[Paper Review] Dynamic Monopolies in Reversible Bootstrap Percolation

Clemens Jeger, Ahad N. Zehmakan|arXiv (Cornell University)|May 5, 2018

Stochastic processes and statistical mechanics26 references4 citations

TL;DR

This paper resolves an open problem on dynamic monopolies in reversible $r$-bootstrap percolation on the $d$-dimensional torus $\mathbb{T}_n^d$, providing tight asymptotic bounds on the minimum size of a monotone dynamo for all $r \in [1, 2d]$. It establishes that for small $r$ ($1 \leq r \leq d$), the minimum monotone dynamo size is $\frac{2}{r}\binom{d}{r-1}n^{r-1} \pm \Theta(n^{r-2})$, and for large $r$ ($d+1 \leq r \leq 2d$), it is $2(1 - \frac{d}{r})n^d \pm \Theta(n^{d-1})$, generalizing prior results and settling long-standing conjectures.

ABSTRACT

We study an extremal question for the (reversible) $r-$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 reversible $r$-bootstrap percolation, each vertex gets active if it has at least $r$ active neighbors, and inactive otherwise. We consider the following question on the $d$-dimensional torus: how many vertices should be initially active so that the whole graph becomes active? Our results settle an open problem by Balister, Bollobás, Johnson, and Walters and generalize the results by Flocchini, Lodi, Luccio, Pagli, and Santoro.

Motivation & Objective

• To resolve an open problem posed by Balister, Bollobas, Johnson, and Walters on the minimum size of a monotone dynamo in reversible $r$-bootstrap percolation on the $d$-dimensional torus.
• To generalize and extend prior results by Flocchini et al. on dynamic monopolies in reversible $r$-bootstrap percolation.
• To provide tight asymptotic bounds on the size of the smallest monotone dynamo for all values of $r$ in $[1, 2d]$.
• To establish a complete characterization of the minimum size of monotone dynamos in the majority model on $\mathbb{T}_n^d$, resolving a gap in the literature.

Proposed method

• Constructs explicit configurations of active vertices using structured vertex sets $S$ and $H$ to ensure monotonic percolation.
• Uses induction and parity-based arguments to show that active sets $B_j$ propagate through the torus in alternating rounds.
• Applies combinatorial and graph-theoretic bounds on edge counts and minimum degrees to derive lower bounds on dynamo size.
• Leverages properties of $2d$-regular graphs and neighborhood structures in the torus to prove tight asymptotic bounds.
• Introduces a novel vertex set $S$ defined via modular linear forms to control neighbor counts and ensure stability.
• Combines structural construction with extremal graph arguments to prove both upper and lower bounds for all $r$-ranges.

Experimental results

Research questions

• RQ1 What is the minimum size of a monotone dynamic monopoly in reversible $r$-bootstrap percolation on the $d$-dimensional torus for $1 \leq r \leq d$?
• RQ2 What is the minimum size of a monotone dynamic monopoly in reversible $r$-bootstrap percolation on the $d$-dimensional torus for $d+1 \leq r \leq 2d$?
• RQ3 What is the minimum size of a monotone dynamic monopoly in the majority model on the $d$-dimensional torus?
• RQ4 How do the bounds for monotone dynamos in reversible $r$-bootstrap percolation compare to those in the standard $r$-bootstrap percolation?
• RQ5 Can the previously open problem on monotone dynamo size in reversible $r$-bootstrap percolation be fully resolved for all $r$?

Key findings

• For small $r$ ($1 \leq r \leq d$), the minimum size of a monotone dynamo in reversible $r$-bootstrap percolation on $\mathbb{T}_n^d$ is $\frac{2}{r}\binom{d}{r-1}n^{r-1} \pm \Theta(n^{r-2})$, which matches the upper bound and closes a gap in prior work.
• For large $r$ ($d+1 \leq r \leq 2d$), the minimum size of a monotone dynamo is $2(1 - \frac{d}{r})n^d \pm \Theta(n^{d-1})$, resolving the open problem posed by Balister et al.
• The minimum size of a monotone dynamo in the majority model on $\mathbb{T}_n^d$ is $(1 - \frac{d}{d+2})n^d \pm \Theta(n^{d-1})$, providing the first tight asymptotic bound for general $d \geq 1$.
• The lower bound of $(1 - \frac{d}{d+2})n^d$ for the majority model is proven via degree and edge-counting arguments in $2d$-regular graphs.
• The construction of the active set $S \cup H$ ensures monotonic percolation and achieves the asymptotically optimal size, with $|S| \leq (1 - \frac{d}{d+2})n^d + \Theta(n^{d-1})$.
• The paper establishes that the bounds for monotone dynamos in reversible $r$-bootstrap percolation are asymptotically tight and generalize previous results for specific $r$ and $d$.