Bootstrap Percolation Research Articles

Selective bootstrap percolation

A new class of bootstrap percolation models in which particle culling occurs only for certain numbers of nearest neighbours is introduced and studied on a Bethe lattice. Upon increasing the density of initial configuration they undergo multiple hybrid (or mixed-order) phase transitions, showing that such intriguing phase behaviours may also appear in fully homogeneous environments, provided that culling is selective rather than cumulative. The idea immediately extends to facilitation dynamics, suggesting a simple way to construct one-component models of multiple glasses and glass-glass transitions as well as more general coarse-grained models of complex cooperative dynamics.

Nucleation and growth in two dimensions

We consider a dynamical process on a graph G, in which vertices are infected (randomly) at a rate which depends on the number of their neighbors that are already infected. This model includes bootstrap percolation and first‐passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph , significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.

Cluster approach to phase transitions from fluid to amorphous solids: gels, glasses and granular materials

Based on various results present in the literature, we elaborate a unifying cluster percolation approach to interpret the dynamical arrest occurring in amorphous materials such as those of the gel, glass and granular variety. In the case of the sol-gel transition, this cluster approach predicts scaling laws relating dynamical exponents to critical random percolation exponents. Interestingly, in the mean-field such relations coincide with those predicted by the schematic continuous mode coupling theory, known as model A. More appropriate to describe the molecular glass transition is the schematic discontinuous mode coupling theory known as model B. In this case a similar cluster approach and a diffusing defect mechanism predicts scaling laws, relating dynamical exponents to the static critical exponents of the bootstrap percolation. In finite dimensions, the glass theory based on the random first order transition suggests that the mode coupling theory transition is only a crossover towards an ideal glass transition characterised by the divergence of cooperative rearranging regions. Interestingly, this scenario can also be mapped onto a mixed order percolation transition, where the order parameter jumps discontinuously at the transition, while the mean cluster size and the linear cluster dimension diverge. A similar mixed order percolation transition seems to apply to the jamming transition as well.

The second term for two-neighbour bootstrap percolation in two dimensions

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 breakthrough, Holroyd determined the sharp threshold for percolation in this model, and his bounds were subsequently sharpened further by Gravner and Holroyd, and by Gravner, Holroyd and Morris. In this paper we strengthen the lower bound of Gravner, Holroyd and Morris by proving that the critical probability $p_c\big( [n]^2,2 \big)$ for percolation in the two-neighbour model on $[n]^2$ satisfies \[p_c\big( [n]^2,2 \big) = \frac{\pi^2}{18\log n} - \frac{\Theta(1)}{(\log n)^{3/2}}\,.\] The proof of this result requires a very precise understanding of the typical growth of a critical droplet, and involves a number of technical innovations. We expect these to have other applications, for example, to the study of more general two-dimensional cellular automata, and to the $r$-neighbour process in higher dimensions.

Bootstrap Percolation in Directed Inhomogeneous Random Graphs

Bootstrap percolation is a process that is used to describe 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 specied by assigning two distinct weights to each vertex, describing the tendency of it to receive edges from or to send edges to other vertices. Under the mild assumption that the limiting degree distribution of the graph is integrable we determine the typical fraction of infected vertices. Our model allows us to study a variety of settings, in particular the prominent case in which the degree distribution has an unbounded variance. As a second main contribution, we quantify the notion of "systemic risk", that is, we characterize to what extent tiny initial infections can propagate to large parts of the graph through a cascade, and discover novel features that make graphs prone/resilient to initially small infections.

Byzantine-resilient distributed observers for LTI systems

Consider a linear time-invariant (LTI) dynamical system monitored by a network of sensors, modeled as nodes of an underlying directed communication graph. We study the problem of collaboratively estimating the state of the system when certain nodes are compromised by adversaries. Specifically, we consider a Byzantine adversary model, where a compromised node possesses complete knowledge of the system dynamics and the network, and can deviate arbitrarily from the rules of any prescribed algorithm. We first characterize certain fundamental limitations of any distributed state estimation algorithm in terms of the measurement and communication structure of the nodes. We then develop an attack-resilient, provably correct state estimation algorithm that admits a fully distributed implementation. To characterize feasible network topologies that guarantee success of our proposed technique, we introduce a notion of ‘strong-robustness’ that captures both measurement and communication redundancy. Finally, by drawing connections to bootstrap percolation theory, we argue that given an LTI system and an associated sensor network, the ‘strong-robustness’ property can be checked in polynomial time.

