[Paper Review] Majority bootstrap percolation on the random graph G(n,p)

Sigurður Örn Stefánsson, Thomas Vallier|arXiv (Cornell University)|Mar 24, 2015

Stochastic processes and statistical mechanics4 references3 citations

TL;DR

This paper studies majority bootstrap percolation on the Erdős–Rényi random graph $G_{n,p}$, where vertices activate if more than half their neighbors are active. It establishes that percolation (activation of almost all vertices) occurs only when the initial active set exceeds $\frac{1}{2}n$, with a sharp threshold at $A(0) = \frac{1}{2}n + o(n)$, and shows that for $p \ll \frac{1}{n}$, no significant activation spreads, while for $p \gg \frac{1}{n}$, a phase transition occurs at $A(0) \approx \frac{1}{2}n$. The results extend to proportional bootstrap percolation with general $\alpha \in (0,1)$.

ABSTRACT

Majority bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have more active neighbours than inactive neighbours become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $n$ (tending to $\infty$), the size $A(0)=A_0(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We prove that the process cannot percolate for $A(0) = o(n)$. We study the process for $A(0) = θn$ and every range of $p$ and show that the model exhibits different behaviours for different ranges of $p$. For very small $p \ll \frac{1}{n}$, the activation does not spread significantly. For large $p \gg \frac{1}{n}$ then we see a phase transition at $A(0) \simeq \frac{1}{2}n$. In the case $p= \frac{c}{n}$, the activation propagates to a significantly larger part of the graph but (the process does not percolate) a positive part of the graph remains inactive.

Motivation & Objective

• To understand the conditions under which activation spreads to almost all vertices in majority bootstrap percolation on $G_{n,p}$.
• To determine the critical threshold for initial active set size $A(0)$ that triggers full or near-full percolation.
• To characterize the phase transition behavior across different ranges of edge probability $p = p(n)$.
• To extend results to proportional bootstrap percolation with general activation threshold $\alpha \in (0,1)$.

Proposed method

• Analyzes the stochastic evolution of active vertices using branching process approximations and coupling arguments.
• Uses concentration inequalities and tail bounds on binomial random variables to estimate the probability of vertex activation at each step.
• Applies stochastic domination techniques to bound the number of vertices that remain inactive after activation rounds.
• Derives asymptotic bounds on the final active set size $A^*$ using probabilistic methods and limit theorems as $n \to \infty$.
• Establishes a sharp threshold for percolation by analyzing the behavior of $A(0)$ relative to $\frac{1}{2}n$ and $\sqrt{n/p}$.
• Considers both fixed initial set size $A(0) = \theta n$ and random initial activation with probability $q(n)$, showing equivalence in asymptotic behavior.

Experimental results

Research questions

• RQ1 What is the critical size of the initial active set $A(0)$ required for percolation in majority bootstrap percolation on $G_{n,p}$?
• RQ2 How does the edge probability $p$ affect the spread of activation, particularly in the regimes $p \ll \frac{1}{n}$, $p = \frac{c}{n}$, and $p \gg \frac{1}{n}$?
• RQ3 Does the process percolate when $A(0) = \frac{1}{2}n + x\sqrt{n/p}$ for finite $x$?
• RQ4 What is the asymptotic behavior of the final active set size $A^*$ when $p = \frac{c}{n}$, and does it reach $n - o(n)$?
• RQ5 Can the results be extended to proportional bootstrap percolation with general activation threshold $\alpha \in (0,1)$?

Key findings

• For $p = o(1/n)$, the process is subcritical: no significant activation spreads, and $A^* = A(0) + o_p(n)$ with high probability.
• For $p \gg 1/n$, a sharp phase transition occurs at $A(0) \approx \frac{1}{2}n$, where $A^* = n - o_p(n)$ if $A(0) > \frac{1}{2}n + o(\sqrt{n/p})$.
• When $p = \frac{c}{n}$, activation spreads to a strictly positive fraction $\theta^* n$ with $\theta < \theta^* < 1$, but does not percolate fully.
• The critical threshold for percolation is $A_c = \frac{1}{2}n + o_p(n)$, and the transition window is of order $\sqrt{n/p}$.
• For $A(0) = \frac{1}{2}n + x\sqrt{n/p}$, the probability of percolation is conjectured to converge to a non-degenerate limit $\phi(x) \in (0,1)$ as $n \to \infty$, though this remains open.
• The model exhibits a universal threshold at $\alpha = \frac{1}{2}$ for proportional bootstrap percolation, consistent with results on the hypercube and global cascade models.