Strong-majority bootstrap percolation on regular graphs with low dissemination threshold
Résumé
Consider the following model of strong-majority bootstrap percolation on a graph. Let r ≥ 1 be some integer, and p ∈ [0, 1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every step of the process, each vertex v of degree deg(v) becomes active if at least (deg(v) + r)/2 of its neighbours are active. Given any arbitrarily small p > 0 and any integer r, we construct a family of d = d(p, r)-regular graphs such that with high probability all vertices become active in the end. In particular, the case r = 1 answers a question and disproves a conjecture of Rapaport, Suchan, Todinca and Verstraëte [38].
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