Bootstrap Percolation in High Dimensions

Abstract

Abstract. In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V (G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n] d , for arbitrary functions n = n(t), d = d(t) and r = r(t), as t → ∞. The main question is to determine the critical probability p c ([n] d , r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d log n. The bootstrap process has been extensively studied on [n] d when d is a fixed constant and 2 r d, and in these cases p c ([n] d , r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined p c ([2] d , 2) up to a constant factor, and Balogh, Bollobás and Morris determined2+ε , and gave much sharper bounds for the hypercube. Here we prove the following result: let λ be the smallest positive root of the equation

Constant Thresholds Can Make Target Set Selection Tractable

“…A parameterized reduction from a parameterized problem P to another parameterized problem P is a function that, given an instance (x, k), computes in f (k) · s O(1) time an instance (x , k ) (with k only depending on k) such that (x, k) is a yes-instance of P if and only if (x , k ) is a yes-instance of P . The basic complexity class for fixed-parameter intractability is called W [1] and there is good complexity-theoretic reason to believe that W [1]-hard problems are not fpt [9,12,23].…”

“…c Springer Bootstrap Percolation [1] (where the threshold of each vertex is k or r, respectively), and so-called dynamic monopolies [24] (where the threshold of a vertex v with degree deg(v) equals deg(v)/2 -in the following this condition is referred to as majority thresholds). Besides being a problem of considerable graph-theoretic interest, TSS is also motivated by applications in areas such as social network analysis [5,17] and distributed computing [24].…”

Graph bootstrap percolation

“…The proof is quite different from that of Theorem 1, and uses ideas from the study of 2-neighbour bootstrap percolation on [n] d (see [3,4], or the more recent improvements in [8,26,27]). …”

“…To be precise, if V (G) = [n] and the elements of A ⊂ V (G) are chosen independently at random, each with probability p, then one aims to determine the value p c of p = p(n) at which percolation becomes likely. Sharp bounds on p c have recently been determined in several cases of particular interest, such as [n] d (see [5,6,7,8,26,27]), on a large family of 'twodimensional' graphs [19], on trees [10,21], and on various types of random graph [11,29]. In each case, it was shown that the critical probability has a sharp threshold.…”

Constant Thresholds Can Make Target Set Selection Tractable

“…The basic complexity class for fixed-parameter intractability is called W [1] and there is good complexity-theoretic reason to believe that W [1]-hard problems are not fpt [9,12,23].…”