Abstract
We investigate the natural situation of the dissemination of information on various graph classes starting with a random set of informed vertices called active. Initially active vertices are chosen independently with probability p, and at any stage in the process, a vertex becomes active if the majority of its neighbours are active, and thereafter never changes its state. This process is a particular case of bootstrap percolation. We show that in any cubic graph, with high probability, the information will not spread to all vertices in the graph if \(p<\frac{1}{2}\) . We give families of graphs in which information spreads to all vertices with high probability for relatively small values of p.
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Authors acknowledge the support of CONICYT via Anillo en Redes ACT08 (I.R., K.S.), Fondecyt 1090156 (I.R.), ECOS-CONICYT (I.R., I.T.), Fondap on Applied Mathematics (I.R.), French ANR projects STAL-DEC-OPT and ALADDIN (I.T.) and an Alfred P. Sloan Fellowship (J.V.).
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Rapaport, I., Suchan, K., Todinca, I. et al. On Dissemination Thresholds in Regular and Irregular Graph Classes. Algorithmica 59, 16–34 (2011). https://doi.org/10.1007/s00453-009-9309-0
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DOI: https://doi.org/10.1007/s00453-009-9309-0