---
title: Majority Bootstrap Percolation Thresholds
url: https://www.emergentmind.com/topics/majority-bootstrap-percolation
type: topic
---

# Majority Bootstrap Percolation Thresholds

Majority bootstrap percolation is a monotone infection process on a finite graph in which vertices, once infected or active, never recover, and new infections occur when the local infected population exceeds a majority threshold. In the random-initial-set model one typically infects each vertex independently with probability \(p\), studies the percolation probability \(\Phi(p,G)\), and defines the critical probability \(p_c(G)\) by the condition \(\Phi(p,G)\ge 1/2\). The subject includes several closely related update rules: majority bootstrap percolation, strict majority bootstrap percolation, and strong-majority or \(r\)-majority bootstrap percolation. Across graph families the model exhibits markedly different thresholds, ranging from the constant limit \(p_c=1/4\) on the \(r\)-wheel [1308.3831], through transitions near \(1/2\) on Erdős–Rényi graphs [1508.02671, 1503.07029], to a universal square-root-log correction on broad classes of high-dimensional graphs [2406.17486, 2507.06597].

## 1. Definitions, threshold conventions, and variants

On a finite graph \(G=(V,E)\), majority bootstrap percolation is commonly formulated by starting from an infected set \(A_0\subseteq V\) and iterating
\[
A_i \;=\; A_{i-1}\;\cup\;\bigl\{\,v\in V\:\bigl|\:|N(v)\cap A_{i-1}|\;>\;\tfrac{d(v)}2\}\,,
\]
where \(N(v)\) is the neighbourhood of \(v\) and \(d(v)=|N(v)|\); one says that \(A_0\) percolates if \(\bigcup_i A_i=V\) [2406.17486]. In a Bernoulli initialisation, each vertex lies in \(A_0\) independently with probability \(p\), and one writes
\[
\Phi(p,G)\;=\;\Pr\bigl(A_0\text{ percolates on }G\bigr),\qquad
p_c(G)\;=\;\inf\bigl\{p\in(0,1):\Phi(p,G)\ge\tfrac12\bigr\}
\]
[2406.17486, 2507.06597].

The papers considered here use three related update thresholds. In strict majority bootstrap percolation, a passive vertex \(v\) becomes active exactly when
\[
\bigl|\{w\in N(v): w\text{ is active}\}\bigr|
\;\ge\;\frac{\deg(v)+1}{2}
\]
[1308.3831]. In strong-majority or \(r\)-majority bootstrap percolation, an inactive vertex becomes active when
\[
\#\{\text{active neighbours of }v\}\;\ge\;\Bigl\lceil\frac{\deg(v)+r}{2}\Bigr\rceil
\]
[1503.08310]. Some majority-bootstrap papers use the equivalent condition “more infected than non-infected neighbours,” while others write the update rule with \(\ge d(v)/2\) rather than \(>d(v)/2\) [1508.02671, 2507.06597]. This difference is part of the model specification, not a contradiction: the literature distinguishes majority, strict majority, and shifted-threshold variants.

A second distinction concerns the randomisation of the initial condition. Some works fix the initial size, for example \(|I_0|=m\) on \(G(n,p)\) [1508.02671], whereas others take a Bernoulli-\(p\) or Bernoulli-\(q\) initial set [2406.17486, 1503.07029, 2507.06597]. In every case the process is monotone, stabilises after finitely many rounds, and the central question is whether the final infected set equals \(V\).

## 2. The \(r\)-wheel and the constant threshold \(1/4\)

Kiwi et al. study strict majority bootstrap percolation on the \(r\)-wheel \(WH_n(r)\), obtained from the ring \(R_n(r)\) by adding one universal vertex adjacent to every ring vertex [1308.3831]. Here \(R_n(r)\) is the cycle on \(n\) vertices in which each vertex is joined to its \(r\) nearest neighbours on each side. Writing \(p_w(n,r,p)\) for the probability that strict-majority percolation on \(WH_n(r)\) activates every vertex, they define
\[
p_t\bigl(WH(r)\bigr)
\;=\;
\inf\Bigl\{p\in[0,1]:\lim_{n\to\infty}p_w(n,r,p)=1\Bigr\},
\]
and prove the main limit theorem
\[
p_t\bigl(WH(r)\bigr)\;\longrightarrow\;\frac14
\qquad (r\to\infty).
\]
Equivalently, if \(p>1/4\), then for large \(r\) percolation occurs with probability arbitrarily close to \(1\) as \(n\to\infty\), whereas if \(p<1/4\), then the probability of percolation is bounded away from \(1\) [1308.3831].

