Bootstrap percolation on a graph with random and local connections
We study a bootstrap percolation on a graph which is a superposition of the random graph $G_{n,p}$ and a $d$-dimensional lattice. Bootstrap percolation is a process of spread of ``activation''
on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r \geq 2$ active neighbours become active as well.
We study the size of the final active set in the limit when $n \rightarrow \infty $. We show that an addition of local edges in the case $d=1$ makes the phase transition (between complete percolation and a very limited growth) even sharper when compared with the bootstrap percolation on random graph $G_{n,p}$.