Random disease on the square grid
proceedings of the eighth international conference on Random structures and algorithms
Lamplighters, Diestel–Leader Graphs, Random Walks, and Harmonic Functions
Combinatorics, Probability and Computing
Majority bootstrap percolation on the hypercube
Combinatorics, Probability and Computing
Bootstrap percolation in high dimensions
Combinatorics, Probability and Computing
Triggering cascades on undirected connected graphs
Information Processing Letters
Random Structures & Algorithms
Bounding the sizes of dynamic monopolies and convergent sets for threshold-based cascades
Theoretical Computer Science
What i tell you three times is true: bootstrap percolation in small worlds
WINE'12 Proceedings of the 8th international conference on Internet and Network Economics
Complex contagion and the weakness of long ties in social networks: revisited
Proceedings of the fourteenth ACM conference on Electronic commerce
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Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability $p$, independently of each other, and a deterministic spreading rule with a fixed parameter $k$: if a vacant site has at least $k$ occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on ${\mathbb Z}^d$; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of $p$ for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than $k$, then the critical probability is 1, while it is $1-1/k$ on the $k$-ary tree. A related result is that in any rooted tree $T$ there is a way of erasing $k$ children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree $T'$ has branching number $\mbox{\rm br}(T')\leq \max\{\mbox{\rm br}(T)-k,\,0\}$. We also prove that on any $2k$-regular non-amenable graph, the critical probability for the $k$-rule is strictly positive.