In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and converges to 0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n denotes the n-th iterate of the exponential function, and the threshold lambda equals pi^2/6 for all d.
In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability threshold for a fairly general class of models. In our proofs we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the "easiest growth direction" in the model. In contrast to the anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metatstability threshold for anisotropic bootstrap percolation can be equal to that of i…
Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" bootstrap percolation model: the neighborhood of a point (m,n) is the set \[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\] At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof …
In the bootstrap percolation model, sites in an $L$ by $L$ square are initially independently declared active with probability $p$. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as $p \to 0$ and $L \to \infty$ simultaneously of the probability $I(L,p)$ that the entire square is eventually active. We prove that $I(L,p) \to 1$ if $\liminf p \log L > λ$, and $I(L,p) \to 0$ if $\limsup p \log L < λ$, where $λ= π^2/18$. We prove the same behaviour, with the same threshold $λ$, for the probability $J(L,p)$ that a site is …
We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized two-dimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization bounds for Holroyd, Liggett, and Romik's k-percolation models, and in the process provide a unified and improved treatment of existing results for bootstrap, modified bootstrap, and Frobose percolation. Furthermore, we prove improved asymptotic bounds for the generating functions of partitions without k-gaps, …
In this note we analyze an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.
In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z^d of dimension d>=3 with threshold r=2, we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the d=2 case, where Gravner and McDonald proved that the cri…
Bootstrap percolation is a wide class of monotone cellular automata with random initial state. In this work we develop tools for studying in full generality one of the three `universality' classes of bootstrap percolation models in two dimensions, termed subcritical. We introduce the new notion of `critical densities' serving the role of `difficulties' for critical models, but adapted to subcritical ones. We characterise the critical probability in terms of these quantities and successfully apply this link to prove new and old results for concrete models such as DTBP and Spiral as well as a ge…
We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z^2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p and pi^2/9 + C sqrt p (-log p)^3. This corrects previo…
In bootstrap percolation it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (that is, sharp and sharper thresholds) for the percolation threshold in general is a hard question. In the case of two-dimensional anisotropic models, sometimes correction terms can be obtained from inversion in a relatively simple manner.
In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For…
Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $ω_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $ω_n$ at time n converges exponentially fast to a final configuration $\barω$, and that the limiting measure corresponding to $\barω$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $β$, $η$, $ν$ and $γ$, and of …
We prove that there exist natural generalizations of the classical bootstrap percolation model on $\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Van Enter (in the case $d=r=2$) and Schonmann (for all $d \geq r \geq 2$) proved that $r$-neighbour bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^d$ for every choice of the parameters $d \geq r \geq 2$: that is, an initial set of density $p$ almost surely percolates $\mathbb{Z}^d$ for every $p>0$. These results effec…
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b>theta>1, the limiting density q=q(p) of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the system linge…
We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is pruned from the network. The heterogeneous-$k$-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds $k_i$ are $1$ with probability $f$ or $k \geq 3$ with probability $(1-f)$, the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolati…
We analyze the metastable states near criticality of the bootstrap percolation on Galton-Watson trees. We find that, depending on the exact choice of the offspring distribution, it is possible to have several distinct metastable states, with varying scaling of their duration while approaching criticality.
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 neighbours are infected. We say that percolation occurs if eventually all vertices in $G$ become infected. In this paper we study majority bootstrap percolation on the Erdős-Rényi random graph $G(n,p)$ above the connectivity threshold. Perhaps surprisingly, the results obtained for small $p$ are comparable to the results for the hypercube obtained by Balogh, Boll…
Answering questions of Itai Benjamini, we show that the event of complete occupation in 2-neighbour bootstrap percolation on the d-dimensional box [n]^d, for d\geq 2, at its critical initial density p_c(n), is noise sensitive, while in k-neighbour bootstrap percolation on the d-regular random graph G_{n,d}, for 2\leq k\leq d-2, it is insensitive. Many open problems remain.
We study the distribution of the percolation time $T$ of two-neighbour bootstrap percolation on $[n]^2$ with initial set $A\sim\mathrm{Bin}([n]^2,p)$. We determine $T$ with high probability up to a constant factor for all $p$ above the critical probability for percolation, and to within a $1+o(1)$ factor for a large range of $p$.
Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $\mathbb{Z}^2\times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $θ$ exhibits a sharp phase transition, while odd $θ$ yields a gradual percolation tran…
Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family of lower bounds on the critical probability for $r$-neighbour bootstrap percolation on Galton--Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollobás, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution.