A central reduction replaces full-wheel percolation by a ring event. Let \(p_R(n,r,p)\) denote the probability that, on \(R_n(r)\), strictly more than \(n/2\) vertices are active after stabilisation. Lemma 2 gives
\[
\liminf_{n\to\infty}p_R(n,r,p)
\;\le\;
\liminf_{n\to\infty}p_w(n,r,p)
\;\le\;
\limsup_{n\to\infty}p_w(n,r,p)
\;\le\;
p+(1-p)\limsup_{n\to\infty}p_R(n,r,p),
\]
so the wheel threshold is controlled by the ring. The next step is to analyse the final state of a typical ring vertex. If \(X_0(n,r)\) is the indicator that vertex \(0\) is active after stabilisation, then symmetry and Lemma 3 yield
\[
2\,\mathbb{E}_p[X_0(n,r)]-1
\;\le\;
p_R(n,r,p)
\;\le\;
2\,\mathbb{E}_p[X_0(n,r)].
\]
Thus the threshold is detected by whether \(\mathbb{E}_p[X_0]\) remains below or rises above \(1/2\).

The lower bound \(p_t\ge 1/4\) for \(r\to\infty\) is obtained by showing that if \(p<1/4\), then for sufficiently large \(r\),
\[
\mathbb{E}_p[X_0(n,r)] \;<\;\frac14.
\]
The argument splits according to the distance to the nearest “wall,” meaning a run of \(r+1\) passive vertices, and combines a Markov-chain estimate with a Chernoff-bound argument to show that eventual activation of the origin is exponentially unlikely in the relevant regimes. The upper bound \(p_t\le 1/4\) uses a coupling to a three-state process on \(\mathbb Z\) with “walls,” “spreaders,” and “empty” sites, together with a block decomposition of length \(\ell=2r+1\). A wall-block has probability at most \(\ell q^{r+1}\), while Lemma 11 constructs many spreader-blocks by counting “Dyck-word”-like configurations of cardinality \(\simeq C\cdot 4^r\). A renewal-process argument then gives \(\mathbb{E}_p[X_0]\to 1\), hence \(p_R\to 1\), and therefore wheel percolation with high probability [1308.3831].

The paper explicitly notes a broader perspective: adding a universal vertex to a low-dimensional base graph can dramatically lower the strict-majority threshold, with the \(r\)-wheel providing a tractable interpolation between a one-dimensional ring and a star. It also highlights the wall-versus-spreader dichotomy and the use of Dyck-word counts as reusable techniques [1308.3831].

## 3. Erdős–Rényi graphs: sparse, Poisson, dense, and connectivity-threshold regimes

For \(G=G(n,p)\), the majority-bootstrap process is sensitive both to the edge density and to the scale of the initial infected set. In the formulation of [1503.07029], one fixes an initial active set \(A(0)\subseteq [n]\) of size \(A_0=A_0(n)\), updates by the rule that an inactive vertex activates when it has more active than inactive neighbours, and writes \(A^*=\lim_{t\to\infty}A(t)\) for the final active set. Percolation means \(A^*=V\), while “almost percolates” means \(|V\setminus A^*|=o(n)\).

Three regimes are distinguished in [1503.07029]. If \(p\ll 1/n\), then with high probability
\[
A^*=A_0(1+o(1)),
\]
so activation does not spread significantly. If \(p=c/n\) with fixed \(c>0\), then the process can enlarge the active set but does not percolate: for sublinear \(A_0=o(n)\), one has \(P(A^*>(1+\varepsilon)A_0)\to 0\) for every \(\varepsilon>0\), while for linear \(A_0=\theta n\) one gets
\[
A^*=\theta^* n + o(n),
\]
where \(\theta^*\in(0,1)\) is the smallest solution of the fixed-point equation stated in the paper, and in particular \(\theta^*<1\). In the dense regime \(np\to\infty\) with \(p=o(1)\), the paper identifies a phase transition at \(A_0\simeq n/2\): if \(A_0=o(n)\), then \(A^*=o(n)\); if \(A_0=\theta n\) with fixed \(\theta\in(0,1)\), then \(A^*=\theta n+o(n)\); and if
\[
\lim_{n\to\infty}\frac{A_0-n/2}{np}=+\infty,
\]
then with high probability
\[
A^*=n-o(n).
\]
The paper summarises this by saying that in the dense regime the critical threshold is \(A_0\approx n/2\) with a window of width \(\Theta(np)\) [1503.07029].

