Majority Bootstrap Percolation Thresholds

• Majority bootstrap percolation is a monotone infection process where a vertex becomes permanently active once more than half its neighbors are infected.
• The topic examines various update rules—including strict and r-majority—across diverse graph families such as rings, Erdős–Rényi graphs, and high-dimensional grids.
• Advanced methods like probabilistic coupling, renewal arguments, and concentration bounds are used to uncover universal thresholds and critical windows in complex networks.

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 (Kiwi et al., 2013), through transitions near $1/2$ on Erdős–Rényi graphs (Holmgren et al., 2015, Stefánsson et al., 2015), to a universal square-root-log correction on broad classes of high-dimensional graphs (Collares et al., 2024, Collares et al., 9 Jul 2025).

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

$\Phi(p,G)$0

where $\Phi(p,G)$1 is the neighbourhood of $\Phi(p,G)$2 and $\Phi(p,G)$3; one says that $\Phi(p,G)$4 percolates if $\Phi(p,G)$5 (Collares et al., 2024). In a Bernoulli initialisation, each vertex lies in $\Phi(p,G)$6 independently with probability $\Phi(p,G)$7, and one writes

$\Phi(p,G)$8

(Collares et al., 2024, Collares et al., 9 Jul 2025).

The papers considered here use three related update thresholds. In strict majority bootstrap percolation, a passive vertex $\Phi(p,G)$9 becomes active exactly when

$p_c(G)$0

(Kiwi et al., 2013). In strong-majority or $p_c(G)$1-majority bootstrap percolation, an inactive vertex becomes active when

$p_c(G)$2

(Mitsche et al., 2015). Some majority-bootstrap papers use the equivalent condition “more infected than non-infected neighbours,” while others write the update rule with $p_c(G)$3 rather than $p_c(G)$4 (Holmgren et al., 2015, Collares et al., 9 Jul 2025). 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 $p_c(G)$5 on $p_c(G)$6 (Holmgren et al., 2015), whereas others take a Bernoulli-$p_c(G)$7 or Bernoulli-$p_c(G)$8 initial set (Collares et al., 2024, Stefánsson et al., 2015, Collares et al., 9 Jul 2025). In every case the process is monotone, stabilises after finitely many rounds, and the central question is whether the final infected set equals $p_c(G)$9.

2. The $\Phi(p,G)\ge 1/2$0-wheel and the constant threshold $\Phi(p,G)\ge 1/2$1

Kiwi et al. study strict majority bootstrap percolation on the $\Phi(p,G)\ge 1/2$2-wheel $\Phi(p,G)\ge 1/2$3, obtained from the ring $\Phi(p,G)\ge 1/2$4 by adding one universal vertex adjacent to every ring vertex (Kiwi et al., 2013). Here $\Phi(p,G)\ge 1/2$5 is the cycle on $\Phi(p,G)\ge 1/2$6 vertices in which each vertex is joined to its $\Phi(p,G)\ge 1/2$7 nearest neighbours on each side. Writing $\Phi(p,G)\ge 1/2$8 for the probability that strict-majority percolation on $\Phi(p,G)\ge 1/2$9 activates every vertex, they define

$r$0

and prove the main limit theorem

$r$1

Equivalently, if $r$2, then for large $r$3 percolation occurs with probability arbitrarily close to $r$4 as $r$5, whereas if $r$6, then the probability of percolation is bounded away from $r$7 (Kiwi et al., 2013).

A central reduction replaces full-wheel percolation by a ring event. Let $r$8 denote the probability that, on $r$9, strictly more than $p_c=1/4$0 vertices are active after stabilisation. Lemma 2 gives

$p_c=1/4$1

so the wheel threshold is controlled by the ring. The next step is to analyse the final state of a typical ring vertex. If $p_c=1/4$2 is the indicator that vertex $p_c=1/4$3 is active after stabilisation, then symmetry and Lemma 3 yield

$p_c=1/4$4

Thus the threshold is detected by whether $p_c=1/4$5 remains below or rises above $p_c=1/4$6.

The lower bound $p_c=1/4$7 for $p_c=1/4$8 is obtained by showing that if $p_c=1/4$9, then for sufficiently large $r$0,

$r$1

The argument splits according to the distance to the nearest “wall,” meaning a run of $r$2 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 $r$3 uses a coupling to a three-state process on $r$4 with “walls,” “spreaders,” and “empty” sites, together with a block decomposition of length $r$5. A wall-block has probability at most $r$6, while Lemma 11 constructs many spreader-blocks by counting “Dyck-word”-like configurations of cardinality $r$7. A renewal-process argument then gives $r$8, hence $r$9, and therefore wheel percolation with high probability (Kiwi et al., 2013).

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 $1/2$0-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 (Kiwi et al., 2013).