This paper analyzes various questions pertaining to bootstrap percolation on the $d$-dimensional Hamming torus where each node is open with probability $p$ and the percolation threshold is 2. For each $d'<d$ we find the critical exponent for the event that a $d'$-dimensional subtorus becomes open and compute the limiting value of its probability under the critical scaling. For even $d'$, we use the Chen-Stein method to show that the number of $d'$-dimensional subtori that become open can be approximated by a Poisson random variable.
A graph $G$ percolates in the $K_{r,s}$-bootstrap process if we can add all missing edges of $G$ in some order such that each edge creates a new copy of $K_{r,s}$, where $K_{r,s}$ is the complete bipartite graph. We study $K_{r,s}$-bootstrap percolation on the Erdős-Rényi random graph, and determine the percolation threshold for balanced $K_{r,s}$ up to a logarithmic factor. This partially answers a question raised by Balogh, Bollobás, and Morris. We also establish a general lower bound of the percolation threshold for all $K_{r,s}$, with $r\geq s \geq 3$.
We study the percolation time of the $r$-neighbour bootstrap percolation model on the discrete torus $(\Z/n\Z)^d$. For $t$ at most a polylog function of $n$ and initial infection probabilities within certain ranges depending on $t$, we prove that the percolation time of a random subset of the torus is exactly equal to $t$ with high probability as $n$ tends to infinity. Our proof rests crucially on three new extremal theorems that together establish an almost complete understanding of the geometric behaviour of the $r$-neighbour bootstrap process in the dense setting. The special case $d-r=0$ o…
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 determine $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely. In this pape…
We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: $f$, the fraction of vertices initially activated, and $p$, the fraction of undamaged vertices in the graph. We observe two transitions: the giant active component appears continuously at a first threshold. There may also be a second, discontinuous, hybrid transition at a higher threshold. Avalanches of activations increase in size as this second critical point is approached, finally diverging at this threshold. We describe the existence of a special …
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-neighbour bootstrap percolation on the d-dimensional discrete torus T_n^d. Moreover, we show that for certain ranges of p = p(n)…
In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom's North-East-Center majority rule cellular automaton.
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 in random graphs when the vertex thresholds are nonuniform and provide an efficient algorithm. This result also helps solv…
Consider a graph $G$ and an initial random configuration, where each node is black with probability $p$ and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least $r$ black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a $d$-dimensional torus and identify the threshold values.
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 percolation starts by random and independent "activation" of nodes with a fixed probability $p$, followed by a deterministi…
In the polluted bootstrap percolation model, vertices of the cubic lattice $\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,q\to 0$. We show that this density converges to $1$ if $q \ll p^3(\log p^{-1})^{-3}$ for both standard and modified rules. Our pri…
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 $i \in \{1,2,3\}$, where $a_1\le a_2\le a_3$. Suppose we infect any healthy vertex $x\in [L]^3$ already having $a_3+1$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the critical length for percolation up to a constant factor in the exponent, for all triples $(a_1,a_2,a_3)$. To do so, we introduce a new algorithm called the bea…
We consider the Erdös--Rényi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size $A_n^*$ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables $\{\frac{n- A_n^*}{f(n)}\}_{n\geq 1}$ with explicit rate functions and allowing the scaling function $f$ to vary in the widest possible rang…
Graph bootstrap percolation, introduced by Bollobás in 1968, is a cellular automaton defined as follows. Given a "small" graph $H$ and a "large" graph $G = G_0 \subseteq K_n$, in consecutive steps we obtain $G_{t+1}$ from $G_t$ by adding to it all new edges $e$ such that $G_t \cup e$ contains a new copy of $H$. We say that $G$ percolates if for some $t \geq 0$, we have $G_t = K_n$. For $H = K_r$, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollobás and Morris considered graph bootstrap perc…
We study qualitative properties of two-dimensional freezing cellular automata with a binary state set initialized on a random configuration. If the automaton is also monotone, the setting is equivalent to bootstrap percolation. We explore the extent to which monotonicity constrains the possible asymptotic dynamics by proving two results that do not hold in the subclass of monotone automata. First, it is undecidable whether the automaton almost surely fills the space when initialized on a Bernoulli random configuration with density $p$, for some/all $0 < p < 1$. Second, there exists an automato…
We analyze the bootstrap percolation process on the stochastic block model (SBM), a natural extension of the Erdös--Rényi random graph that allows representing the "community structure" observed in many real systems. In the SBM, nodes are partitioned into subsets, which represent different communities, and pairs of nodes are independently connected with a probability that depends on the communities they belong to. Under mild assumptions on system parameters, we prove the existence of a sharp phase transition for the final number of active nodes and characterize sub-critical and super-critical …
