Related papers: Optimal and Near-Optimal Constructions for Bootstr…
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 $r$…
In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…
Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots,…
Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of…
In this paper we investigate the critical probability $p_c(Q_n,r)$ for bootstrap percolation with the infection threshold $r$ on the $n$-dimensional hypercube $Q_n$ with vertex set $V(Q_n)=\{0,1\}^n$ and edges connecting the pairs at…
On a geometric model for complex networks (introduced by Krioukov et al.) we investigate the bootstrap percolation process. This model consists of random geometric graphs on the hyperbolic plane having $N$ vertices, a dependent version of…
The $r$-edge bootstrap percolation on a graph is an activation process of the edges. The process starts with some initially activated edges and then, in each round, any inactive edge whose one of endpoints is incident to at least $r$ active…
In this paper, we study the k-neighbor bootstrap percolation process on the d-dimensional grid [n]^d, and show that the minimum number of initial vertices that percolate is (1-d/k)n^d + O(n^{d-1})$ when d<=k<=2d. This confirms a conjecture…
Following Bradonji\'c and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the $2$-dimensional torus. In this model, the expected number of vertices of the graph is $n$, and the expected degree of a…
Majority bootstrap percolation on a graph $G$ is an epidemic process defined in the following manner. Firstly, an initially infected set of vertices is selected. Then step by step the vertices that have more infected than non-infected…
We investigate bootstrap percolation with infection threshold $r> 1$ on the binomial $k$-uniform random hypergraph $H_k(n,p)$ in the regime $n^{-1}\ll n^{k-2}p \ll n^{-1/r}$, when the initial set of infected vertices is chosen uniformly at…
Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of…
The $r$-neighbour bootstrap process is an update rule for the states of vertices in which `uninfected' vertices with at least $r$ `infected' neighbours become infected and a set of initially infected vertices is said to \emph{percolate} if…
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…
Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,3\}$,…
We study the following bootstrap percolation process: given a connected graph $G$, a constant $\rho \in [0, 1]$ and an initial set $A \subseteq V(G)$ of \emph{infected} vertices, at each step a vertex~$v$ becomes infected if at least a…
We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori…
Bootstrap Percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least $r$ previously infected vertices. If…
Metastability thresholds lie at the heart of bootstrap percolation theory. Yet proving precise lower bounds is notoriously hard. We show that for two of the most classical models, two-neighbour and Frob\"ose, upper bounds are sharp to…
Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollob\'as in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph…