geometry. ...

##
###
Bootstrap Percolation on Geometric Inhomogeneous Random Graphs *

unpublished

*Geometric*

*inhomogeneous*

*random*

*graphs*(GIRGs) are a model for scale-free networks with underlying geometry. ... We study

*bootstrap*

*percolation*

*on*these

*graphs*, which is a process modelling the spread of an infection of vertices starting within a (small) local region. ... Pr [{u, v} ∈ E | w u , w v , x v ] = Θ min w u w v n , 1 . (4) 147:8

*Bootstrap*

*Percolation*

*on*

*Geometric*

*Inhomogeneous*

*Random*

*Graphs*Note in particular that the right hand side of (4) is independent of ...

##
###
Bootstrap percolation in random geometric graphs
[article]

2021
*
arXiv
*
pre-print

Following Bradonjić and Saniee, we study a model of

arXiv:2110.12166v1
fatcat:ja27g6o2brek5ggj6sku7vu53m
*bootstrap**percolation**on*the Gilbert*random**geometric**graph**on*the 2-dimensional torus. ... We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional*bootstrap**percolation*model*on*the circle. ... Acknowledgements Research*on*this project was done while the second author visited the first author in Umeå in Spring 2018 with financial support from STINT Initiation grant IB 2017-7360, which the authors ...##
###
Selective Bootstrap Percolation
[article]

2019
*
arXiv
*
pre-print

A new class of

arXiv:1911.01674v1
fatcat:frvbpio4ine4pkt5casfpoirry
*bootstrap**percolation*models in which particle culling occurs only for certain numbers of nearest neighbours is introduced and studied*on*a Bethe lattice. ... The idea immediately extends to facilitation dynamics, suggesting a simple way to construct*one*-component models of multiple glasses and glass-glass transitions as well as more general coarse-grained models ... Selective*Bootstrap**Percolation*In the more general selective*bootstrap**percolation*(SBP) we now introduce, S is an arbitrary subset of non-negative integers less than f . ...##
###
Normalization Phenomena in Asynchronous Networks
[chapter]

2015
*
Lecture Notes in Computer Science
*

This includes in particular

doi:10.1007/978-3-662-47666-6_55
fatcat:dl5a6amsznhoxorugfkx7jwxwu
*inhomogeneous**random*networks, as given by Chung-Lu*graphs*with degree parameter β ∈ (2, 3). ... tiny) set of active vertices, it only*percolates*to a certain level that depends only*on*the activation threshold and the ratio of inhibitory to excitatory vertices. ...*Bootstrap**percolation*has also been studied*on*a variety of*graphs*such as trees [5] ,*random*regular*graphs*[6] , Erdős-Rényi*graphs*[17] , and power-law*graphs*[2] . ...##
###
Percolation since Saint-Flour
[article]

2012
*
arXiv
*
pre-print

This is a short survey of work

arXiv:1207.0373v1
fatcat:qkilkcc4zre6bjrbjr67aq36au
*on**percolation*and first-passage*percolation*since the publication (in 1996 and 1984, respectively) of the two authors' Saint-Flour notes*on*these topics. ... Acknowledgements We thank Raphaël Cerf, Michael Damron, Christophe Garban, and Chuck Newman for their comments*on*a draft of this article. ...*Percolation**on*non-amenable*graphs*. In the last 20 years, there has been a good deal of interest in*random*processes*on**graphs*whose surface/volume ratio does not tend to zero. ...##
###
Cooperative heterogeneous facilitation: Multiple glassy states and glass-glass transition

2012
*
Physical Review E
*

*on*Bethe lattices,

*random*

*graphs*and complex networks. ... The formal structure of glass singularities in the mode-coupling theory (MCT) of supercooled liquids dynamics is closely related to that appearing in the analysis of heterogeneous

*bootstrap*

*percolation*... The existence of two transitions in

*bootstrap*

*percolation*was first discovered by Fontes and Schonmann [26] in homogeneous trees and then found in Erdős-Rényi

*graphs*and complex networks in [27, 28] ...