Geometric inhomogeneous random graphs (GIRGs) are a model for scale-free networks with underlying geometry. We study bootstrap percolation on these graphs, which is a process modelling the spread of an infection of vertices starting within a (small) local region. We show that the process exhibits a phase transition in terms of the initial infection rate in this region. We determine the speed of the process in the supercritical case, up to lower order terms, and show that its evolution is fundamentally influenced by the underlying geometry. For vertices with given position and expected degree, …
We analyze the bootstrap percolation process on the stochastic block model (SBM), a natural extension of the Erdős--Rényi random graph that incorporates the community structure observed in many real systems. In the SBM, nodes are partitioned into two subsets, which represent different communities, and pairs of nodes are independently connected with a probability that depends on the communities they belong to. Under mild assumptions on the system parameters, we prove the existence of a sharp phase transition for the final number of active nodes and characterize the sub-critical and the super-cr…
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 Chung-Lu model. The process starts with infection rate $p=p(N)$. Each uninfected vertex with at least $\mathbf{r}\geq 1$ infected neighbors becomes infected, remaining so forever. We identify a function $p_c(N)=o(1)$ such that a.a.s.\ when $p\gg p_c(N)$ the infection spreads to a positive fraction of vertices, whereas when $p\ll p_c(N)$ the process cannot…
Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor i…
Recent experimental studies of living neural networks reveal that their global activation induced by electrical stimulation can be explained using the concept of bootstrap percolation on a directed random network. The experiment consists in activating externally an initial random fraction of the neurons and observe the process of firing until its equilibrium. The final portion of neurons that are active depends in a non linear way on the initial fraction. The main result of this paper is a theorem which enables us to find the asymptotic of final proportion of the fired neurons in the case of r…
Bootstrap percolation is a process that is used to model the spread of an infection on a given graph. In the model considered here each vertex is equipped with an individual threshold. As soon as the number of infected neighbors exceeds that threshold, the vertex gets infected as well and remains so forever. We perform a thorough analysis of bootstrap percolation on a novel model of directed and inhomogeneous random graphs, where the distribution of the edges is specified by assigning two distinct weights to each vertex, describing the tendency of it to receive edges from or to send edges to o…
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 random from all sets of given size. We establish a threshold such that if there are less vertices in the initial set of infected vertices, then whp only a few additional vertices become infected, while if the initial set of infected vertices exceeds the threshold then whp almost every vertex becomes infected. In addition, for $k=2$, we show that the probabil…
Consider the following model of strong-majority bootstrap percolation on a graph. Let r be some positive integer, and p in [0,1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every step of the process, each vertex v of degree deg(v) becomes active if at least (deg(v)+r)/2 of its neighbours are active. Given any arbitrarily small p>0 and any integer r, we construct a family of d=d(p,r)-regular graphs such that with high probability all vertices become active in the end. In particular, the case r=1 answers a question and disproves a conjec…
We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: \[ p_c\big( [L]^2,\mathcal{N}_{(1,2)},3 \big) \; = \; \frac{(\log \log L)^2}{12\log L} \, - \, \frac{\log \log L \, \log \log \log L}{ 3\log L} + \frac{\left(\log \frac{9}{2} + 1 \pm o(1) \right)\log \log L}{6\log L}. \] We note…
Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph $H$, and a set $G \subset E(K_n)$ of initially `infected' edges, we infect, at each time step, a new edge $e$ if there is a copy of $H$ in $K_n$ such that $e$ is the only not-yet infected edge of $H$. We say that $G$ percolates in the $H$-bootstrap process if eventually every edge of $K_n$ is infected. The extremal questions for this model, when $H$ is the complete graph $K_r$, were solved (independently) by Alon, Kalai and Frankl almost thirty ye…
Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: `healthy' or `infected' and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. Vertices are initially infected independently at random and the central question is whether all vertices eventually becom…
We study bootstrap percolation with the threshold parameter $θ\geq 2$ and the initial probability $p$ on infinite periodic trees that are defined as follows. Each node of a tree has degree selected from a finite predefined set of non-negative integers and starting from any node, all nodes at the same graph distance from it have the same degree. We show the existence of the critical threshold $p_f(θ) \in (0,1)$ such that with high probability, (i) if $p > p_f(θ)$ then the periodic tree becomes fully active, while (ii) if $p < p_f(θ)$ then a periodic tree does not become fully active. We also de…
Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim π^2/(18 log n) as n \to \infty. Here we sharpen this result, prov…
In this paper we study the strict majority bootstrap percolation process on graphs. Vertices may be active or passive. Initially, active vertices are chosen independently with probability p. Each passive vertex becomes active if at least half of its neighbors are active (and thereafter never changes its state). If at the end of the process all vertices become active then we say that the initial set of active vertices percolates on the graph. We address the problem of finding graphs for which percolation is likely to occur for small values of p. Specifically, we study a graph that we call r-whe…