On connectivity, conductance and bootstrap percolation for a random k‐out, age‐biased graph

A uniform attachment graph (with parameter k), denoted Gn,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v − 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of Gn,k to show that the conductance of Gn,k is of order . We also study the bootstrap percolation on Gn,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.

An Improved Upper Bound for Bootstrap Percolation in All Dimensions

AbstractIn r-neighbour bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbours. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V(G) will eventually become infected. Improving a result of Balogh, Bollobás and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n]d and d ⩾ r ⩾ 2. We show that for all d ⩾ r ⩾ 2 there exists a constant cd,r > 0 such that if n is sufficiently large, then $$p_c (\left[ n \right]^d ,{\rm{ }}r){\rm{\le }}\left( {\frac{{\lambda (d,r)}}{{\log _{(r - 1)} (n)}} - \frac{{c_{d,r} }}{{(\log _{(r - 1)} (n))^{3/2} }}} \right)^{d - r + 1} ,$$where λ(d, r) is an exact constant and log(k) (n) denotes the k-times iterated natural logarithm of n.

Gaussian Message Passing-Based Cooperative Localisation with Bootstrap Percolation Scheme in Dense Networks

In dense networks, the tremendous computational complexity and communication overhead of cooperative localisation are the two main bottlenecks that limit practical application. In this study, we introduce a bootstrap percolation scheme into Gaussian message passing-based cooperative localisation for precise positioning, aimed at reducing the system overhead. Considering the uncertainty information and geometric distribution of neighbours, an approximate collinear detection criterion is proposed to detect the possible flip ambiguities in cooperative localisation. According to the detection result and our connection constraint, agents are divided into three categories and are approximated by different distribution families. A message passing rule is designed to control the propagation direction from high precision to low precision, thereby mitigating potential error propagation. Additionally, a layer-by-layer positioning mechanism is established where the agents are located gradually. Analytical and simulation results indicate that when the ranging standard deviation is 0·2 m, 89·3% of the agent nodes can be located within 0·4 m using the proposed algorithm. Compared with the Hybrid Sum-Product Algorithm over A Wireless Network (H-SPAWN), this ratio is increased by 8·4% and the computational complexity is reduced by 72%.

Bootstrap percolation with inhibition

We study a variant of the classical bootstrap percolation process on Erdős Rényi random graphs. The graphs we consider have inhibitory vertices obstructing the diffusion of activity and excitatory vertices facilitating it. We study both a synchronous and an asynchronous version of the process. Both begin with a small initial set of active vertices, and the activation spreads to all vertices for which the number of excitatory active neighbors exceeds the number of inhibitory active neighbors by a certain amount. We show that in the synchronous process, inhibitory vertices may cause unstable behavior: tiny changes in the size of the starting set can dramatically influence the size of the final active set. We further show that in the asynchronous model the process becomes stable and stops with an active set containing a nontrivial deterministic constant fraction of all vertices. Moreover, we show that percolation occurs significantly faster asynchronously than synchronously.

Fate of the Hybrid Transition of Bootstrap Percolation in Physical Dimension.

Bootstrap, or k-core, percolation displays on the Bethe lattice a mixed first- and second-order phase transition with both a discontinuous order parameter and diverging critical fluctuations. I apply the recently introduced M-layer technique to study corrections to mean-field theory showing that at all orders in the loop expansion the problem is equivalent to a spinodal with quenched disorder. This implies that the mean-field hybrid transition does not survive in physical dimension. Nevertheless, its critical properties as an avoided transition make it a proxy of the avoided mode-coupling-theory critical point of supercooled liquids.

Dynamic monopolies in two-way bootstrap percolation

We study an extremal question for the (two-way) r−bootstrap percolation processes. Given a graph and an initial configuration where each vertex is active or inactive, in the r−bootstrap percolation process the following rule is applied in discrete-time rounds: each vertex gets active if it has at least r active neighbors, and an active vertex stays active forever. In the two-way r−bootstrap percolation, each vertex gets active if it has at least r active neighbors, and inactive otherwise. We consider the following question on thed-dimensional torus: how many vertices must be initially active so that the whole graph becomes active? Our results settle an open problem by Balister et al. (2010) and generalize the results by Flocchini et al. (2004).

Newman-Ziff algorithm for the bootstrap percolation: Application to the Archimedean lattices

We propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the Newman-Ziff algorithm of the classical percolation (M.E.J. Newman and R.M. Ziff (2000) [27]). Using these algorithms and the finite-size-scaling, we calculated with high precision the percolation threshold and critical exponents in the eleven two-dimensional Archimedean lattices. We present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation, and show that they have the same critical exponents as the classical percolation within error bars in two dimensions. We conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation.

