Bootstrap percolation on the random regular graph

Abstract

ABSTRACT:The k-parameter bootstrap percolation on a graph is a model of an interacting particle system, which can also be viewed as a variant of a cellular automaton growth process with threshold k ≥ 2. At the start, each of the graph vertices is active with probability p and inactive with probability 1 − p, independently of other vertices. Presence of active vertices triggers a bootstrap percolation process controlled by a recursive rule: an active vertex remains active forever, and a currently inactive vertex becomes active when at least k of its neighbors are active. The basic problem is to identify, for a given graph, p − and p + such that for p < p − ( p > p + resp.) the probability that all vertices are eventually active is very close to 0 (1 resp.). The bootstrap percolation process is a deterministic process on the space of subsets of the vertex set, which is easy to describe but hard to analyze rigorously in general. We study the percolation on the random d-regular graph, d ≥ 3, via analysis of the process on the multigraph counterpart of the graph. Here, thanks to a "principle of deferred decisions," the percolation dynamics is described by a surprisingly simple Markov chain. Its generic state is formed by the counts of currently active and nonactive vertices having various degrees of activation capabilities. We replace the chain by a deterministic dynamical system, and use its integrals to show-via exponential supermartingales-that the percolation process undergoes relatively small fluctuations around the deterministic trajectory. This allows us to show existence of the phase transitionis of order n −1/2 for k < d − 1, and n −εn , (ε n ↓ 0, ε n log n → ∞), for k = d − 1.Note that p * is the same as the critical probability of the process on the corresponding infinite regular tree.

Efficient control of epidemics over random networks

“…This result was first proved in [1], see also Theorem 5.1 in [14] which corresponds exactly to our Theorem 4 in this particular setting.…”

On percolation in random graphs with given vertex degrees

“…Bootstrap percolation on the random regular graph G(n, d) with fixed vertex degree d was studied by Balogh and Pittel [1]. We can recover a large part of their results from Theorem 4.5.…”

Graph bootstrap percolation

“…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.…”