Holmgren–Juškevičius–Kettle analyse a more refined regime above the connectivity threshold, assuming
\[
(1+\varepsilon)\log n \le p(1-p)n,
\]
so in particular \(G(n,p)\) is whp connected and \(p\gg (\log n)/n\) [1508.02671]. They write
\[
d=\frac{np}{1-p},
\]
fix an initial infected set of size \(m\), and identify a critical initial size
\[
m_c
\;=\;
\frac n2
\;-\;
\frac n2 \sqrt{\frac{\log d}{d}+\tau(n)},
\]
where the second term is of order \(\sqrt{(\log d)/d}\) and \(\tau(n)\) is a smaller correction of order \(n\log\log d/\sqrt{d\log d}\). The main theorem states that for the corresponding parametrisation of \(m\), if \(\lambda>1/2\) then percolation occurs whp, whereas if \(\lambda<0\) then percolation fails whp [1508.02671].

The same paper also considers a Bernoulli-\(q\) initialisation. When
\[
p\gg \frac{(\log\log\log n)^2}{\log n}
\]
and
\[
q = \frac12 - \frac12\sqrt{\frac{\log d}{d} + \frac{\theta}{\sqrt n}},
\]
the full-percolation probability satisfies
\[
P'_q \to \Phi(2\theta),
\]
where \(\Phi\) is the standard Gaussian cdf. Thus, in this supercritical regime, the transition window has width \(\Theta(1/\sqrt n)\) and becomes asymptotically Gaussian [1508.02671]. The paper explicitly compares this with the hypercube, where the critical density is also \(1/2\) minus a correction of order \(\sqrt{(\log(\text{degree}))/\text{degree}}\).

Together, these works show that on \(G(n,p)\) majority bootstrap percolation does not have a single universal threshold formula. Instead, the model has qualitatively different behaviours in the sparse, critical Poisson, dense, and connectivity-threshold regimes [1503.07029, 1508.02671].

## 4. Universal behaviour on high-dimensional geometric graphs

A different picture emerges on high-dimensional geometric graphs. The 2024 work [2406.17486] isolates a family \(\mathcal H(K)\) of graphs that “look locally like a high-dimensional grid” and satisfy six structural properties: locally \(K\)-almost regular, bounded backwards expansion, typical local structure, projection or fractal self-symmetry, separation, and exponential order. The examples listed in the paper include Cartesian products of bounded-size factors, hence grids, tori, and Hamming graphs, as well as the middle-layer graph \(M_n\), the odd graph \(O_n\), and the folded hypercube \(\tilde Q_n\) [2406.17486].

For a sequence \((G_n)\subset \mathcal H(K)\) with \(\delta(G_n)\to\infty\), the main theorem gives a universal two-term threshold. Writing \(\delta_n=\delta(G_n)\) and \(\Delta_n=\Delta(G_n)\), for every fixed \(\varepsilon>0\),
\[
p<\frac12-\Bigl(\frac12+\varepsilon\Bigr)\sqrt{\frac{\log \delta_n}{\delta_n}}
\quad\Longrightarrow\quad
\Phi(p,G_n)\to 0,
\]
and
\[
p>\frac12-\Bigl(\frac12-\varepsilon\Bigr)\sqrt{\frac{\log \Delta_n}{\Delta_n}}
\quad\Longrightarrow\quad
\Phi(p,G_n)\to 1.
\]
In particular, for \(d\)-regular graphs in the class with \(d\to\infty\), the paper states a sharp-threshold expansion with leading terms
\[
p_c(G_n)\approx \frac12-\frac12\sqrt{\frac{\log d}{d}},
\]
and the width of the critical window is
\[
o\!\Bigl(\sqrt{\frac{\log d}{d}}\Bigr)
\]
[2406.17486].

