Bootstrap percolation on Galton–Watson trees

B. Bollobas, K. Gunderson, C. Janson, M. Przykucki

Electronic Journal of Probability 19, 27 (2014)

#percolation#graphtheory#cellularautomata

Download the PDF

LQ placeholderIn bootstrap percolation, a node is infected when a sufficient number of neighbours are infected.

In bootstrap percolation, a node is infected when a sufficient number of neighbours are infected.

Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number r, the r-neighbour bootstrap process is an update rule for vertices of a graph in one of two states: ‘infected’ or ‘healthy’. In consecutive rounds, each healthy vertex with at least r infected neighbours becomes itself infected. Percolation is said to occur if every vertex is eventually infected. Usually, the starting set of infected vertices is chosen at random, with all vertices initially infected independently with probability p. In that case, given a graph G and infection threshold r, a quantity of interest is the critical probability, pc(G, r), at which percolation becomes likely to occur. In this paper, we look at infinite trees and, answering a problem posed by Balogh, Peres and Pete, we show that for any b ≥ r and for any  > 0 there exists a tree T with branching number br(T) = b and critical probability pc(T, r) < . However, this is false if we limit ourselves to the well-studied family of Galton–Watson trees. We show that for every r ≥ 2 there exists a constant cr > 0 such that if T is a Galton–Watson tree with branching number br(T) = b ≥ r then pc > f(b,cr, r). We also show that this bound is sharp up to a factor of O(b) by giving an explicit family of Galton–Watson trees with critical probability bounded from above by some constant Cr > 0.

LQ placeholder

Degree-correlations in a bursting dynamic network model

F. Vanni, P. Barucca

Journal of Economic Interaction and Coordination

LQ placeholder

Scale of non-locality for a system of n particles

S. Talaganis, I. Teimouri

Sub. to Physical Review D

LQ placeholder

Changes to Gate Closure and its impact on wholesale electricity prices: The case of the UK

A. Facchini, A. Rubino, G. Caldarelli, G. Liddo

Energy Policy

LQ placeholder

How much can we influence the rate of innovation?

T. Fink, M. Reeves

Science Advances

LQ placeholder

The statistical physics of real-world networks

G. Cimini, T. Squartini, F. Saracco, D. Garlaschelli, A. Gabrielli, G. Caldarelli

Nature Reviews Physics

LQ placeholder

PopRank: Ranking pages’ impact and users’ engagement on Facebook

A. Zaccaria, M. Vicario, W. Quattrociocchi, A. Scala, L. Pietronero

PLoS ONE

128 / 128 papers