Majority bootstrap percolation on the random graph G(n,p)

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 $A(0) = \theta n$ 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.