The proof architecture separates the supercritical and subcritical sides. On the \(1\)-statement side, a local-determination lemma shows that if
\[
p=\frac12-c\sqrt{\frac{\log d}{d}}
\qquad\text{with } c<\frac12,
\]
then for each vertex \(x\),
\[
\Pr\bigl(x\in A_2(m)\bigr)\ge \frac34+o(1),
\]
even under a slightly stronger threshold; this is then bootstrapped to round \(5\), where the failure probability is \(\exp(-\Omega(d))\), and to round \(11\), where it becomes super-exponentially small. On the \(0\)-statement side, the paper introduces a dominating process \(\mathrm{Boot}_2(\gamma)\), in which the first two rounds use relaxed thresholds \(d/2-2\gamma\) and \(d/2-\gamma\), and proves that round-2 and round-3 infections are unlikely enough that whp the process stalls before infecting the whole graph. Chernoff/Hoeffding bounds, anti-concentration, local-binomial coupling, and the graph-theoretic properties P1–P6 are the essential inputs [2406.17486].

The significance of this result is the universality claim itself: the same two-term expansion that appears on the hypercube persists across a broad class of high-dimensional graphs, and the phase transition is bounded away from \(1/2\) [2406.17486].

## 5. Superexponential-order extensions, the permutahedron, and irregular high-dimensional graphs

The 2025 paper [2507.06597] presents a complementary framework in which an exponential-order class \(\mathcal F(K)\) and a strengthened superexponential-order class \(\mathcal H(K)\) are treated separately. For the superexponential class, the exponential-size condition is replaced by
\[
|V(G)|\;\le\;\exp\!\Bigl(\delta^{3/2}\,\tfrac{\log\log\delta}{\log^2\delta}\Bigr),
\]
and the local-structure assumptions are strengthened. The exact theorem states that for fixed \(\lambda<-1/4\) and \(\lambda'>1/2\), if
\[
f(\ell,\lambda)
=\frac12-\frac12\sqrt{\frac{\log \ell}{\ell}+\lambda\,\frac{\log\log \ell}{\sqrt{\ell\log \ell}}},
\]
then
\[
\Phi(p,G)\to 0 \quad\text{if}\quad p<f\!\bigl(\delta(G),\lambda\bigr),
\qquad
\Phi(p,G)\to 1 \quad\text{if}\quad p>f\!\bigl(\Delta(G),\lambda'\bigr).
\]
For \(n\)-regular graphs this yields
\[
p_c(G)=\frac12-\frac12\sqrt{\frac{\log n}{n}
\;+\;O\!\Bigl(\frac{\log\log n}{\sqrt{n\log n}}\Bigr)},
\]
recovering the optimal barrier \(\lambda>-1/4\) [2507.06597].

A principal motivating example is the permutahedron \(P_n\), the Cayley graph of \(S_{n+1}\) generated by adjacent transpositions. It is \(n\)-regular and has \(|V(P_n)|=(n+1)!\approx \exp(n\log n)\). For
\[
p(n,\lambda)=\frac12-\frac12\sqrt{\frac{\log n}{n}+\lambda\,\frac{\log\log n}{\sqrt{n\log n}}},
\]
the paper proves
\[
\Phi\bigl(p(n,\lambda),P_n\bigr)\to
\begin{cases}
0,&\lambda<-\tfrac14,\\[2mm]
1,&\lambda>\tfrac12.
\end{cases}
\]
Hence
\[
p_c(P_n)=\frac12-\frac12\sqrt{\frac{\log n}{n}
\;+\;O\!\Bigl(\frac{\log\log n}{\sqrt{n\log n}}\Bigr)}.
\]
The paper states that this refines the hypercube result by pushing the lower barrier to \(\lambda=-1/4\) [2507.06597].

The same work also treats an explicit irregular example, the Cartesian product of \(n\) copies of the star \(K_{1,q}\). If
\[
G_n = H\times H\times\cdots\times H \qquad (n\text{ times}),
\]
with \(H=K_{1,q}\), then
\[
\delta(G_n)=n,\qquad \Delta(G_n)=qn,\qquad |V(G_n)|=(q+1)^n.
\]
Nevertheless the threshold statement is the same:
\[
\Phi\bigl(p(n,\lambda),G_n\bigr)\to
\begin{cases}
0,&\lambda<-\tfrac14,\\[2mm]
1,&\lambda>\tfrac12,
\end{cases}
\]
and the paper emphasises that the second-order term is governed by the minimum degree \(\delta=n\), entirely independently of \(q\) [2507.06597].