Insights into bootstrap percolation: Its equivalence with k-core percolation and the giant component.

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-core percolation and vice versa, if the functionality thresholds in both processes satisfy a complementary relation. Another interesting problem in bootstrap and k-core percolation is the fraction of nodes belonging to their giant connected components P_{∞b} and P_{∞c}, respectively. We solve this problem analytically for arbitrary randomly connected graphs and arbitrary threshold distributions, and we show that P_{∞b} and P_{∞c} are not complementary. Our theoretical results coincide with computer simulations in the limit of very large graphs. In bootstrap percolation, we show that when using the branching theory to compute the size of the giant component, we must consider two different types of links, which are related to distinct spanning branches of active nodes.

Towards a universality picture for the relaxation to equilibrium of kinetically constrained models

Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata. In the two-dimensional lattice $\mathbb{Z}^{2}$, there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behavior. Much less is known for their nonmonotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM, each vertex is resampled (independently) at rate one by tossing a $p$-coin iff it can be infected in the next step by the bootstrap model. In particular, an infection can also heal, hence the nonmonotonicity. Besides the connection with bootstrap percolation, KCM have an interest in their own as they feature some of the most striking features of the liquid/glass transition, a major and still largely open problem in condensed matter physics. In this paper, we pave the way towards proving universality results for the characteristic time scales of KCM. Our novel and general approach gives the right tools to establish a close connection between the critical scaling of characteristic time scales for KCM and the scaling of the critical length in critical bootstrap models. When applied to the Fredrickson–Andersen $k$-facilitated models in dimension $d\ge2$, among the most studied KCM, and to the Gravner–Griffeath model, our results are close to optimal.

Metastable behavior of bootstrap percolation on Galton-Watson trees

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.

A PROOF OF ANDREWS’ CONJECTURE ON PARTITIONS WITH NO SHORT SEQUENCES

Our main result establishes Andrews’ conjecture for the asymptotic of the generating function for the number of integer partitions of$n$without$k$consecutive parts. The methods we develop are applicable in obtaining asymptotics for stochastic processes that avoid patterns; as a result they yield asymptotics for the number of partitions that avoid patterns.Holroyd, Liggett, and Romik, in connection with certain bootstrap percolation models, introduced the study of partitions without$k$consecutive parts. Andrews showed that when$k=2$, the generating function for these partitions is a mixed-mock modular form and, thus, has modularity properties which can be utilized in the study of this generating function. For$k>2$, the asymptotic properties of the generating functions have proved more difficult to obtain. Using$q$-series identities and the$k=2$case as evidence, Andrews stated a conjecture for the asymptotic behavior. Extensive computational evidence for the conjecture in the case$k=3$was given by Zagier.This paper improved upon early approaches to this problem by identifying and overcoming two sources of error. Since the writing of this paper, a more precise asymptotic result was established by Bringmann, Kane, Parry, and Rhoades. That approach uses very different methods.

Polluted bootstrap percolation with threshold two in all dimensions

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 $${\mathbb {Z}}^d$$ of dimension $$d\ge 3$$ 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 critical parameter is $$q/{p^2}$$ .

A modified bootstrap percolation on a random graph coupled with a lattice

In this paper a random graph model GZN2,pd is introduced, which is a combination of fixed torus grid edges in (Z∕NZ)2 and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices u,v∈(Z∕NZ)2 with graph distance d on the torus grid is pd=c∕Nd, where c is some constant. We show that, whp, the diameter D(GZN2,pd)=Θ(logN). Moreover, we consider a modified non-monotonous bootstrap percolation on GZN2,pd. We prove the presence of phase transitions in mean-field approximation and provide fairly sharp bounds on the error of the critical parameters.

Contagious sets in a degree‐proportional bootstrap percolation process

We study the following bootstrap percolation process: given a connected graph G, a constant ρ ∈ [0,1] and an initial set A⊆V(G) of infected vertices, at each step a vertex v becomes infected if at least a ρ‐proportion of its neighbors are already infected (once infected, a vertex remains infected forever). Our focus is on the size hρ(G) of a smallest initial set which is contagious, meaning that this process results in the infection of every vertex of G. Our main result states that every connected graph G on n vertices has hρ(G) < 2ρn or hρ(G) = 1 (note that allowing the latter possibility is necessary because of the case , as every contagious set has size at least one). This is the best‐possible bound of this form, and improves on previous results of Chang and Lyuu and of Gentner and Rautenbach. We also provide a stronger bound for graphs of girth at least five and sufficiently small ρ, which is asymptotically best‐possible.