Janson, Svante and Kozma, Róbert and Ruszinkó, Miklós and Sokolov, Yury (2016) Bootstrap percolation on a random graph coupled with a lattice. ELECTRONIC JOURNAL OF COMBINATORICS. ISSN 10971440 (In Press)

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Abstract
In this paper a random graph model $G_{\mathbb{Z}^2_N,p_d}$ is introduced, which is a combination of fixed torus grid edges in $(\mathbb{Z}/N \mathbb{Z})^2$ and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices $u,v\in(\mathbb{Z}/N \mathbb{Z})^2$ with graph distance $d$ on the torus grid is $p_d=c/Nd$, where $c$ is some constant. We show that, {\em whp}, the diameter $D(G_{\mathbb{Z}^2_N,p_d})=\Theta (\log N)$. Moreover, we consider nonmonotonous bootstrap percolation on $G_{\mathbb{Z}^2_N,p_d}$. We prove the presence of phase transitions in meanfield approximation and provide fairly sharp bounds on the error of the critical parameters. Our model addresses interesting mathematical questions of nonmonotonous bootstrap percolation, and it is motivated by recent results of brain research.
Item Type:  Article 

Subjects:  Q Science / természettudomány > QA Mathematics / matematika 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  02 Jan 2017 15:20 
Last Modified:  02 Jan 2017 15:20 
URI:  http://real.mtak.hu/id/eprint/44124 
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