Extremal Bounds for Bootstrap Percolation in the Hypercube

Abstract

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 vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A 0 percolates.We prove a conjecture of Balogh and Bollobás which says that, for fixed r and d → ∞, every percolating set in the d-dimensional hypercube has cardinality at least 1+o(1) r d r−1 . We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when r = 3, we prove that the minimum cardinality of a percolating set in the d-dimensional hypercube is d(d+3) 6 + 1 for all d ≥ 3.

Contagious sets in a degree‐proportional bootstrap percolation process

“…The case r = 2 was settled by Balogh and Bollobás . Recently exact asymptotics for the hypercube were determined by Morrison and Noel . This proved a conjecture of Balogh and Bollobás and improved earlier bounds of Balogh, Bollobás and Morris .…”

Dynamic monopolies in two-way bootstrap percolation

“…Building on their results, Hambardzumyan, Hatami, and Qian [17] proved the upper bound of d r−1 n r−1 , which leaves an r process dynamo monotone dynamo r−BP (small r) Our results are marked in bold. However, the lower bound for r−BP (small r) comes from [21], and some of our results (as will be discussed in the text) were known for d = 2 and r = d, d + 1 respectively by [6,13] and [3].…”

“…In this section we study (monotone) dynamos in T d n for 1 ≤ r ≤ d. We first give the minimum size of r−BP dynamos and monotone reversible r−BP dynamos. We present our constructive upper bounds in Theorem 2.4, and relying on prior results by Morrison and Noel [21] we provide matching lower bounds. Then, building on these results we study the minimum size of reversible r−BP dynamos.…”

“…by Schonmann [26] and Balogh and Pete [6], it was formally defined and studied in the seminal work by Kempe, Kleinberg, and Tardos [20] and independently by Peleg [22], motivated from fault-local mending in distributed systems. There is a massive body of work concerning the minimum size of a (monotone) dynamo for (reversible) r−BP and majority model in different classes of graphs, for instance hypercube [5,17,21], the binomial random graph [9,10,12], random regular graphs [16], power-law graphs [11], planar graphs [23], and many others. Motivated from the literature of cellular automata and a range of applications in statistical physics special attention has been devoted to the d-dimensional torus T d n ; the d−dimensional torus T d n is the graph with vertex set [n] d , where two vertices are adjacent if and only if they differ by 1 mod n in exactly one coordinate (see Definition 1.6).…”

“…[21]) For all d ≥ 1 and 1 ≤ r ≤ d it holds that m(T d n , r) ≥ 1 r d r−1 n r−1 . For d ≥ 1 and 1 ≤ r ≤ 2d let ω be a monotone dynamo in T d n and reversible r−BP.…”