相关论文: Minimal percolating sets in bootstrap percolation
The $r$-neighbour bootstrap process describes an infection process on a graph, where we start with a set of initially infected vertices and an uninfected vertex becomes infected as soon as it has $r$ infected neighbours. An inital set of…
In this paper we fill in a fundamental gap in the extremal bootstrap percolation literature, by providing the first proof of the fact that for all $d \geq 1$, the size of the smallest percolating sets in $d$-neighbour bootstrap percolation…
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…
We improve the status of the problem of determining minimum-sized percolating sets in $a \times b \times c$ grids under the $3$-neighbour process. Using several new constructions, we show that optimal percolating sets exist whenever…
For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts with a set of infected vertices and, in each time step, every vertex with at least $r$ infected neighbours becomes infected. The initial infection percolates if every…
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…
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…
In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…
The $r$-bond bootstrap percolation process on a graph $G$ begins with a set $S$ of infected edges of $G$ (all other edges are healthy). At each step, a healthy edge becomes infected if at least one of its endpoints is incident with at least…
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…
We consider a classic model known as bootstrap percolation on the $n \times n$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has…
The $r$-neighbor bootstrap percolation is a graph infection process based on the update rule by which a vertex with $r$ infected neighbors becomes infected. We say that an initial set of infected vertices propagates if all vertices of a…
This paper is dedicated to the study of the interaction between dynamical systems and percolation models, with views towards the study of viral infections whose virus mutate with time. Recall that r-bootstrap percolation describes a…
For any integer $r\geqslant0$, the $r$-neighbor bootstrap percolation on a graph is an activation process of the vertices. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least…
We study a new geometric bootstrap percolation model, line percolation, on the $d$-dimensional integer grid $[n]^d$. In line percolation with infection parameter $r$, infection spreads from a subset $A\subset [n]^d$ of initially infected…
We consider the $r$-neighbor bootstrap percolation process on the graph with vertex set $V=\{0,1\}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots,k$, where $k\ge 2$. We find asymptotics of the critical probability of…
We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r…
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…
In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain…