%0 Book Section %1 Bradonjic2014Bootstrap %A Bradonjić, Milan %A Saniee, Iraj %B 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO) %C Philadelphia, PA %D 2014 %E Sedgewick, Robert %E Ward, Mark D. %I Society for Industrial and Applied Mathematics %K percolation tree-graphs bootstrap %P 89--96 %R 10.1137/1.9781611973761.8 %T Bootstrap Percolation on Periodic Trees %U http://dx.doi.org/10.1137/1.9781611973761.8 %X We study bootstrap percolation with the threshold parameter \$2\$ and the initial probability \$p\$ on infinite periodic trees that are defined as follows. Each node of a tree has degree selected from a finite predefined set of non-negative integers and starting from any node, all nodes at the same graph distance from it have the same degree. We show the existence of the critical threshold \$p\_f(þeta) (0,1)\$ such that with high probability, (i) if \$p > p\_f(þeta)\$ then the periodic tree becomes fully active, while (ii) if \$p < p\_f(þeta)\$ then a periodic tree does not become fully active. We also derive a system of recurrence equations for the critical threshold \$p\_f(þeta)\$ and compute these numerically for a collection of periodic trees and various values of \$þeta\$, thus extending previous results for regular (homogeneous) trees. %@ 978-1-61197-376-1

@inbook{Bradonjic2014Bootstrap, abstract = {{We study bootstrap percolation with the threshold parameter \$\theta \geq 2\$ and the initial probability \$p\$ on infinite periodic trees that are defined as follows. Each node of a tree has degree selected from a finite predefined set of non-negative integers and starting from any node, all nodes at the same graph distance from it have the same degree. We show the existence of the critical threshold \$p\_f(\theta) \in (0,1)\$ such that with high probability, (i) if \$p > p\_f(\theta)\$ then the periodic tree becomes fully active, while (ii) if \$p < p\_f(\theta)\$ then a periodic tree does not become fully active. We also derive a system of recurrence equations for the critical threshold \$p\_f(\theta)\$ and compute these numerically for a collection of periodic trees and various values of \$\theta\$, thus extending previous results for regular (homogeneous) trees.}}, added-at = {2019-06-10T14:53:09.000+0200}, address = {Philadelphia, PA}, archiveprefix = {arXiv}, author = {Bradonji\'{c}, Milan and Saniee, Iraj}, biburl = {https://www.bibsonomy.org/bibtex/27c0cbf966985a88cf53bf139336b9cf8/nonancourt}, booktitle = {2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)}, citeulike-article-id = {12827284}, citeulike-linkout-0 = {http://dx.doi.org/10.1137/1.9781611973761.8}, citeulike-linkout-1 = {http://arxiv.org/abs/1311.7449}, citeulike-linkout-2 = {http://arxiv.org/pdf/1311.7449}, day = 22, doi = {10.1137/1.9781611973761.8}, editor = {Sedgewick, Robert and Ward, Mark D.}, eprint = {1311.7449}, interhash = {ce3c003c3242f18e552818df1f8a7389}, intrahash = {7c0cbf966985a88cf53bf139336b9cf8}, isbn = {978-1-61197-376-1}, keywords = {percolation tree-graphs bootstrap}, month = dec, pages = {89--96}, posted-at = {2013-12-11 14:47:13}, priority = {2}, publisher = {Society for Industrial and Applied Mathematics}, timestamp = {2019-08-23T10:58:50.000+0200}, title = {{Bootstrap Percolation on Periodic Trees}}, url = {http://dx.doi.org/10.1137/1.9781611973761.8}, year = 2014 }