Strict majority bootstrap percolation in the r-wheel

In the strict Majority Bootstrap Percolation process each passive vertex v becomes active if at least ⌈ 2 ⌉ of its neighbors are active (and thereafter never changes its state). We address the problem of finding graphs for which a small proportion of initial active vertices is likely to eventually make all vertices active. We study the problem on a ring of n vertices augmented with a “central” vertex u. Each vertex in the ring, besides being connected to u, is connected to its r closest neighbors to the left and to the right. We prove that if vertices are initially active with probability p > 1/4 then, for large values of r, percolation occurs with probability arbitrarily close to 1 as n → ∞. Also, if p < 1/4, then the probability of percolation is bounded away from 1.