This irregular example is important because the general upper and lower bounds use \(\Delta(G)\) and \(\delta(G)\), respectively. A plausible implication is that regularity is not merely technical: in irregular graphs, identifying the true threshold may require information beyond the extremal degrees.

## 6. Extremal low-threshold constructions, conjectural boundaries, and open questions

Mitsche, Pérez-Giménez, and Prałat study strong-majority bootstrap percolation on regular graphs and prove an extremal existence result of a very different kind: for any arbitrarily small \(p>0\) and any integer \(r\), there exists a family of \(d=d(p,r)\)-regular graphs on which the \(r\)-majority process disseminates asymptotically almost surely [1503.08310]. Their construction starts from the “line-coupled” toroidal grid \(\mathcal L(n,k)\), then adds \(r\) uniformly random perfect matchings to obtain \(\mathcal L^*(n,k,r)\), a \((4k+r+2)\)-regular graph. With
\[
k=\Bigl\lceil\frac{1000}{p}\ln\frac1p\Bigr\rceil,
\]
one gets
\[
d(p,r)=4k+r+2
=O\!\Bigl(\tfrac{\ln(1/p)}{p}\Bigr),
\qquad
r\le \tfrac p{20}\,k=O(\ln(1/p)).
\]
The proof has two phases: deterministic growth plus percolation in the underlying grid, and then a random-matching argument ruling out stable collections of inactive blocks. A switching-counting lemma shows that for a uniformly random admissible \(r\)-tuple of matchings there is asymptotically almost surely no non-trivial stable collection of the required size, forcing full dissemination [1503.08310].

For strict majority, the case \(r=1\) yields a direct consequence: \(\inf p_c=0\), which answers a question and disproves a conjecture of Rapaport–Suchan–Todinca–Verstraëte [1503.08310]. This result should be read together with the universal high-dimensional threshold results: the latter describe \(p_c(G)\) for structured graph families, whereas the former shows that among all regular graphs the dissemination threshold can be driven to \(0\) by choosing the host graph appropriately.

Several open directions are explicitly identified in the 2025 survey [2507.06597]. One is the dependence of the upper and lower critical-window bounds on \(\Delta(G)\) and \(\delta(G)\): for highly irregular graphs, it is not known in general whether the true threshold may lie strictly between these two bounds. Another is the optimal third-order term on the hypercube: the paper states that one does not know whether there is a sharp constant \(\lambda^*\in[-2,1/2]\) such that \(\Phi(p(n,\lambda),Q^n)\) jumps precisely at \(\lambda=\lambda^*\), and records the conjecture \(\lambda^*=1/2\). Further questions concern extensions beyond majority to \(r\)-neighbor rules for \(r\ge 3\), and whether the “cherry–projection” axioms can be weakened, perhaps replaced by purely spectral-expansion conditions [2507.06597].

The current landscape can be summarised by a few representative threshold statements.

| Graph family / model | Threshold phenomenon | Source |
|---|---|---|
| \(r\)-wheel, strict majority | \(p_t(WH(r))\to 1/4\) as \(r\to\infty\) | [1308.3831] |
| \(G(n,p)\), dense regime \(np\to\infty\) | transition at \(A_0\approx n/2\), window \(\Theta(np)\) | [1503.07029] |
| \(G(n,p)\) above connectivity threshold | \(m_c=\frac n2-\frac n2\sqrt{(\log d)/d+\tau(n)}\) | [1508.02671] |
| high-dimensional geometric graphs | \(p_c\approx \frac12-\frac12\sqrt{(\log d)/d}\) in the regular case | [2406.17486] |
| permutahedron and star products | \(\lambda<-\tfrac14\) gives non-percolation, \(\lambda>\tfrac12\) gives percolation | [2507.06597] |
| regular graphs, strong-majority | for any fixed \(p>0,r\), some \(d\)-regular families disseminate a.a.s. | [1503.08310] |

Taken together, these results show that majority bootstrap percolation is governed by a combination of local threshold geometry and global graph structure. On some families the decisive scale is a constant such as \(1/4\); on random graphs it is an initial-size threshold near \(n/2\); and on high-dimensional graphs it is the universal square-root-log correction to \(1/2\). The available theorems therefore describe not one critical phenomenon, but a collection of sharply different regimes linked by a common monotone-dynamics framework.

Source: https://www.emergentmind.com/topics/majority-bootstrap-percolation