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August 2020 The cone percolation model on Galton–Watson and on spherically symmetric trees
Valdivino V. Junior, Fábio P. Machado, Krishnamurthi Ravishankar
Braz. J. Probab. Stat. 34(3): 594-612 (August 2020). DOI: 10.1214/19-BJPS441

Abstract

We study a rumor model from a percolation theory and branching process point of view. It is defined according to the following rules: (1) at time zero, only the root (a fixed vertex of the tree) is declared informed, (2) at time $n+1$, an ignorant vertex gets the information if it is, at a graph distance, at most $R_{v}$ of some its ancestral vertex $v$, previously informed. We present relevant lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual. We work with (homogeneous and non-homogeneous) Galton–Watson branching trees and spherically symmetric trees which includes homogeneous and $k$-periodic trees. We also present bounds for the expected size of the connected component in the subcritical case for homogeneous trees and homogeneous Galton–Watson branching trees.

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Valdivino V. Junior. Fábio P. Machado. Krishnamurthi Ravishankar. "The cone percolation model on Galton–Watson and on spherically symmetric trees." Braz. J. Probab. Stat. 34 (3) 594 - 612, August 2020. https://doi.org/10.1214/19-BJPS441

Information

Received: 1 October 2018; Accepted: 1 April 2019; Published: August 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07232914
MathSciNet: MR4124542
Digital Object Identifier: 10.1214/19-BJPS441

Rights: Copyright © 2020 Brazilian Statistical Association

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Vol.34 • No. 3 • August 2020
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