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Bootstrap percolation on geometric inhomogeneous random graphs [article]

Christoph Koch, Johannes Lengler
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.  ... 
arXiv:1603.02057v3 fatcat:tqmel5pwozaw3ephumaf4yyhkm

Bootstrap Percolation on Geometric Inhomogeneous Random Graphs *

Christoph Koch, Johannes Lengler
unpublished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  ... 
fatcat:avpdleneffevrbtngktfnbxoci

Bootstrap percolation in random geometric graphs [article]

Victor Falgas-Ravry, Amites Sarkar
2021 arXiv   pre-print
Following Bradonjić and Saniee, we study a model of 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  ... 
arXiv:2110.12166v1 fatcat:ja27g6o2brek5ggj6sku7vu53m

Selective Bootstrap Percolation [article]

Mauro Sellitto
2019 arXiv   pre-print
A new class of 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 .  ... 
arXiv:1911.01674v1 fatcat:frvbpio4ine4pkt5casfpoirry

Normalization Phenomena in Asynchronous Networks [chapter]

Amin Karbasi, Johannes Lengler, Angelika Steger
2015 Lecture Notes in Computer Science  
This includes in particular 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] .  ... 
doi:10.1007/978-3-662-47666-6_55 fatcat:dl5a6amsznhoxorugfkx7jwxwu

Percolation since Saint-Flour [article]

Geoffrey R. Grimmett, Harry Kesten
2012 arXiv   pre-print
This is a short survey of work 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.  ... 
arXiv:1207.0373v1 fatcat:qkilkcc4zre6bjrbjr67aq36au

Cooperative heterogeneous facilitation: Multiple glassy states and glass-glass transition

Mauro Sellitto
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]  ... 
doi:10.1103/physreve.86.030502 pmid:23030856 fatcat:sbo2h2pkxbdjjbhmt3loh5q54m

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

Ana P. Millán, Giacomo Gori, Federico Battiston, Tilman Enss, Nicolò Defenu
2020 arXiv   pre-print
We propose our model as a tool to probe universal behaviour 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.  ... 
arXiv:2006.10421v2 fatcat:ihousv2fg5edfbqtfxredmeaoi

A modified bootstrap percolation on a random graph coupled with a lattice

Svante Janson, Robert Kozma, Miklós Ruszinkó, Yury Sokolov
2019 Discrete Applied Mathematics  
Moreover, we consider non-monotonous 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] .  ... 
doi:10.1016/j.dam.2018.11.006 fatcat:k5r7wqy47faxlacugbdfywktuy

Strong-majority bootstrap percolation on regular graphs with low dissemination threshold [article]

Dieter Mitsche, Xavier Pérez-Giménez, Paweł Prałat
2015 arXiv   pre-print
Consider the following model of strong-majority 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] .  ... 
arXiv:1503.08310v1 fatcat:yd3kccw5xbhn5ez3mlvthyefme

High-dimensional bootstrap processes in evolving simplicial complexes [article]

Nikolaos Fountoulakis, Michał Przykucki
2019 arXiv   pre-print
We study 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] .  ... 
arXiv:1910.10139v1 fatcat:sp5k6uyp5neenbxk5a5wkhalqa

Bootstrap percolation and the geometry of complex networks [article]

Elisabetta Candellero, Nikolaos Fountoulakis
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  ... 
arXiv:1412.1301v2 fatcat:bqkmlb6x7val7mygpfrc5zlhpu

Bootstrap percolation and the geometry of complex networks

Elisabetta Candellero, Nikolaos Fountoulakis
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  ... 
doi:10.1016/j.spa.2015.08.005 fatcat:yuuorjomgnf3jox3ibb6h5wjta

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

Janko Gravner, David Sivakoff
2018 arXiv   pre-print
We also establish a gradual transition for 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] .  ... 
arXiv:1807.10323v1 fatcat:rpevadkkxnfyrnidix2lt3frai

Bootstrap Percolation in Directed Inhomogeneous Random Graphs

Thilo Meyer-Brandis, Nils Detering, Konstantinos Panagiotou
2019 Electronic Journal of Combinatorics  
We perform a thorough analysis of 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  ... 
doi:10.37236/7832 fatcat:5a23caquozeo7kyfcmz3gwof6q
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