Lower bounds for bootstrap percolation on Galton-Watson trees

In the $r$-neighbour bootstrap process on a graph $G$, vertices are infected (in each time step) if they have at least $r$ already-infected neighbours. Motivated by its close connections to models from statistical physics, such as the Ising model of ferromagnetism, and kinetically constrained spin models of the liquid-glass transition, the most extensively-studied case is the two-neighbour bootstrap process on the two-dimensional grid $[n]^2$. Around 15 years ago, in a major ...

By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they have at least $r\in\mathbb{N}$ previously infected neighbors. When the set $A$ is chosen at random, the main aim is to determine the critical probability $p_c(G,r)$ at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been ext...

We investigate the behaviour of $r$-neighbourhood bootstrap percolation on the binomial $k$-uniform random hypergraph $H_k(n,p)$ for given integers $k\geq 2$ and $r\geq 2$. In $r$-neighbourhood bootstrap percolation, infection spreads through the hypergraph, starting from a set of initially infected vertices, and in each subsequent step of the process every vertex with at least $r$ infected neighbours becomes infected. For our analysis the set of initially infected vertices i...

We obtain new lower bounds on the critical points for various models of oriented percolation. The method is to provide a stochastic domination of the percolation processes by multitype Galton-Watson trees. This can be apply to the classical bond and site oriented percolation on Z^2 , but also on other lattices such as inhomogeneous ones, and on dimension three.

In r-neighbour bootstrap percolation on the vertex set of a graph G, vertices are initially infected independently with some probability p. At each time step, the infected set expands by infecting all uninfected vertices that have at least r infected neighbours. We study the distribution of the time t at which all vertices become infected. Given t = t(n) = o(log n/log log n), we prove a sharp threshold result for the probability that percolation occurs by time t in d-neighbou...

The $r$-neighbour bootstrap percolation process on a graph $G$ starts with an initial set $A_0$ of "infected" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ eventually becomes infected, then we say that $A_0$ percolates. We prove a conjecture of Balogh and Bollob\'as which says that, for fixed $r$ and $d\to\infty$, ev...

Bootstrap percolation has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of collective action and cultural fads in human societies. It is defined on an (arbitrary) network of interacting agents whose state is determined by the state of their neighbors according to a threshold rule. In a typical setting, bootstrap per...

In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid $[n]^d$. The elements of the set A are usually chosen independently, with some density p, and the main question is to dete...

For fixed $r\geq 2$, we consider bootstrap percolation with threshold $r$ on the Erd\H{o}s-R\'enyi graph ${\cal G}_{n,p}$. We identify a threshold for $p$ above which there is with high probability a set of size $r$ which can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants. As an application of our results, we also obtain an upper bound for the threshold for $K_4$-boots...

In this paper we focus on $r$-neighbor bootstrap percolation, which is a process on a graph where initially a set $A_0$ of vertices gets infected. Now subsequently, an uninfected vertex becomes infected if it is adjacent to at least $r$ infected vertices. Call $A_f$ the set of vertices that is infected after the process stops. More formally set $A_t:=A_{t-1}\cup \{v\in V: |N(v)\cap A_{t-1}|\geq r\}$, where $N(v)$ is the neighborhood of $v$. Then $A_f=\bigcup_{t>0} A_t$. We de...