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

For $1/2$1, the majority-bootstrap process is sensitive both to the edge density and to the scale of the initial infected set. In the formulation of (Stefánsson et al., 2015), one fixes an initial active set $1/2$2 of size $1/2$3, updates by the rule that an inactive vertex activates when it has more active than inactive neighbours, and writes $1/2$4 for the final active set. Percolation means $1/2$5, while “almost percolates” means $1/2$6.

Three regimes are distinguished in (Stefánsson et al., 2015). If $1/2$7, then with high probability

$1/2$8

so activation does not spread significantly. If $1/2$9 with fixed $G=(V,E)$0, then the process can enlarge the active set but does not percolate: for sublinear $G=(V,E)$1, one has $G=(V,E)$2 for every $G=(V,E)$3, while for linear $G=(V,E)$4 one gets

$G=(V,E)$5

where $G=(V,E)$6 is the smallest solution of the fixed-point equation stated in the paper, and in particular $G=(V,E)$7. In the dense regime $G=(V,E)$8 with $G=(V,E)$9, the paper identifies a phase transition at $A_0\subseteq V$0: if $A_0\subseteq V$1, then $A_0\subseteq V$2; if $A_0\subseteq V$3 with fixed $A_0\subseteq V$4, then $A_0\subseteq V$5; and if

$A_0\subseteq V$6

then with high probability

$A_0\subseteq V$7

The paper summarises this by saying that in the dense regime the critical threshold is $A_0\subseteq V$8 with a window of width $A_0\subseteq V$9 (Stefánsson et al., 2015).

Holmgren–Juškevičius–Kettle analyse a more refined regime above the connectivity threshold, assuming

$\Phi(p,G)$00

so in particular $\Phi(p,G)$01 is whp connected and $\Phi(p,G)$02 (Holmgren et al., 2015). They write

$\Phi(p,G)$03

fix an initial infected set of size $\Phi(p,G)$04, and identify a critical initial size

$\Phi(p,G)$05

where the second term is of order $\Phi(p,G)$06 and $\Phi(p,G)$07 is a smaller correction of order $\Phi(p,G)$08. The main theorem states that for the corresponding parametrisation of $\Phi(p,G)$09, if $\Phi(p,G)$10 then percolation occurs whp, whereas if $\Phi(p,G)$11 then percolation fails whp (Holmgren et al., 2015).

The same paper also considers a Bernoulli-$\Phi(p,G)$12 initialisation. When

$\Phi(p,G)$13

and

$\Phi(p,G)$14

the full-percolation probability satisfies

$\Phi(p,G)$15

where $\Phi(p,G)$16 is the standard Gaussian cdf. Thus, in this supercritical regime, the transition window has width $\Phi(p,G)$17 and becomes asymptotically Gaussian (Holmgren et al., 2015). The paper explicitly compares this with the hypercube, where the critical density is also $\Phi(p,G)$18 minus a correction of order $\Phi(p,G)$19.

Together, these works show that on $\Phi(p,G)$20 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 (Stefánsson et al., 2015, Holmgren et al., 2015).

4. Universal behaviour on high-dimensional geometric graphs

A different picture emerges on high-dimensional geometric graphs. The 2024 work (Collares et al., 2024) isolates a family $\Phi(p,G)$21 of graphs that “look locally like a high-dimensional grid” and satisfy six structural properties: locally $\Phi(p,G)$22-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 $\Phi(p,G)$23, the odd graph $\Phi(p,G)$24, and the folded hypercube $\Phi(p,G)$25 (Collares et al., 2024).

For a sequence $\Phi(p,G)$26 with $\Phi(p,G)$27, the main theorem gives a universal two-term threshold. Writing $\Phi(p,G)$28 and $\Phi(p,G)$29, for every fixed $\Phi(p,G)$30,

$\Phi(p,G)$31

and

$\Phi(p,G)$32

In particular, for $\Phi(p,G)$33-regular graphs in the class with $\Phi(p,G)$34, the paper states a sharp-threshold expansion with leading terms

$\Phi(p,G)$35

and the width of the critical window is

$\Phi(p,G)$36

(Collares et al., 2024).

The proof architecture separates the supercritical and subcritical sides. On the $\Phi(p,G)$37-statement side, a local-determination lemma shows that if

$\Phi(p,G)$38

then for each vertex $\Phi(p,G)$39,

$\Phi(p,G)$40

even under a slightly stronger threshold; this is then bootstrapped to round $\Phi(p,G)$41, where the failure probability is $\Phi(p,G)$42, and to round $\Phi(p,G)$43, where it becomes super-exponentially small. On the $\Phi(p,G)$44-statement side, the paper introduces a dominating process $\Phi(p,G)$45, in which the first two rounds use relaxed thresholds $\Phi(p,G)$46 and $\Phi(p,G)$47, 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 (Collares et al., 2024).

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 $\Phi(p,G)$48 (Collares et al., 2024).

