Bootstrap percolation on infinite trees and non-amenable groups

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...

Bootstrap percolation is an often used model to study the spread of diseases, rumors, and information on sparse random graphs. The percolation process demonstrates a critical value such that the graph is either almost completely affected or almost completely unaffected based on the initial seed being larger or smaller than the critical value. To analyze intervention strategies we provide the first analytic determination of the critical value for basic bootstrap percolation ...

Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with probability $p$ between any pair of vertices. Bootstrap percolation on a random graph is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have ...

Consider independent bond percolation with retention probability p on a spherically symmetric tree Gamma. Write theta_Gamma(p) for the probability that the root is in an infinite open cluster, and define the critical value p_c=inf{p:theta_Gamma(p)>0}. If theta_Gamma(p_c)=0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p_c, as demonstrated recently by Haggstrom, Peres and Steif. Here we relate this phenomenon to the...

K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been suggested in recent years that they could be complementary. In this manuscript we provide a rigorous analysis that shows that for any degree and threshold distributions heterogeneous bootstrap percolation can be mapped into heterogeneous k...

We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs...

In the Constrained-degree percolation model on a graph $(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open at time $U_e$, it succeeds if both its end-vertices would have degrees at most $k-1$. We prove a phase transition theorem for this model on the square lattice $\mathbb{L}^2$, as well as on the d-ary regular tree. We also prove that on the sq...

We consider Bernoulli (bond) percolation with parameter $p$ on the Cayley tree of order $k$. We introduce the notion of zebra-percolation that is percolation by paths of alternating open and closed edges. In contrast with standard percolation with critical threshold at $p_c= 1/k$, we show that zebra-percolation occurs between two critical values $p_{{\rm c},1}$ and $p_{{\rm c},2}$ (explicitly given). We provide the specific formula of zebra-percolation function.

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 percolation for $r=2,3$. In the deterministic setting, we obtain several results for the size of the smallest percolating set for $k\ge 2$, including the exact values for $k=2$ and $2\le r\le 6$.

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...