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Jigsaw percolation on random hypergraphs [article]

Béla Bollobás, Oliver Cooley, Mihyun Kang, Christoph Koch
2017 arXiv   pre-print
The jigsaw percolation process on graphs was introduced by Brummitt, Chatterjee, Dey, and Sivakoff as a model of collaborative solutions of puzzles in social networks.  ...  In particular, we determine the asymptotic order of the critical threshold probability for percolation when both hypergraphs are chosen binomially at random.  ...  Jigsaw percolation is a deterministic process on clusters of vertices that evolves in discrete time.  ... 
arXiv:1603.07883v2 fatcat:cbdep4l6fbhhxigq6hreul7k74

Multi-coloured jigsaw percolation on random graphs [article]

Oliver Cooley, Abraham Gutiérrez
2017 arXiv   pre-print
We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities.  ...  The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle.  ...  Acknowledgements We would like to thank Christoph Koch for his helpful comments on an earlier version of this paper, including pointing out the simple proof of Claim 12.  ... 
arXiv:1712.00992v1 fatcat:dop33tah55bbrdirmzq5au7e7a

The sharp threshold for jigsaw percolation in random graphs [article]

Oliver Cooley, Tobias Kapetanopoulos, Tamás Makai
2019 arXiv   pre-print
We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are 'jointly connected'.  ...  Bollobás, Riordan, Slivken and Smith proved that when the two graphs are independent binomial random graphs, whether the jigsaw process percolates undergoes a phase transition when the product of the two  ...  Bollobás, Cooley, Kang and Koch [3] proved a generalisation to k-uniform hypergraphs and a jigsaw percolation process on the j-sets for each 1 ≤ j ≤ k − 1.  ... 
arXiv:1809.01907v2 fatcat:5su4yahvrfd3neaic26gxjsb5u

The Size of the Giant Joint Component in a Binomial Random Double Graph

Mark Jerrum, Tamás Makai
2021 Electronic Journal of Combinatorics  
We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices.  ...  A superficially similar percolation model is jigsaw percolation introduced by Brummitt, Chatterjee, Dey and Sivakoff [4] .  ...  In addition, extensions to multi-coloured random graphs [6] and random hypergraphs [2] exist.  ... 
doi:10.37236/8846 fatcat:5wuz7r6nn5cp7pcrqjgulakbsu

A Network Model of Interpersonal Alignment in Dialog

Alexander Mehler, Andy Lücking, Petra Weiß
2010 Entropy  
Since the linguistic levels are interconnected, alignment is, according to the IAM, supposed to percolate through these levels.  ...  level and on the level of situation models [9] .  ...  series from random ones.  ... 
doi:10.3390/e12061440 fatcat:maz7k53wtrhadn64ta7dssp4lu

Bivariate fluctuations for the number of arithmetic progressions in random sets [article]

Yacine Barhoumi-Andréani, Christoph Koch, Hong Liu
2019 arXiv   pre-print
Given p∈[0,1] we denote by [n]_p the random subset of [n] which includes every number with probability p, independently of one another.  ...  The focus lies on sparse random subsets, i.e. when p=p(n)=o(1) as n→+∞. Let X_ℓ denote the number of distinct arithmetic progressions of length ℓ which are contained in [n]_p.  ...  on random hypergraphs.  ... 
arXiv:1902.04176v1 fatcat:t2l6yapypbdnpp3rmxrorvdtdy