##
###
Complex networks with tuneable dimensions as a universality playground
[article]

2020
*
arXiv
*
pre-print

We propose our model as a tool to probe universal behaviour

arXiv:2006.10421v2
fatcat:ihousv2fg5edfbqtfxredmeaoi
*on**inhomogeneous*structures and comment*on*the possibility that the universal behaviour of correlated models*on*such networks mimics the*one*... However, for interacting*inhomogeneous*systems described by complex networks a clear understanding of the relevant parameters for universality is still missing. ... of*inhomogeneous**graphs*. ...##
###
A modified bootstrap percolation on a random graph coupled with a lattice

2019
*
Discrete Applied Mathematics
*

Moreover, we consider non-monotonous

doi:10.1016/j.dam.2018.11.006
fatcat:k5r7wqy47faxlacugbdfywktuy
*bootstrap**percolation**on*G_Z^2_N,p_d. ... In this paper a*random**graph*model G_Z^2_N,p_d is introduced, which is a combination of fixed torus grid edges in (Z/N Z)^2 and some additional*random**ones*. ... introduction*Bootstrap**percolation**on*lattices has been extensively investigated in the last decades, and recently it has been considered*on*the classical Erdős-Rényi*random**graph*G n,p as well [14] . ...##
###
Strong-majority bootstrap percolation on regular graphs with low dissemination threshold
[article]

2015
*
arXiv
*
pre-print

Consider the following model of strong-majority

arXiv:1503.08310v1
fatcat:yd3kccw5xbhn5ez3mlvthyefme
*bootstrap**percolation**on*a*graph*. Let r be some positive integer, and p in [0,1]. ... Given any arbitrarily small p>0 and any integer r, we construct a family of d=d(p,r)-regular*graphs*such that with high probability all vertices become active in the end. ... Finally, extensions to*inhomogeneous**random**graphs*were considered by Amini, Fountoulakis and Panagiotou in [3] . ...##
###
High-dimensional bootstrap processes in evolving simplicial complexes
[article]

2019
*
arXiv
*
pre-print

We study

arXiv:1910.10139v1
fatcat:sp5k6uyp5neenbxk5a5wkhalqa
*bootstrap**percolation*processes*on**random*simplicial complexes of some fixed dimension d ≥ 3. ... We introduce new vertices*one*by*one*, all equipped with a*random*weight from a fixed distribution μ. ... In a different context, such a phenomenon was also shown in*inhomogeneous**random**graphs*[3, 18] as well as in*random**graphs**on*the hyperbolic plane [12] . ...##
###
Bootstrap percolation and the geometry of complex networks
[article]

2015
*
arXiv
*
pre-print

*On*a

*geometric*model for complex networks (introduced by Krioukov et al.) we investigate the

*bootstrap*

*percolation*process. ... This model consists of

*random*

*geometric*

*graphs*

*on*the hyperbolic plane having N vertices, a dependent version of the Chung-Lu model. The process starts with infection rate p=p(N). ...

*Random*

*geometric*

*graphs*

*on*the hyperbolic plane and

*inhomogeneous*

*random*

*graphs*The most common representations of the hyperbolic plane are the upper-half plane representation {z = x + iy : y > 0} as well ...

##
###
Bootstrap percolation and the geometry of complex networks

2016
*
Stochastic Processes and their Applications
*

*On*a

*geometric*model for complex networks (introduced by Krioukov et al.) we investigate the

*bootstrap*

*percolation*process. ... This model consists of

*random*

*geometric*

*graphs*

*on*the hyperbolic plane having N vertices, a dependent version of the Chung-Lu model. The process starts with infection rate p = p(N ). ...

*Random*

*geometric*

*graphs*

*on*the hyperbolic plane and

*inhomogeneous*

*random*

*graphs*The most common representations of the hyperbolic plane are the upper-half plane representation {z = x + iy : y > 0} as well ...

##
###
Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square
[article]

2018
*
arXiv
*
pre-print

We also establish a gradual transition for

arXiv:1807.10323v1
fatcat:rpevadkkxnfyrnidix2lt3frai
*bootstrap**percolation**on*Z^2× K_n. The main tool is heterogeneous*bootstrap**percolation**on*Z^2. ... The initially occupied set is*random*, given by a uniform product measure with a low density p. Our main focus is*on*this process*on*the product*graph*Z^2× K_n^2, where K_n is a complete*graph*. ...*on**inhomogeneous**geometric**random**graphs*[KL] . ...##
###
Bootstrap Percolation in Directed Inhomogeneous Random Graphs

2019
*
Electronic Journal of Combinatorics
*

We perform a thorough analysis of

doi:10.37236/7832
fatcat:5a23caquozeo7kyfcmz3gwof6q
*bootstrap**percolation**on*a novel model of directed and*inhomogeneous**random**graphs*, where the distribution of the edges is specied by assigning two distinct weights to ...*Bootstrap**percolation*is a process that is used to describe the spread of an infection*on*a given*graph*. In the model considered here each vertex is equipped with an individual threshold. ...*Bootstrap**Percolation*for Finitary Vertex Type Sequences In this section we study*bootstrap**percolation*in directed*inhomogeneous**random**graphs*with so-called finitary vertex sequences that are defined ...
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