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

The 2025 paper (Collares et al., 9 Jul 2025) presents a complementary framework in which an exponential-order class $\Phi(p,G)$49 and a strengthened superexponential-order class $\Phi(p,G)$50 are treated separately. For the superexponential class, the exponential-size condition is replaced by

$\Phi(p,G)$51

and the local-structure assumptions are strengthened. The exact theorem states that for fixed $\Phi(p,G)$52 and $\Phi(p,G)$53, if

$\Phi(p,G)$54

then

$\Phi(p,G)$55

For $\Phi(p,G)$56-regular graphs this yields

$\Phi(p,G)$57

recovering the optimal barrier $\Phi(p,G)$58 (Collares et al., 9 Jul 2025).

A principal motivating example is the permutahedron $\Phi(p,G)$59, the Cayley graph of $\Phi(p,G)$60 generated by adjacent transpositions. It is $\Phi(p,G)$61-regular and has $\Phi(p,G)$62. For

$\Phi(p,G)$63

the paper proves

$\Phi(p,G)$64

Hence

$\Phi(p,G)$65

The paper states that this refines the hypercube result by pushing the lower barrier to $\Phi(p,G)$66 (Collares et al., 9 Jul 2025).

The same work also treats an explicit irregular example, the Cartesian product of $\Phi(p,G)$67 copies of the star $\Phi(p,G)$68. If

$\Phi(p,G)$69

with $\Phi(p,G)$70, then

$\Phi(p,G)$71

Nevertheless the threshold statement is the same: $\Phi(p,G)$72 and the paper emphasises that the second-order term is governed by the minimum degree $\Phi(p,G)$73, entirely independently of $\Phi(p,G)$74 (Collares et al., 9 Jul 2025).

This irregular example is important because the general upper and lower bounds use $\Phi(p,G)$75 and $\Phi(p,G)$76, 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 $\Phi(p,G)$77 and any integer $\Phi(p,G)$78, there exists a family of $\Phi(p,G)$79-regular graphs on which the $\Phi(p,G)$80-majority process disseminates asymptotically almost surely (Mitsche et al., 2015). Their construction starts from the “line-coupled” toroidal grid $\Phi(p,G)$81, then adds $\Phi(p,G)$82 uniformly random perfect matchings to obtain $\Phi(p,G)$83, a $\Phi(p,G)$84-regular graph. With

$\Phi(p,G)$85

one gets

$\Phi(p,G)$86

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 $\Phi(p,G)$87-tuple of matchings there is asymptotically almost surely no non-trivial stable collection of the required size, forcing full dissemination (Mitsche et al., 2015).

For strict majority, the case $\Phi(p,G)$88 yields a direct consequence: $\Phi(p,G)$89, which answers a question and disproves a conjecture of Rapaport–Suchan–Todinca–Verstraëte (Mitsche et al., 2015). This result should be read together with the universal high-dimensional threshold results: the latter describe $\Phi(p,G)$90 for structured graph families, whereas the former shows that among all regular graphs the dissemination threshold can be driven to $\Phi(p,G)$91 by choosing the host graph appropriately.

Several open directions are explicitly identified in the 2025 survey (Collares et al., 9 Jul 2025). One is the dependence of the upper and lower critical-window bounds on $\Phi(p,G)$92 and $\Phi(p,G)$93: 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 $\Phi(p,G)$94 such that $\Phi(p,G)$95 jumps precisely at $\Phi(p,G)$96, and records the conjecture $\Phi(p,G)$97. Further questions concern extensions beyond majority to $\Phi(p,G)$98-neighbor rules for $\Phi(p,G)$99, and whether the “cherry–projection” axioms can be weakened, perhaps replaced by purely spectral-expansion conditions (Collares et al., 9 Jul 2025).

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

Graph family / model	Threshold phenomenon	Source
$p_c(G)$00-wheel, strict majority	$p_c(G)$01 as $p_c(G)$02	(Kiwi et al., 2013)
$p_c(G)$03, dense regime $p_c(G)$04	transition at $p_c(G)$05, window $p_c(G)$06	(Stefánsson et al., 2015)
$p_c(G)$07 above connectivity threshold	$p_c(G)$08	(Holmgren et al., 2015)
high-dimensional geometric graphs	$p_c(G)$09 in the regular case	(Collares et al., 2024)
permutahedron and star products	$p_c(G)$10 gives non-percolation, $p_c(G)$11 gives percolation	(Collares et al., 9 Jul 2025)
regular graphs, strong-majority	for any fixed $p_c(G)$12, some $p_c(G)$13-regular families disseminate a.a.s.	(Mitsche et al., 2015)

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 $p_c(G)$14; on random graphs it is an initial-size threshold near $p_c(G)$15; and on high-dimensional graphs it is the universal square-root-log correction to $p_c(G)$16. The available theorems therefore describe not one critical phenomenon, but a collection of sharply different regimes linked by a common monotone-dynamics framework.